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Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, in my previous post, I was thinking about variance of the sum of two random variables. But many friends would find it interesting to find a hermite polynomial series expression for the density of the sum of to random variables that can be used for further analytics. Here I try to write the equations for that.

Suppose our first asset/random variable is $Y_1$ with weight $w_1$ and hermite representation (upto fourth hermite polynomial) of the first asset/random variable is given as

$\, Y_1\, = \, a_0 \, + \, a_1 \, Z_1 \, + \, a_2 \, \big[ {Z_1}^2 -1 \big]\, + \, a_3 \, \big[{Z_1}^3 -3 \, Z_1\big]\, + \, a_4 \, \big[{Z_1}^4 \, -3 \, {Z_1}^2 \, +3 \big]\,$

our second asset/random variable is $Y_2$ with weight $w_2$ and hermite representation (upto fourth hermite polynomial) of the second asset/random variable is given as

$\, Y_2 \, = \, b_0 \, + \, b_1 \, Z_2 \, + \, b_2 \, \big[ {Z_2}^2 -1 \big]\, + \, b_3 \, \big[{Z_2}^3 -3 \, Z_2\big]\, + \, b_4 \, \big[{Z_2}^4 \, -3 \, {Z_2}^2 \, +3 \big]\,$

and we want to find hermite series expression for the sum random variable where

$\, Y_3 \, = w_1 \, Y_1 \, + \, w_2 \, Y_2$

Let us first describe the hermite form of this summed random variable $Y_3$ and then we would relate its hermite coefficients to hermite coefficients of $Y_1$ and $Y_2$. This hermite form of $Y_3$ with yet unknown coefficients is given as

$\, Y_3 \, = \, c_0 \, + \, c_1 \, Z_3 \, + \, c_2 \, \big[ {Z_3}^2 -1 \big]\, + \, c_3 \, \big[{Z_3}^3 -3 \, Z_3\big]\, + \, c_4 \, \big[{Z_3}^4 \, -3 \, {Z_3}^2 \, +3 \big]\,$

Now we write the coefficients of $Y_3$ in terms of coefficients of $Y_1$ and $Y_2$
Zeroth coefficient is mean in this expansion and is added linearly. Coefficients of higher order hermite polynomials are added in sum squared fashion but accounting for sign.

$c_0 \, =\, w_1\, a_0 \, + \,w_2 \, b_0 \,$

Expressions below are very simple but since higher hermite coefficients can have both negative and positive signs, you have to be careful in squared sum that when a large negative coefficient is squared and added to a small positive coefficient squared, the result is a smaller negative squared. We take its absolute value, take its squared root and assign it a negative sign.

Please note that coefficient of first hermite polynomial coefficients would almost always be positive. I have never encountered a situation in which coefficient of first hermite polynomial was negative otherwise the derivative of random variable with respect to standard normal will possibly take a negative value which is not allowed for densities. All hermite polynomials after first hermite polynomial can have negative signs for their coefficients.
This is really a simple squared sum but accounting for sign makes the resulting expression a bit complicated.

$c_1 \, = \sqrt{ \,{w_1}^2 \, {a_1}^2 \, + \, {w_2}^2 \,{b_1}^2 + \, 2 \, \rho_1 \, w_1 \, w_2 \,a_1 \, b_1} \,$

$c_2 \, = sgn({sgn(a_2) \,{w_1}^2 \, {a_2}^2 \, + \, sgn(b_2) {w_2}^2 \,{b_2}^2 + \, 2 \, \rho_2 \, w_1 \, w_2 \,a_2 \, b_2}) \\ *\, \sqrt{ |{sgn(a_2) \,{w_1}^2 \, {a_2}^2 \, + \, sgn(b_2) {w_2}^2 \,{b_2}^2 + \, 2 \, \rho_2 \, w_1 \, w_2 \,a_2 \, b_2}|} \,$

$c_3 \, = sgn({sgn(a_3) \,{w_1}^2 \, {a_3}^2 \, + \, sgn(b_3) {w_2}^2 \,{b_3}^2 + \, 2 \, \rho_3 \, w_1 \, w_2 \,a_3 \, b_3}) \\ *\, \sqrt{ |{sgn(a_3) \,{w_1}^2 \, {a_3}^2 \, + \, sgn(b_3) {w_2}^2 \,{b_3}^2 + \, 2 \, \rho_3 \, w_1 \, w_2 \,a_3 \, b_3}|} \,$

$c_4 \, = sgn({sgn(a_4) \,{w_1}^2 \, {a_4}^2 \, + \, sgn(b_4) {w_2}^2 \,{b_4}^2 + \, 2 \, \rho_4 \, w_1 \, w_2 \,a_4 \, b_4}) \\ *\, \sqrt{ |{sgn(a_4) \,{w_1}^2 \, {a_4}^2 \, + \, sgn(b_4) {w_2}^2 \,{b_4}^2 + \, 2 \, \rho_4 \, w_1 \, w_2 \,a_4 \, b_4}|} \,$
Since a formal expression taking into account signs is unnecessarily complicated, I am writing simpler expression without correlations for friends to gain intuition.
If there are zero correlations between hermite polynomials, we get simpler expressions that are given in case of zero correlations as

$c_1 \, =\sqrt{ \,{w_1}^2 \, {a_1}^2 \, + {w_2}^2 \,{b_1}^2 }\,$

$c_2 \, = sgn(w_1 \,a_2\, + \, w_2 \, b_2) \, \sqrt{ |{sgn(a_2) \,{w_1}^2 \, {a_2}^2 \, + \, sgn(b_2) {w_2}^2 \,{b_2}^2 }|} \,$

$c_3 \, = sgn(w_1 \,a_3\, + \, w_2\, b_3) \, \sqrt{ |{sgn(a_3) \,{w_1}^2 \, {a_3}^2 \, + \, sgn(b_3) {w_2}^2 \,{b_3}^2 }|} \,$

$c_4 \, = sgn(w_1 \,a_4\, + \, w_2 \, b_4) \, \sqrt{ |{sgn(a_4) \,{w_1}^2 \, {a_4}^2 \, + \, sgn(b_4) {w_2}^2 \,{b_4}^2 }|} \,$
.
.
Friends, sorry about this above post. Please disregard above post 1785. I confused signs while recalling from a recent case when there where some correlations. The corrected post is given below. Sorry again since many times I write on wilmott, I do not make any notes and write from memory and confused signs with a recent case where there were some correlations.

Friends, in my previous post, I was thinking about variance of the sum of two random variables. But many friends would find it interesting to find a hermite polynomial series expression for the density of the sum of to random variables that can be used for further analytics. Here I try to write the equations for that.

Suppose our first asset/random variable is $Y_1$ with weight $w_1$ and hermite representation (upto fourth hermite polynomial) of the first asset/random variable is given as

$\, Y_1\, = \, a_0 \, + \, a_1 \, Z_1 \, + \, a_2 \, \big[ {Z_1}^2 -1 \big]\, + \, a_3 \, \big[{Z_1}^3 -3 \, Z_1\big]\, + \, a_4 \, \big[{Z_1}^4 \, -3 \, {Z_1}^2 \, +3 \big]\,$

our second asset/random variable is $Y_2$ with weight $w_2$ and hermite representation (upto fourth hermite polynomial) of the second asset/random variable is given as

$\, Y_2 \, = \, b_0 \, + \, b_1 \, Z_2 \, + \, b_2 \, \big[ {Z_2}^2 -1 \big]\, + \, b_3 \, \big[{Z_2}^3 -3 \, Z_2\big]\, + \, b_4 \, \big[{Z_2}^4 \, -3 \, {Z_2}^2 \, +3 \big]\,$

and we want to find hermite series expression for the sum random variable where

$\, Y_3 \, = w_1 \, Y_1 \, + \, w_2 \, Y_2$

Let us first describe the hermite form of this summed random variable $Y_3$ and then we would relate its hermite coefficients to hermite coefficients of $Y_1$ and $Y_2$. This hermite form of $Y_3$ with yet unknown coefficients is given as

$\, Y_3 \, = \, c_0 \, + \, c_1 \, Z_3 \, + \, c_2 \, \big[ {Z_3}^2 -1 \big]\, + \, c_3 \, \big[{Z_3}^3 -3 \, Z_3\big]\, + \, c_4 \, \big[{Z_3}^4 \, -3 \, {Z_3}^2 \, +3 \big]\,$

Now we write the coefficients of $Y_3$ in terms of coefficients of $Y_1$ and $Y_2$
Zeroth coefficient is mean in this expansion and is added linearly. Coefficients of higher order hermite polynomials are added in sum squared fashion but accounting for sign.

$c_0 \, =\, w_1\, a_0 \, + \,w_2 \, b_0 \,$

Expressions below are very simple but since higher hermite coefficients can have both negative and positive signs, you have to be careful in squared sum that when a large negative coefficient is squared and added to a small positive coefficient squared, the result is a smaller negative squared. We take its absolute value, take its squared root and assign it a negative sign.

Please note that coefficient of first hermite polynomial coefficients would almost always be positive. I have never encountered a situation in which coefficient of first hermite polynomial was negative otherwise the derivative of random variable with respect to standard normal will possibly take a negative value which is not allowed for densities. All hermite polynomials after first hermite polynomial can have negative signs for their coefficients.
This is really a simple squared sum but accounting for sign makes the resulting expression a bit complicated.

$c_1 \, = \sqrt{ \,{w_1}^2 \, {a_1}^2 \, + \, {w_2}^2 \,{b_1}^2 + \, 2 \, \rho_1 \, w_1 \, w_2 \,a_1 \, b_1} \,$

$c_2 \, = \sqrt{ \,{w_1}^2 \, {a_2}^2 \, + \, {w_2}^2 \,{b_2}^2 + \, 2 \, \rho_2 \, w_1 \, w_2 \,a_2 \, b_2} \,$

$c_3 \, = \, \sqrt{ \,{w_1}^2 \, {a_3}^2 \, + \, {w_2}^2 \,{b_3}^2 + \, 2 \, \rho_3 \, w_1 \, w_2 \,a_3 \, b_3} \,$

$c_4 \, =\, \sqrt{ \,{w_1}^2 \, {a_4}^2 \, + \, {w_2}^2 \,{b_4}^2 + \, 2 \, \rho_4 \, w_1 \, w_2 \,a_4 \, b_4} \,$
Since a formal expression taking into account signs is unnecessarily complicated, I am writing simpler expression without correlations for friends to gain intuition.
If there are zero correlations between hermite polynomials, we get simpler expressions that are given in case of zero correlations as

$c_1 \, =\sqrt{ \,{w_1}^2 \, {a_1}^2 \, + {w_2}^2 \,{b_1}^2 }\,$

$c_2 \, = \, \sqrt{ \,{w_1}^2 \, {a_2}^2 \, + \, {w_2}^2 \,{b_2}^2 } \,$

$c_3 \, = \, \sqrt{ \,{w_1}^2 \, {a_3}^2 \, + \, {w_2}^2 \,{b_3}^2 }\,$

$c_4 \, = \, \sqrt{ {w_1}^2 \, {a_4}^2 \, + \,{w_2}^2 \,{b_4}^2 }\,$
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, I will continue to write small posts to explain the calculation of correlations/covariances, regression example and calibration of moments to coefficients of hermite series so that certain numbers of moments of the Hermite series representation of the random variable match the moments calculated from the data. When a hermite representation has been found, the density of the random variable is pinned down and can easily be found.

Over the weekend, I will also give references to posts where I have posted some important matlab program or where I explained some idea related to Hermite series representation and related analytics.

If you have not been following my work, you will still be able to learn most of the important things now within a few days.

Since I got the information that motherboard of my second laptop  also has problems, I am trying to install my working programs on this old computer to keep working with my programming for a week or so until I buy a new laptop again.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, in next three four days, I will explain all major basics about my research and point out related matlab programs and also try to explain main programs.

But before I start explaining everything in more detail, I want to point out relationship between Z-series and hermite polynomial series. This is important to understand since in my programs, I continued to switch from Z-series representation of densities to hermite polynomial representation of the densities of random variables.

Below is a Z-series representation of the density of a random variable.
$\, X \, = \, c_0 \, + \, c_1 \, Z + \, c_2 \, Z^2 + \, c_3 \, Z^3 + \, c_4 \, Z^4 + \, c_5 \, Z^5$

Below is a hermite-series representation of the density of a random variable.

$\, X \, = \, ch_0 \, + \, ch_1 \, Z + \, ch_2 \, (Z^2 - \, 1) + \, ch_3 \, (Z^3 \, - 3 \, Z)+ \, ch_4 \,( Z^4 \, - 6 \, Z^2 \, + 3) + \, ch_5 \, (Z^5 \, - 10 \, Z^3 \, + 15 \, Z)$

Though both series look very different, they are both same order polynomials and share the same span of (fifth order here in above equations) same order polynomials. So every random variable that can be represented by a Z-series of certain order can also be represented by a hermite series of the same order (since they share the same span) and vice-versa. However the coefficients chosen on different order terms would be different. The fixed relationship between these coefficients can be found by simple algebra. Below I have given programs that have the relationships between coefficients of hermite series and Z-series hard-coded so that they point to same random variable.

A hermite series representation of the density of random variables is very suitable for analytics like addition of random variables, calculation of correlations, variances and covariances.
But there are certain things you cannot do with a hermite series representation and you have to use equivalent Z-series for that. For example once you have a hermite series representation of a random variable and you want to find hermite series representation of a function of the random variable (like for example square-root of the random variable or any other function), you would have to do that through Taylor series in Z-series representation.
For example, suppose you want to add a lognormal random variable and a heavy-tailed CEV random variable and then you want to find hermite representation of the  square-root of the resulting sum. First you would add the lognormal and CEV random variable through taking square-root of the squared sum of the coefficients of respective orthogonal polynomials in orthogonal polynomial  hermite representation of the lognormal and CEV variable. This would give us hermite representation of the sum of the two random variables. But in order to find  the (density of) square-root (or any other function) of the resultant variable, you would have to first convert to Z-series representation of the resultant sum and then do  a Taylor expansion of resultant Z-series and then re-convert it back to hermite polynomial representation.

Making friends familiar with my programs.
Below is the simple program to convert Coefficients of a Z-series to coefficients of equivalent hermite series so that both of them would represent the same random variable.

The function header is
[aH0,aH] = ConvertZCoeffsToHCoeffs(a0,a,SeriesOrder)

Here scalar value a0 represents the first coefficient of the Z-series. a is an array of coefficients that represents coefficients of non-zero powers of the Z-series. SeriesOrder means the order of the input Z-series and it also determines the length of input array of coefficients a.

The output of program is a scalar value  aH0 which is the coefficient of zerothe hermite term (hence the mean) and aH is an array of length Series order and represents the coefficients of hermite polynomial of same order as index of the array. H in aH0 and aH is used to know that this is hermite series coefficients.

function [aH0,aH] = ConvertZCoeffsToHCoeffs(a0,a,SeriesOrder)

if(SeriesOrder==3)
aH0=a0+a(2);
aH(1)=a(1)+3*a(3);
aH(2)=a(2);
aH(3)=a(3);
end

if(SeriesOrder==4)
aH0=a0+a(2)+3*a(4);
aH(1)=a(1)+3*a(3);
aH(2)=a(2)+6*a(4);
aH(3)=a(3);
aH(4)=a(4);
end

if(SeriesOrder==5)
aH0=a0+a(2)+3*a(4);
aH(1)=a(1)+3*a(3)+15*a(5);
aH(2)=a(2)+6*a(4);
aH(3)=a(3)+10*a(5);
aH(4)=a(4);
aH(5)=a(5);
end

if(SeriesOrder==6)
aH0=a0+a(2)+3*a(4)+15*a(6);
aH(1)=a(1)+3*a(3)+15*a(5);
aH(2)=a(2)+6*a(4)+45*a(6);
aH(3)=a(3)+10*a(5);
aH(4)=a(4)+15*a(6);
aH(5)=a(5);
aH(6)=a(6);
end

if(SeriesOrder==7)
aH0=a0+a(2)+3*a(4)+15*a(6);
aH(1)=a(1)+3*a(3)+15*a(5)+105*a(7);
aH(2)=a(2)+6*a(4)+45*a(6);
aH(3)=a(3)+10*a(5)+105*a(7);
aH(4)=a(4)+15*a(6);
aH(5)=a(5)+21*a(7);
aH(6)=a(6);
aH(7)=a(7);
end

end
.
The function below converts coefficients of a Hermite series to Coefficients of a Z-series. Its function header is
function [a0,a] = ConvertHCoeffsToZCoeffs(aH0,aH,SeriesOrder)

It takes as its arguments first a scalar value that is coefficient of the Zeroth order hermite and then an array of coefficients of hermite polynomials stored in aH which has a length equal to series order.
And it returns a scalar value a0 which is the coefficient of the Zeroth order power of Z(this is also equal to median) and an array (of length series order) representing the coefficients of non-zero powers of the array ranging from one to series order.
.
.
function [a0,a] = ConvertHCoeffsToZCoeffs(aH0,aH,SeriesOrder)

if(SeriesOrder==3)
a0=aH0-aH(2);
a(1)=aH(1)-3*aH(3);
a(2)=aH(2);
a(3)=aH(3);
end

if(SeriesOrder==4)
a0=aH0-aH(2)+3*aH(4);
a(1)=aH(1)-3*aH(3);
a(2)=aH(2)-6*aH(4);
a(3)=aH(3);
a(4)=aH(4);
end

if(SeriesOrder==5)
a0=aH0-aH(2)+3*aH(4);
a(1)=aH(1)-3*aH(3)+15*aH(5);
a(2)=aH(2)-6*aH(4);
a(3)=aH(3)-10*aH(5);
a(4)=aH(4);
a(5)=aH(5);
end

if(SeriesOrder==5)
a0=aH0-aH(2)+3*aH(4);
a(1)=aH(1)-3*aH(3)+15*aH(5);
a(2)=aH(2)-6*aH(4);
a(3)=aH(3)-10*aH(5);
a(4)=aH(4);
a(5)=aH(5);
end
% if(SeriesOrder==5)
% a0=aH0-aH(2)-3*aH(4);
% a(1)=aH(1)-3*aH(3)-15*aH(5);
% a(2)=aH(2)-6*aH(4);
% a(3)=aH(3)-10*aH(5);
% a(4)=aH(4);
% a(5)=aH(5);
% end
if(SeriesOrder==6)
a0=aH0-aH(2)+3*aH(4)-15*aH(6);
a(1)=aH(1)-3*aH(3)+15*aH(5);
a(2)=aH(2)-6*aH(4)+45*aH(6);
a(3)=aH(3)-10*aH(5);
a(4)=aH(4)-15*aH(6);
a(5)=aH(5);
a(6)=aH(6);
end

if(SeriesOrder==7)
a0=aH0-aH(2)+3*aH(4)-15*aH(6);
a(1)=aH(1)-3*aH(3)+15*aH(5)-105*aH(7);
a(2)=aH(2)-6*aH(4)+45*aH(6);
a(3)=aH(3)-10*aH(5)+105*aH(7);
a(4)=aH(4)-15*aH(6);
a(5)=aH(5)-21*aH(7);
a(6)=aH(6);
a(7)=aH(7);
end

end
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

First important thing to learn is how to find the actual density of a random variable once its hermite series or equivalently Z-series is given.

Suppose the Z-series representation of a random variable is given as

$\, X \, = \, c_0 \, + \, c_1 \, Z + \, c_2 \, Z^2 + \, c_3 \, Z^3 + \, c_4 \, Z^4 + \, c_5 \, Z^5$    Equation(1)

and we want to find its density.

First we take a grid in Z ranging from -5 to +5 with equal intervals. nth point on the grid is denoted as $Z_n$ and there are a total of N points.
We find $X(Z_n)$ with respect to every point on the grid and also calculate $\frac{dX}{dZ}$ at nth point either by finite differences or by differentiating the expression for X and then evaluating it at $Z_n$. We use Equation(1) for these calculations.

now
$p(X(Z_n)) \, = \, p(Z_n) \, \frac{dZ_n}{dX(Z_n)}$
Here p(Z_n) is standard normal density at nth grid point that was earlier calculated.

The above equation is calculated on the enire Z-grid and you can find the probability density at every point $X_n$ and plot it.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

I will come towards the details of moments calibrations to hermite series to find the coefficients of hermite series later. First we suppose that we have found coefficients of hermite series already so that the coefficients represent a random variable through matching its moments.

We suppose that we have already determined the coefficients of our hermite series or equivalently its Z-series.
Below is a Z-series representation of the density of a random variable.

$\, X \, = \, c_0 \, + \, c_1 \, Z + \, c_2 \, Z^2 + \, c_3 \, Z^3 + \, c_4 \, Z^4 + \, c_5 \, Z^5$    Equation(1)

When we make an observation of the variable, we know its value in X-coordinates. But in most calculations, and analytics, we would need to know the underlying value of Z. This underlying value of Z is the value that when used in RHS of Equation(1), makes the result on LHS equal to observed value of X.

We will need the value of Z underlying observations for analytics like calculations of correlations between various hermits polynomials and regression based predictions.

This underlying value of Z is found by Newton root-finding. To find the initial starting guess for Newtons's method, I used inverse series from Mathematica which is an expansion of Z in terms of powers of X. The program takes an array of values of observations of the same random variable and does Newton-root finding. Then it checks for convergence of each observation so that it matches the calculated underlying Z-values. The values that have not converged are given to a second function that finds the underlying Z-value through bisection method.

Here is the matlab(please do not worry about matlab since I rarely used any matlab specific function and it is mostly arithmetic coded in matlab) program.

header of the function is

function [Z] = CalculateZgivenXAndZSeriesC3(X,c0,c)

The function takes in an array of observations for which we have to find an array of underlying Z-values. c0 is a scalar input to function and represents the first coefficient of zeroth poer of Z-series representation of the random variable and c is an array of coefficients of higher powers of Z-series.

This function will only work for Z-series of third order of highest power only equal to three.

.
function [Z] = CalculateZgivenXAndZSeriesC3(X,c0,c)

%Below is starting guess for Newton from inverse Series of third order. We are inverting
Z=(X-c0)/c(1)-(c(2) * (X-c0).^2)/c(1).^3+(((2* c(2).^2)./c(1).^2-c(3)./c(1)).* (X-c0).^3)./c(1).^3;

%Below we go into Newton while-loop
mm=0;
while((max(abs(X-(c0+c(1)*Z+c(2)*Z.^2+c(3)*Z.^3)))>.00000001) && (mm<=10))

X0=c0+c(1)*Z+c(2)*Z.^2+c(3).*Z.^3;
dX0dZ=c(1)+2.*c(2)*Z+3.*c(3).*Z.^2;
Z=Z-(X0-X)./(dX0dZ);
mm=mm+1;
end

%Below program checks for convergence and for unconverged values finds answer through bisection method.
if(mm>=10)
pp=0;
for nn=1:length(X)
if( abs(X(nn)-(c0+c(1)*Z+c(2)*Z.^2+c(3)*Z.^3))>.00000001 )
pp=pp+1;
Y(pp)=X(nn);
Index(pp)=nn;
end
end
for ppp=1:pp
ZZ(ppp)=CalculateZgivenXAndZSeriesBisectionC3(Y(ppp),c0,c);
Z(Index(ppp))=ZZ(ppp);
end
end

end
.
.
The program below takes a scalar observation as input and returns its underlying Z-value as result. It also takes a scalar value c0 equal to zeroth-coefficient of Z-series of the random variable and an array of coefficients of first three powers of Z-series.
.
.
.
function [Z]=CalculateZgivenXAndZSeriesBisectionC3(Y,c0,c)

if(Y>c0)
if(Y< c0+c(1)*.5+c(2)*.5^2 +c(3) * .5^3)
Za=0;
Zb=.5;
elseif(Y< c0+c(1)*1+c(2)*1^2 +c(3) * 1^3)
Za=.5;
Zb=1;
elseif(Y< c0+c(1)*1.5+c(2)*1.5^2 +c(3) * 1.5^3)
Za=1;
Zb=1.5;
elseif(Y< c0+c(1)*2+c(2)*2^2 +c(3) * 2^3)
Za=1.5;
Zb=2;
elseif(Y< c0+c(1)*2.5+c(2)*2.5^2 +c(3) * 2.5^3)
Za=2;
Zb=2.5;
elseif(Y< c0+c(1)*3+c(2)*3^2 +c(3) * 3^3)
Za=2.5;
Zb=3;
elseif(Y< c0+c(1)*3.5+c(2)*3.5^2 +c(3) * 3.5^3)
Za=3;
Zb=3.5;
elseif(Y< c0+c(1)*4+c(2)*4^2 +c(3) * 4^3)
Za=3.5;
Zb=4;
elseif(Y< c0+c(1)*4.5+c(2)*4.5^2 +c(3) * 4.5^3)
Za=4;
Zb=4.5;
else
Za=4.5;
Zb=5.5;
end
for mm=1:100
Zmid=.5*Za+.5*Zb;
if(Y< c0+c(1)*Zmid+c(2)*Zmid^2 +c(3) * Zmid^3)
Zb=Zmid;
elseif(Y> c0+c(1)*Zmid+c(2)*Zmid^2 +c(3) * Zmid^3)
Za=Zmid;
end
Z=.5*Za+.5*Zb;
Za
Zb
Z
end

end

if(Y<c0)
if(Y> c0+c(1)*-.5+c(2)*(-.5)^2 +c(3) * (-.5)^3)
Za=-.5;
Zb=0;
elseif(Y> c0+c(1)*(-1)+c(2)*(-1)^2 +c(3) * (-1)^3)
Za=-1;
Zb=-.5;
elseif(Y> c0+c(1)*(-1.5)+c(2)*(-1.5)^2 +c(3) * (-1.5)^3)
Za=-1.5;
Zb=-1;
elseif(Y> c0+c(1)*(-2)+c(2)*(-2)^2 +c(3) * (-2)^3)
Za=-2;
Zb=-1.5;
elseif(Y> c0+c(1)*(-2.5)+c(2)*(-2.5)^2 +c(3) * (-2.5)^3)
Za=-2.5;
Zb=-2.0;
elseif(Y> c0+c(1)*(-3)+c(2)*(-3)^2 +c(3) * (-3)^3)
Za=-3;
Zb=-2.5;
elseif(Y> c0+c(1)*(-3.5)+c(2)*(-3.5)^2 +c(3) * (-3.5)^3)
Za=-3.5;
Zb=-3.0;
elseif(Y> c0+c(1)*(-4)+c(2)*(-4)^2 +c(3) * (-4)^3)
Za=-4;
Zb=-3.5;
elseif(Y> c0+c(1)*(-4.5)+c(2)*(-4.5)^2 +c(3) * (-4.5)^3)
Za=-4.5;
Zb=-4.0;
else
Za=-5.5;
Zb=-4.5;
end
for mm=1:100
Zmid=.5*Za+.5*Zb;
if(Y< c0+c(1)*Zmid+c(2)*Zmid^2 +c(3) * Zmid^3)
Zb=Zmid;
elseif(Y> c0+c(1)*Zmid+c(2)*Zmid^2 +c(3) * Zmid^3)
Za=Zmid;
end
Z=.5*Za+.5*Zb;
Za
Zb
Z
end

end

end
.
.
Using above programs, we calculate the value of Z underlying an observation when coefficients of Z-series are given. The above programs work only for third order polynomial representation of random variable.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Once we find a Z-series representation by fitting first four moment, it would be a third order polynomial.

Suppose we have Z-series of two random variables are given and we want to find correlations between them given a set of joint observations of two variables.
The first variable is
$Y=a_0 + a_1 \, Z_1 + a_2 \, Z_1^2 + a_3 \, Z_1^3$  Eq(A)
The second variable is
$X=b_0 + b_1 \, Z_2 + b_2 \, Z_2^2 + b_3 \, Z_2^3$  Eq(B)

We want to remind the friends that for every Z-series in powers of Z, there is an equivalent Z-series in hermite polynomials given as
$Y=ah_0 + ah_1 \, Z_1 + ah_2 \, (Z_1^2-1) + ah_{3} \, (Z_1^3-3 \, Z_1)$
and
$X=bh_0 + bh_1 \, Z_2 + bh_2 \, (Z_2^2 \, -\, 1) + bh_3 \, (Z_2^3 \, - \, 3 \, Z_2)$

For each data point for every variable, we find out corresponding value of Z by inverting the Z-series relationship given in equations A and B.
In previous post, we found out how to do this and I also gave a program does exactly this to calculate the underlying Z-values given the observations of a random variable when coefficients of its Z-series are known.

Suppose we have a total of P  joint observations of stochastic processes $X$ and Y. Here
$Z_{1,p}$ denotes our standardized Z-value corresponding to pth observation of data of Y and  $Z_{2,p}$ denotes our standardized Z-value corresponding to pth observation of data of explanatory variable $X$
For Finding Correlation between first hermite polynomial components, we write the equation for the correlation between first hermites of both variables as

$\rho_{XY,1} \, = \, \frac{\displaystyle\sum_{p=1}^{P} \, Z_{1,p} \, \, Z_{2,p} \,}{\sqrt{\displaystyle\sum_{p=1}^{P} \, {Z_{1,p}}^2 \,\displaystyle\sum_{p=1}^{P} \, \, {Z_{2,p}}^2 \,}}$

Again, Please notice that for calculation of above equation, we first have to find the specific values of $Z_{1,p}$ and $Z_{2,p}$ that are associated with particular pth observation of $X_p$ and $Y_p$ by some inversion method as we did in previous post.
For Finding Correlation between second hermite polynomials, we write the equation for the correlation between second hermite polynomial of observation Z-values of X and Y as

$\rho_{XY,2} \, = \, \frac{\displaystyle\sum_{p=1}^{P} \, ({Z_{1,p}}^2-1) \, \, ({Z_{2,p}}^2-1) \,}{\sqrt{\displaystyle\sum_{p=1}^{P} \, {(Z_{1,p}^2-1)}^2 \,\displaystyle\sum_{p=1}^{P} \, \, {({Z_{2,p}}^2-1})^2 \,}}$

similarly for nth hermite polynaomial correlations,( even though we have only three hermites in this example) we would write as

$\rho_{XY,n} \, = \, \frac{\displaystyle\sum_{p=1}^{P}\, H_n(Z_1) \, \, H_n(Z_2) \,}{\sqrt{\displaystyle\sum_{p=1}^{P} \, {H_n(Z_1)}^2 \,\displaystyle\sum_{p=1}^{P} \, \, {H_n(Z_2)}^2 \,}}$

Please note that correlations are in joint products of hermite polynomials associated with observation Z-values. Their coefficients are only required for covariances between hermite polynomials.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

After Calculations of Correlations, it is obvious how to calculate covariances between two random variables whose hermite orthogonal representation is known. However, we introduce some nomenclature that would help us clearly understand the regression between random variables in question.

First of all we copy the hermite series of the two random variables from previous post.
The hermite polynomial series of first random variable is given as
$Y=ah_0 + ah_1 \, Z_1 + ah_2 \, (Z_1^2-1) + ah_{3} \, (Z_1^3-3 \, Z_1)$       Equation(1)
and the hermite polynomial series of second random variable is given as
$X=bh_0 + bh_1 \, Z_2 + bh_2 \, (Z_2^2 \, -\, 1) + bh_3 \, (Z_2^3 \, - \, 3 \, Z_2)$  Equation(2)

Here I want to tell friends an interesting thing that first zeroth coefficient in hermite series representation is always mean of the random variable.

We write the equation(1) for Y  dividing it into orthogonal polynomial components as
$Y=\mu_Y + Y_1 + Y_2 + Y_3 \,$
where
$\mu_Y= \, ah_0$ , $\, Y_1 \, = \, ah_1 \, Z_1 \,$ , $\, Y_2 \, = \, ah_2 \, ({Z_1}^2 \, - \, 1) \,$  , $\, Y_3 \, = \, ah_3 \, ({Z_1}^3 \, - \, 3 \, Z_1) \,$

Similarly, we write the equation(2) for X dividing it into orthogonal polynomial components as
$X=\mu_X + X_1 + X_2 + X_3 \,$
where
$\mu_X= \, bh_0$ , $\, X_1 \, = \, bh_1 \, Z_2 \,$ , $\, X_2 \, = \, bh_2 \, ({Z_2}^2 \, - \, 1) \,$  , $\, X_3 \, = \, bh_3 \, ({Z_2}^3 \, - \, 3 \, Z_2) \,$

So total covariance between X and Y would be sum of covariances across various orthogonal polynomial components

$\, COV[X,Y] \, = \, E[\, X\, - \,\mu_X, \, Y\, -\, \mu_Y ] =E[ \,( X_1 + X_2 + X_3 \,) \,( Y_1 + Y_2 + Y_3 \,)] \,$
We notice that there are possible non-zero correlations between X and Y components with the same index. When X and Y do not share any index, their product goes to zero since they belong to orthogonal polynomial of a different order.

So
$\, COV[X,Y] \, =E[ \,( X_1 \, Y_1+ X_2\, Y_2 + X_3 \, Y_3 \,) \,] \,$
$=\,E \Big[ \, \big[bh_1 \,Z_2 \big]\, \big[ah_1 \, Z_1 \big]+ \big[bh_2 \, ({Z_2}^2 \, - \, 1) \big], \big[ah_2 ({Z_1}^2\,-\,1) \big] + \big[bh_3 \, ({Z_2}^3 \, - \, 3 \, Z_2)\big]\, \big[ah_3 ({Z_1}^3\,-\,3 \, Z_1) \big] \Big] \,$
$=\, (ah_1 \, bh_1)\, E\big[Z_2 \,Z_1 \big]\, + \, \, (ah_2 \, bh_2)\, E\big[ ({Z_2}^2 \, - \, 1) \,({Z_1}^2\,-\,1) \big] + \, \, (ah_3 \, bh_3)\,E\big[ \, ({Z_2}^3 \, - \, 3 \, Z_2)\, ({Z_1}^3\,-\,3 \, Z_1) \big] \,$

But we know from previous posts that
$E\big[Z_2 \,Z_1 \big]\,= \rho_{XY,1} \,$
$E\big[ ({Z_2}^2 \, - \, 1) \,({Z_1}^2\,-\,1) \big]\,= 2 \, \rho_{XY,2} \,$
$E\big[ \, ({Z_2}^3 \, - \, 3 \, Z_2)\, ({Z_1}^3\,-\,3 \, Z_1) \big] \,\,= 6 \, \rho_{XY,3} \,$

So our complete covariance equation would become
$\, COV[X,Y] \, =\, ah_1 \, bh_1\, \rho_{XY,1} \,\, + \,2 \, ah_2 \, bh_2\,\rho_{XY,2} \, + \,6 \, ah_3 \, bh_3 \,\rho_{XY,3} \, \,$
$\, COV[X,Y] \, = COV_{XY,1} \, + \, COV_{XY,2} \, + \, COV_{XY,3} \,$

We will like to introduce similar notation for Variances as
$\, Var[Y] \, =\, {ah_1}^2 \, + \, 2 \, {ah_2}^2 \, + \, 6 \, {ah_3}^2 \,$
$=\, Var_{Y,1} \,+\, Var_{Y,2} \,+\, Var_{Y,3} \,$

Similarly for X
$\, Var[X] \, =\, {bh_1}^2 \, + \, 2 \, {bh_2}^2 \, + \, 6 \, {bh_3}^2 \,$
$=\, Var_{X,1} \,+\, Var_{X,2} \,+\, Var_{X,3} \,$
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

We want to learn univariate regression in this post. Please consider this post extension of previous post.

We know classical univariate regression formula is given as
$\, E[Y \,|X] \, = \, \mu_Y \, + \, \frac{COV[X,Y]}{Var[X]} \, (X\, - \, \mu_X) \,$

We modify the above formula to deal with our orthogonal hermite components representation as
$\, E[Y \, | {X_1,X_2,X_3}] \, = \, \mu_Y \, + \, \frac{COV_{XY,1}}{VAR_{X,1}} \, X_1\,+ \, \frac{COV_{XY,2}}{VAR_{X,2}} \, X_2\,+ \, \frac{COV_{XY,3}}{VAR_{X,3}} \, X_3\,$

We have calculated the values of all variables on rhs in above equation in previous post. Substituting those values from previous post into this equation , we get

$\, E[Y \, | {X_1,X_2,X_3}] \, = \, \mu_Y \, + \, \frac{\, ah_1 \, bh_1\, \rho_{XY,1} \,}{\, {bh_1}^2 \,} \,(\, bh_1 \, Z_2 \,)\,+ \, \, \frac{\,2\, ah_2 \, bh_2\, \rho_{XY,2} \,}{\,2\, {bh_2}^2 \,} \,\, (\, bh_2 \, ({Z_2}^2 \, - \, 1) \,)\,+ \, \, \frac{\,6 \, ah_3 \, bh_3\, \rho_{XY,3} \,}{\,6\, {bh_3}^2 \,} \, (\, bh_3 \, ({Z_2}^3 \, - \, 3 \, Z_2) \,)\,$

Simplifying above, we get simple univariate rgression equation with multiple hermite polynomial representation as

$\, E[Y \, | {X_1,X_2,X_3}] \, = \, \mu_Y \, + \, ah_1 \, \rho_{XY,1} \, Z_2 \,+ \, \, ah_2 \, \rho_{XY,2} \, ({Z_2}^2 \, - \, 1) \,+ \, \, ah_3 \, \rho_{XY,3} \, ({Z_2}^3 \, - \, 3 \, Z_2) \,$

Please notice that in above equation, we only have values of hermite polynomials of the regressors and not their coefficients (of hermite polynomials of regressors) in the equation.

where

$X=bh_0 + bh_1 \, Z_2 + bh_2 \, (Z_2^2 \, -\, 1) + bh_3 \, (Z_2^3 \, - \, 3 \, Z_2)$
$=\mu_X + X_1 + X_2 + X_3 \,$
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, I will write another post about multivariate regressions tomorrow.
Only today evening, I was able to install an old version of windows 7 on my computer and installed computational programs on that. My younger brother did all of this for me.
I will start working on my computational work tomorrow. First I expect to re-write the moments calibration to hermite series part of the program using the idea of perturbing parameters while staying on ellipsoid. As soon as that moments calibration work is complete, I will start to solve important public data sets for regression of variables and .post my programs here.
We merely do not merely want to calculate proper calculations, in future research, we want to find out the dynamics behind the change of correlations. I am brainstorming if we could use differential equations with random noise for that purpose. But this is only after we are a bit more familiar with the current work and apply it at a few places in applied setting.
I also wanted to write a post describing how my mind control has increased in many aspects but  I am a bit sleepy and will write the post tomorrow. One of the worse things is that they are freely using gases again in my room that tend to sicken me. Mind control guys are completely resolved to continue my mind control. I will write this post tomorrow.

I will also be writing a small formal paper 10-15 pages using most of the material already posted on this forum. It will describe moments of a random variable calibration to hermite series all the way to regression with hermite poynomial representation of the random variables. In this paper, I do not intend to add any of my earlier research on SDEs.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, I will give detailed information about matching moments of a random variable to find its hermite orthogonal polynomial series expansion later since I am updating the numerical method that does this work.
However matching moments of a random variable to find the coefficients of hermite series is not the only method which can be used to find coefficients of hermite series expansion of the random variable. I have previously discovered another method that finds the density of the given random variable on a grid and then maps a standard normal random variable on this grid of the random variable through equivalent CDF and then extracts coefficients of hermite polynomial series by inner product between each orthogonal hermite polynomial and the given random variable whose hermite series representation has to be found. By mapping standard normal on the density of a given random variable, I mean that we find the CDF of the given random variable along each point on the constructed grid on which its probability density has been calculated and then use norminv function (that maps a CDF to value of standard normal variable at that point) to find the value of standard normal associated with the CDF of random variable. So standard normal random variable is mapped on the grid of original random variable so that both random variables share the same CDF at each grid cell. From the value of standard normal on each grid cell, we can find the values of hermite polynomials of various order associated with that grid cell and then be able to take inner product of hermite polynomials with the original random variable whose hermite polynomial representation has to be found.
This method works well when we have enough samples of the random variable to be able to construct its Parzen windom based density on a grid. I used it to find the hermite polynomial series representation of an asset in stochastic volatility monte carlo setting. On the other hand, you can use matching moments method when only as low as  7-9 samples of the density are available.

I explained this method in post 1688 written on January 22nd this year. https://forum.wilmott.com/viewtopic.php?f=4&t=99702&start=1680#p874060

Here I copy the post from there again for friends.

Friends, here is the explanation about the program. I will first give main equations and then relate them to the program to make it easy for friends to understand the program.
The main idea is to use the orthogonality property of hermite polynomials with respect to standard normal density. Let us assume we have a random variable X and we can represent it as a Z-series. All we know is that our random variable can be represented by a Hermite series version of Z-series(as both are equivalent), we do not know the precise coefficients yet and these coefficients are the ones we want to find out. Let us represent X as a hermite series as

$X \,= ah_0 \, + \, ah_1 \, H_1(Z) \, + ah_2 \, H_2(Z) \, + ... + \, ah_n \, H_n(Z) \,$

We multiply X with first hermite polynomial and integrate with respect to Gaussian density to find

$\displaystyle\int_{-\infty}^{\infty} \, X \, H_1(Z) \, p(Z) \, dZ$
$\displaystyle\int_{-\infty}^{\infty} \, (ah_0 \, + \, ah_1 \, H_1(Z) \, + ah_2 \, H_2(Z) \, + ... + \, ah_n \, H_n(Z) \,) \, H_1(Z) \, p(Z) \, dZ$
$=ah_1$

Similalry taking scalar product of X and second hermite polynomial with respect to Gaussian density would give us

$\displaystyle\int_{-\infty}^{\infty} \, X \, H_2(Z) \, p(Z) \, dZ$
$\displaystyle\int_{-\infty}^{\infty} \, (ah_0 \, + \, ah_1 \, H_1(Z) \, + ah_2 \, H_2(Z) \, + ... + \, ah_n \, H_n(Z) \,) \, H_2(Z) \, p(Z) \, dZ$
$=2! \, ah_2$

Taking scalar product of X with nth hermite polynomial with respect to Gaussian density would give us

$\displaystyle\int_{-\infty}^{\infty} \, X \, H_n(Z) \, p(Z) \, dZ$
$\displaystyle\int_{-\infty}^{\infty} \, (ah_0 \, + \, ah_1 \, H_1(Z) \, + ah_2 \, H_2(Z) \, + ... + \, ah_n \, H_n(Z) \,) \, H_n(Z) \, p(Z) \, dZ$
$=n! \, ah_n$

So basically we have to be able to take scalar product of X and nth hermite polynomial with respect to Gaussian density to find scaled value of coefficient of nth hermite polynomial.

In order to be able to do this, I first constructed a monte carlo density by dividing the valid domain into equal grid spacings of $\Delta X$ and calculated probability mass within each grid cell. Since the grid is uniform width, For each of the P monte carlo paths, we can just use one division per path to assign weight 1/P to appropriate grid cell. Once the iteration of this is complete over all paths, we get total probability mass in each of the grid cell as a result so that summed over all the grid cells probability mass becomes one. If there are M grid cells, probability mass in mth grid cell is given as $\Delta P_m$
Now we associate a value of Z with center of each of the grid cells. For this we take sum of probability mass over all previous grid cells plus half of the probability mass in the current cell where we want to find the value of Z.
This value of Z is found by norminv(sum of all the probability mass up to center of the grid cell)
It is the value of CDF at a point X that associates a corresponding Z to that point. So the value of Z at center of a grid cell is found by inversion of CDF of X at center of that grid cell.
Once we have found $Z_m$ associated with center of mth grid cell, we can replace our inner product integrals (between X and H_n(Z)) with respect to Gaussian density by finite sums over all the grid cells as

$\displaystyle\int_{-\infty}^{\infty} \, X \, H_n(Z) \, p(Z) \, dZ$
$\displaystyle\sum_{m=1}^{M} \, X_m \, H_n(Z_m) \, p(Z_m) \, \Delta Z_m$

But we know that  probability mass in mth cell is given as
$p(Z_m) \, \Delta Z_m=p(X_m) \, \Delta X_m = \Delta P_m$

So we can write as
$\displaystyle\int_{-\infty}^{\infty} \, X \, H_n(Z) \, p(Z) \, dZ$
$\displaystyle\sum_{m=1}^{M} \, X_m \, H_n(Z_m) \, p(Z_m) \, \Delta Z_m$
$\displaystyle\sum_{m=1}^{M} \, X_m \, H_n(Z_m) \, \Delta P_m \,$

which means that
$n! \, ah_n \, = \, \displaystyle\sum_{m=1}^{M} \, X_m \, H_n(Z_m) \, \Delta P_m \,$

All the values in RHS are known . $X_m$ is center of mth grid cell. $Z_m$ is value found by inversion of CDF at $X_m$ and $\Delta P_m \,$ which is the probability mass in mth grid cell was also calculated when we divided P monte carlo paths over the grid cells with 1/P weight.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

https://forum.wilmott.com/viewtopic.php?f=4&t=99702&start=1635#p873552

In the post given in link below, I have discussed another method about how to find Z-series of a random variable when its density is given in closed form for example as a lognormal random variable or as a CEV random variable etc. Once we find Z-series of the given random variable, we can convert it to its equivalent hermite orthogonal polynomials series expansion and then use it for all sort of analytics.

So we have three ways to find hermite orthogonal polynomial representation of a random variable.
1. By finding coefficients of the hermite polynomial series expansion so that moments of hermite series match the data moments of the random variable. This can possibly work well even when we have very few data points/samples from the density of the random variable.
2. By constructing a Parzen-window type density of the random variable when we have a large number of data points from its distribution. We map standard normal random variable on Parzen-window grid of the original random variable through equivalent CDF and then extract coefficients of the hermite polynomial representation of the random variable by doing inner product of the random variable with hermite polynomials of differnet order on the grid.
3. When we are given closed form density of the random variable, we can find its hermite representation by using the method to which I have given the link at the start of this post.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, I have been saying for quite some time that CIA and mind control agencies are only buying time and they have absolutely no intention to let me get off their mind control and start a normal life.
Even though my family became somewhat better with me after my latest research, I am convinced that mind control agencies recently started asking them that my forced and undue medication with antipsychotics has to be increased since I have started doing better intellectual work after my medication decreased. CIA and mind control agencies have desperate urgency that they cannot let this continue that I keep doing good research and keep posting it on internet while exposing their mean and inhuman tactics at the same time.
At home, I remain busy with my work and I rarely have negative thoughts about others despite that my family forced anitpsychotics on me. Even if I have some negative thoughts, I shrug them off and continue my research. And I remain amicable with my family in all respects.
Recently, on the insistence of mind control agents, my mother started taunting me that I was getting sick again and I had refused eating at home and I needed treatment. Just to show her that it was not the case, I took some food (two pieces of mutton meat) and ate them.
I am writing this to tell friends what it means to have food drugged with mind control chemicals and what victim has to go through when he eats drugged food.
A few minutes after eating mutton pieces, (all of a sudden out of the blue) I felt strong resentment and anger about my family that they were helping CIA in my persecution and were openly trying to deceive me. I also had strong feeling of anger that CIA had raped me for more than twenty years. Though I totally controlled my self and showed no emotion at all, I was being very upset inside and my mood was very bad.
Here I will really tell friends that I had to live with my family and over time I learnt to remain perfectly amicable and really have no negative thoughts about them. If I were fighting with them all the time, I could never have gotten any space to do the research work I have been able to do. I really try to concentrate on my work. At this stage I have practically forgiven them and I do not fret about how they behaved in the past and I remain perfectly amicable. Only time I show some resistance is when they try to do something to increase my medication. Really, if I had taken their bad behaviour to heart and kept fighting them, I would never have been able to do any research.
But I want to tell friends that mind control agencies give drugs with "mean neurotransmitters" to my family to add to my food. These drugs induce negative thoughts about my family and this is the purpose behind the drugs so that I would lose composure and start fighting with my family and I would be declared sick and schizophrenic.
I really want to request good Americans to please stop CIA and mind control agencies and ask them to end my persecution and let me live as a normal human being. I want to continue my research and people at CIA have already realized that it was a mistake to let me work on my research on insistence of so many good Americans who earlier protested about my inhuman treatment. They have started using gases in my room again that they had almost stopped a few weeks ago. Today, they drugged water at a store where I bought it and I bought drugged water and brought it home. They were very carefully mapping my path when I went out on motorcycle. I am sure they must have drugged water at several stores that lied on my way. (Though I was lucky that I also had some empty water bottles that I filled from elsewhere and they were good).
I really want to tell friends that CIA has absolutely no intention of ending my mind control and they want to continue my persecution at all cost.
For the context, I am copying some old posts. In these posts I have spoken about role of some bad jews in continuing my mind control. There I apologize to all good jews and tell them that my posts are not meant to hurt them in any way and I do not hole them responsible for this.  I pray for happiness and great future of all good jews who are nice to others and want everybody else to do the same.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, I want to make a special request for help in this post. As I told friends several times, I have been on antipsychotics since 1999 with the exception of a few brief periods of a few months each. Doctor says the reason behind forcing antipsychotics on me are my baseless fears that CIA somehow wants to hurt me and damage my brain. Doctors call me paranoid schizophrenic. Psychiatrists are very aware that I am perfectly healthy but they connive with Pakistan military who ask them to declare me mentally sick, and use force when I argue against their forced treatment. Most friends in US know about mind control have learnt over time that CIA in fact does try to retard innocent civilians who they do not like for whatever reasons. Many people even know complete truth how innocent victims are cornered by adding mind control drugs in the food, beverages and water supply of entire cities on behest of CIA. And when a victim complains about it, he is called schizophrenic and this has become a very useful tactic in the hands of CIA and military agencies to target and victimize people they do not like.
Though I never liked antipsychotics, I have been taking them in one form or another for more than past twenty years. When I was young, I was still able to take antipsychotics and run for a few kilometers everyday and be able to work since I used to have enough energy and stimulation. But I am already 50 years old now and it continues to become very difficult for my body to stand antipsychotic medication now. I am excessively over-weight due to low metabolism rate associated with decades of antipsychotics. My current weight is roughly 108 Kg which is more than thirty kilograms higher than the right weight for my height. I suffer from so many other physical problems due to antipsychotics. For example in the two weeks following an antipsychotic injection (fluanxol) that is given to me every 23 days, my breathing remains disturbed and is remarkably noisy as compared to breathing of an ordinary healthy human. Sometimes, I would even give involuntarily snorting like sounds when breathing. And many times I would have difficulty sleeping at night due to heavy and forced breathing. My family and doctors are not ready to compromise on my antipsychotics so it is useless to argue with them to end the antipsychotics as I have tried this several times to no avail. My heart beat that used to be around 75 more than 20 years ago as I recall, now remained around 120 for past twenty years due to high antipsychotics. These antipsychotics continue to slowly weaken the heart and negatively affect its functioning. I am grateful that I still do not have diabetes but I am sure that I would be diabetic in next few years because of high weight and low metabolism. Continued use of antipsychotics also have a positive correlation with diabetes even when controlled for other factors like weight. Due to continued use of antipsychotics at old age of 50 years, I am rarely upbeat and mostly remain lethargic and highly sedated. I have constant body aches and pains due to weakness caused by continued use of antipsychotics.
I know I can do some good research but it is only possible since this is the only thing I did in past twenty years of my life. Apart from research, there is no bright spot especially regarding my body and physical health. I really want to request American friends to please force your government and military agencies to not force psychiatrists to give me any more antipsychotics. My greatest wish at this part of my life is to be able to live a life without antipsychotics forced on me. I really cannot thank friends enough who could lobby for this or protest to American army against forced antipsychotics on me. Please help me me live a life without antipsychotics and I would be truly grateful to you.

I had earlier told friends that mind control agents and their backers are only buying time and they really want to increase mind control again as soon as the whole thing goes out of limelight.
Now this observation is proving to be exactly true. Mind control agents are again pressurizing my family especially my parents and my sister to do something to help the mind control agents in my mind control. About a month ago, doctor had decreased the period between my antipsychotic injection to one month. I was feeling remarkably better due to doctor's decision. Just about a week or so ago, many colors returned in my eyes and I was able to see the world as I used to before the start of mind control. And I was happy like a child when I would notice the brightness in many shades of colors. When people say that animals cannot see many colors and their shades, I know this firsthand since I have experienced it myself several times in past twenty five years. When I would not notice some shade or see it differently without brightness, I would have some sense that I am missing something in my brain but I would not know what I am missing. But when some parts of my brain would recombine and I would see some shades with brightness and richness again, I would know that I missed this color for several months or years. Just a few months ago, I was in such a bad state that it was difficult for me many times to differentiate many shades of blue and green.
Now after 25 years of my persecution, I am used to colors going out of my eyes and then coming back again after several years. But recently for more than a week, I have been seeing many colors and their shades with special richness as I had not done for many long years. And many times I become happy like a child when I see the world in its original and complete beauty.
But mind control agents are getting upset that they are losing control of my brain and some of my research has also gone out of limelight, therefore mind control agents have started to pressurize my family to find ways to put me back into complete mind control. They want my family to ask the psychiatrist that I am not doing well and he should increase my medication and put me back to 21 days interval between the antipsychotic injections. My sister called today and told me that she had realized that I was not taking proper care of my self and she would call the doctor on her own and ask her to increase my medication again. Mind control agents pressurize my family who are mere puppets in their hands and now my family wants to do everything possible to convince the doctor to increase my medication.
I am perfectly healthy in all respect but Pakistan army pressurizes the doctors to cooperate with CIA and when the psychiatrists also know that victim's family is also playing in their hands, they become too bold in forcing antipsychotic drugs on their victims. Though the current doctor is very aware that I am perfectly healthy and has maltreated me, I am still thankful that he was finally ready to decrease my medication and increase the interval between antipsychotic injections to thirty days. But now mind control agents want to pressurize my family to roll back what psychiatrist has done. I think I have told friends earlier that I am not allowed by my family to choose a psychiatrist of my own choice and every time I tried to approach a psychiatrist of my own choice, they detained me for months with their own psychiatrist.
I was rather hoping that my antipsychotics would completely end very soon but now that seems a relatively remote scenario and I am afraid that my family wants the doctor to undo whatever favor he has already given me.
I want friends that if I have just one wish in my mind, it is to be able to live a life without forced antipsychotics. I would not be willing to accept hundreds of millions of dollars if you pay me that money to continue my antipsychotics.
I really want to request good American people to ask their government and army to stop giving me antipsychotics and end my mind control. Please ask the American army and CIA to not force my family to be inhuman with me. I Want to request those jews and neocons behind me persecution to Please let me live with human dignity if you are ready to accept meas a human being and if it is not beyond your capability to accept a human being of muslim origin getting ahead in life.
I also want to respect good Europeans who sympathise with me to ask their embassies in Pakistan to force the Pakistan army to end mind control of innocent intelligent people on behest of neocons in CIA and also ask the psychiatrists to end the forced persecution of innocent victims.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

Friends, in a previous post I told friends how staff at a lady psychiatrist's hospital started beating me and abused me. During past twenty three years, there have been many many instances of my obvious abuse by different psychiatrists. I will describe some of those instances in this post.
Now back to my obvious abuse by two other psychiatrists. I was detained by this doctor who was running a mental health facility and was not a psychiatrist himself. At that time I did not use to walk very much and my blood pressure used to be somewhat high. Since other antipsychotics had failed to retard me, doctor suggested I take aripperizole injections. These injections affect the heart and also elevate blood pressure but doctor who wanted to retard me had no regard for my health while pretending to cure me. In initial days of my detention, staff at the hospital when checked my blood pressure told me high enough blood pressure in line with what I used to have but after I complained about aripperizole and its effect on blood pressure the staff started doctoring blood pressure and started telling me that my blood pressure was very good or even low. This is a post I wrote on another thread in this forum about it. How to safeguard my research - Page 32 - wilmott.com
.

Amin:

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Later after a few days, I was given a second aripperizole injection and released from detention. An injection was due every fifteen days. My parents were in Saudi Arabia and I was living with my sister. She insisted after fifteen days that I get a new aripperizole injection. I complained about too much side effects and other serious problems but my sister insisted that I must get the injection. She talked to another doctor and the doctor told her that if his blood pressure and pulse are fine, just give him the injection. I asked my sister that my blood pressure must be checked before I get the new injection. We went to a small hospital close to her house and when my blood pressure was checked my lower blood pressure was 110. The hospital staff recommended that I stay at the hospital in their care until my blood pressure decreases. It was a totally random blood perssure check. When hospital staff came to know that I had been given aripperizole injections, they told my sister to not give me those injections any more. My sister asked the staff to relieve me and we came home but I was not given aripperizole injections after that.
But this is interesting that the psychiatrist was initially desperate to prescribe aripperizole despite that I had told him about my heart and blood pressure problems and his staff was asked to doctor blood pressure so that I would accept the injections.
Good thing is that only after stopping aripperizole, I started my treatment with good psychiatrist  doctor Naeem Aftab (who is an American national) and he gave me enough relief that I was able to get a break in my research and solve the problem of high order monte carlo simulations. I would always be thankful to Dr. Naeem Aftab for giving me relief at such a crucial point in my life.

Yet another experience for abuse was with a seasoned psychiatrist who is known for his connections with Pakistan army. When army dictator Pervaiz Musharraf ruled the country, he made this doctor the dean(president) of the university of health sciences which regulates most medical universities in Pakistan. He had treated me for three years in year 2001. He started my treatment without interviewing me and gave me electric shocks at the start to jumpstart the therapy. I was given electric shocks for fifteen days in early 2001. I still recall that doctor would place a wooden wedge between my teeth and then give me an injection and I still recall how numbness would move in my body from feet to the upper body after the injection.
When going to that doctor's psychiatrist facility in 2001 I had to travel from Kot Addu to Rawalpindi. My family had given me some drug in my food and I became unconscious after an hour or so. I still recall how I was begging my younger brother to help me that I do not want to go anywhere and if he could help me but he like everybody else in the family refused. Slowly drug in food showed its effect and I have absolutely no memory of being taken in my father's car despite that it would have taken him at least eight hours to take me to Rawalpindi. Psychiatrist did not interview me and detained me and started giving me electric shocks. I still recall how I defecated in my bed there and they had to wash it.
Anyway, I was taken to this psychiatrist in 2017 again. He was very clearly told that I had been given antipsychotic drug larjactil by another doctor and it backfired and I had extremely severe drug-induced jaundice. But the doctor was so haughty and cavalier and so desperate to retard me that he still insisted that I must get the same drug larjactil and said that it was not necessary that it would cause jaundice every time. Problem for the doctor was that most antipsychotics had failed to retard me and he wanted to give me some potent antipsychotic that had not been well-tried on me. Though the drug was given to me earlier and later discontinued, the doctor wanted to take his chances to retard me and he had no regard for my health. This doctor is a big crony of Pakistan army and really wanted to please the generals and therefore he decided to give me larjactil again. And as would already be obvious to most friends, after two weeks, I started to show all signs of drug-induced jaundice again and my body and eyes were all yellow. My Biluribin shot like anything. And it took more than two months to control jaundice after that. Though my biluribin came back to normal after two months, several less known enzymes in my liver remained disturbed for several years after that and continued to show as abnormal in my liver function test reports.

I have been detained more than 30 times by various psychiatrists during past twenty-five years on orders of Pakistan Army. This is despite that I have never been violent all my life. I have never hit anyone in my adult life and I rarely speak with anyone in a loud voice. I have never done anything like breaking a glass or any utensils that is associated with sick people.
Here, I will recall an incident eight years ago when I myself approached a lady psychiatrist for help and how she abused me, and her staff beat me for twenty minutes and then tied me and started my treatment with antipsychotics again.
I am recalling this from early 2015. They had previously put me on extremely high antipsychotics and my creativity was all gone. I could barely do any intellectual work. At that time, I was able to convince my family to decrease the injections and they had agreed since I had already lost most of my creativity and intellectual capability. I think mind control agencies were also thinking that I had lost my creativity forever and they wanted to see if it was right time to decrease my drugs and still make sure that I would remain intellectually dull after injections had ended.
But I slowly started to work on my research and started to regain my brain. When mind control agencies realized that I had started to do better research, they asked my family to do something again to retard me. My father asked a self-stylized religious saint to come to our house and take care of me. He came to our house with one of his followers and told me that he wanted to give me injections and I would go to sleep for 24 hours, and they would take scans of my brain to treat me, but I continued to beg and plead him, and I was able to convince him to allow me to leave the home in my car. The supposed saint allowed me to leave since he was probably sure that he would get me later as I had no other place to go other than my home and, therefore, I was able to convince him to let me go outside. When I left the house, I made posts about the saint on at least two different forums from outside my house and pleaded for help. Enough people were following my story even at that time and the saint had to leave. I also called my father when I was outside and protested against this.
My family still remained very tense with me, and I was very afraid that they would force antipsychotics injections and mind control drugs on me therefore I stayed one night out of the home. Here is the post I made when I was away on April 14th, 2015, 2:09 pm
https://forum.wilmott.com/viewtopic.php?f=15&t=94796&start=315#p748191
"I am staying out of my home for the night since I had realized that my father will try to force me into detention again and I would request a court tomorrow for stay order to stop my father from hospitalizing me forcefully again and I would request the court that if there has to be a check on my mental health, it has to be done and investigated properly by a psychiatrist of my choice and I should remain free at the same time. I do not want to be at the mercy of my father's chosen psychiatrist as we saw in the past year that they continued to quote mental health laws and forcefully manipulate me into taking injections. If my father forcefully hospitalized me, I would be cutoff from the outer world and they would do whatever they like. My father's self-stylized saint(peer) had already hinted that he wanted to give me injections that would put me to sleep for 24 hours and he had no argument for my mental sickness at all.”

Before my computer was taken away from me, I was able to take notes for two days in the hospital on my computer that I write here. After first two days of medication, I was so extremely weak that I could barely walk and I had extreme difficulty speaking.

“My Diary for past two days.

4/27/2015
I had two sessions with the doctor N previously on the 20th of April and once before at 13 or 14th of April. I remained away from home during this time and second meeting on 20th of April, evening after the night when I remained away from home. I had told the doctor about my circumstances in the first meeting with her. She had said that I looked perfectly fine mentally but she needed to talk to my family. On 20th of April, I took my mother for a second joint session of myself and my and the psychiatrist said, after the meeting, that said she really could not call me schizophrenic but she needed one more sessions and asked me to come to her facility on 27th of April.
When I arrived at the psychiatric facility, a psychologist had asked me to accompany her which I did and on the way to the her room, I asked for the reason, after which she replied that she wanted to take a session of mine. I told her that I already had a session with another psyc hologist and also with a psychiatrist so I would prefer to have a new session with the psychiatrist now. She agreed and I returned to the waiting room. I was sitting on a sofa in the waiting room on 27th at 8:00 PM (PST) in the line to see the psychiatrist when about five guards took control of me, while I was sitting on a sofa in the waiting room to see the doctor, and gave me an injection. I was beaten and my legs and hands were very strictly tied for more than hour. Though the doctor was in her consultation room and a lot of people were waiting to see her, I was told again and again that doctor was not present at the facility.
In the previous two sessions, she had given the impression I was healthy mentally but she used force without interviewing for the third time or seeing me and without requests to comply. If she was given any special information, she should have talked to me about it and she should have verified with me any change in circumstances.
One injection was given around 8 PM when I arrived for third session with the doctor N. I really think that the doctor was there since there was a large rush of visitors and her SUV was parked in the hospital but everybody among the staff continued to say that doctor did not come to the facility this evening.
I was kept tied for more than one hour. I was taken tied from the waiting room to my room. All these rooms have video cameras so you can ask Dr. N for a video of the events though she might make the excuse that cameras were not working. One of the staff had mentioned when they brought me into a patient room in the facility, that everything was watched by cameras.I would request people familiar with the matter to request her to show them footage of the cameras. Of course, there could be many excuses that cameras were not working but I insist that cameras were working.
Good Mental health facilities try to avoid treating humans like this.
Another injection was given at night before I slept.
and my internet USB device was also taken. They simply took possession of everything out of my pockets and also took my laptop.
When I asked why they gave me injections, one of the staff told me you were ?hyper? and I was totally surprised since I was quietly sitting on the sofa in the waiting lounge for the doctor. I was told that I would stay in the hospital for three days. I am sure this matter can be resolved who is lying and who is truthful once you look apt video footage of cameras in most rooms.

4/28/2015
One injection was given in the morning. One injection was given around noon. All injections were IM (given in the muscle.)
Unknown pills(some pills in the morning +5 pills around noon + 3 pills at night) were given and I was told,? We will tell you know the names of these pills only when you leave.?
Computer was returned today but I was told that we did not take your mobile and they continued to say we never took my mobile. My mobile was never returned( despite I always keep my mobile on me.)
I was also not given my EVO USB device and I was told you cannot write anything on internet. However the USB device was returned to my mother at night.
I was told today by the psychologist that doctor N had gone to America and will come back after seven days and only then she would be able to see you. I was told we are adjusting your medicine and see what was a good medicine. I was told they were discussing the drugs with her on phone and skype There were several conflicting statements about where Dr. N was. Somebody said she was just out of city and had gone to Islamabad.
When I asked my mother who is living with me at the hospital, I wanted a second opinion, because of beating and other irresponsible behavior of the psychiatrist, my mother would make all sort of excuses while the staff would say that you cannot leave the hospital until Dr. N arrives who is currently out of country.
Earlier today, I was told that I would live at the hospital for fifteen days and later I was told that I would live at the facility for seven days. When I was tied, somebody had told me that I would live in the hospital for three days only.
My mother told me please do not write anything wrong on internet otherwise, they would take your USB forcefully like Dr. A had done more than one year ago. The purpose was to threaten me to not write about any maltreatment so that the doctor would continue to freely manipulate me.
7-8 pills were given in the afternoon and similar number of pills were given at night. At many times during the day, I could barely walk and had extreme difficulty speaking.”

Friends, I have previously described several horror experiences with psychiatrists but there are many more such experiences that I have not posted on this thread. Today I am going to describe a continuing experience in which the psychiatrist did everything in his power to retard me.
This is from year 2004/2005. I had a sound logic at that time and I was working from Lahore at a job with a foreign Japanese company that was paying me \$4000 per month. I was still on high antipsychotics and when high dosage of normal antipsychotics did not work, in order to retard me, the psychiatrist employed a procedure that is not known in psychiatric practice in any country. The psychiatrist would give me some drugs in a glucose drip every two/three days in the operation theater. When I would be given the glucose drip, I would gradually start to lose consciousness and when half of the drip bottle would have gone into my blood, I would literally be unconscious, and I would later be taken unconscious to my hospital bed by the hospital staff and I would wake up several hours later on the bed in my hospital room and I would be extremely weak.
It was such a frightening experience that I would shudder from horror as the day of this treatment would come closer and continued to beg the psychiatrist to stop this treatment. The psychiatrist would say that he was giving me vitamins and vitamins were always harmless. He insisted that vitamins had to be given to me. He never told me what drugs he gave me that made me unconscious and never discussed it with me other than mentioning vitamins. The psychiatrist continued to perform this procedure for several weeks after sometimes second or usually third day. Most people cannot imagine how it would feel when the drug in glucose drip would go into my body and I would lose my senses. To this day I shudder from complete horror and recall the extremely sick feeling when I would slowly lose consciousness.
Now when I look back, I realize that they were giving me mind control drugs in the drip to take the intelligent neurotransmitters out of my body. They continued to try to take intelligent neurotransmitters out of my body again and again through this procedure for several weeks. I was completely helpless before my family and the psychiatrist and had to go through such unlawful medical procedures.
As friends would have realized that purpose of mind control agencies was always to retard me since neocon scumbags of pentagon and their backers were not capable of appreciating my humanity and did everything in their power to retard me with an iron fist.
I have seen continuing pain, trauma and torture in my life only because I had some special neurotransmitters that people with hatred and venom were not ready to let me have since they considered it a threat to let a Muslim have those neurotransmitters.

After starting mind control in Lahore in early 2000's, bad jews (again apologies to good jews) were very emboldened how easy it was to manipulate army and government in Pakistan to start mind control of intelligent people in Pakistan. It was a time when Pervaiz Musharraf who was an army dictator ruled the country. He was very ready to please the CIA in order to get help from American army to prolong his illegal dictatorship in Pakistan. He fixed in place all the required mind control infrastructure in all large cities of Pakistan to please American army who was playing in hand of bad jews to damage the Muslim majority country.
When Americans rightly blame our country for terrorism, they must not forget that many neocon generals in American army have played in the hand of bad jews and have made every effort to systematically destroy talent in Pakistan on a very large scale using mind control technologies to keep it backward forever.
Mind control in Pakistan is ensured by top Pakistani army generals since they are given several hundred million dollars by US army every year and the top brass of Pakistan army divides this money among themselves with bigger fish getting larger share and smaller fish getting a smaller share. Therefore, mind control infrastructure in Pakistan is very detailed and thorough.
From my own experience, there is very large mind control activity in Lahore and Islamabad, the two cities where I have lived during the past twenty years. There are at least more than 500 mind control targets in Pakistan most of whom are computer science majors.
Mind control in Pakistan and many other places increased dramatically like never before. We have in Lahore a huge building called Arfa Towers where all the facilities for technology and IT companies were provided to attract new firms and entrepreneurs and later Arfa Towers became a scene of mind control carnage since tens of aspiring computer scientists and entrepreneurs were retarded and it became very easy for mind control agencies to find the talent since the facility attracted talent and retard it. Time between 2008 and 2016 was like black death for talent in technology in our country (I am not talking about pseudo-talent that American embassy continued to support and showcase heroically in the meantime.)  I still recoil with horror when I look back and recall how difficult it was to get good drinking water in Lahore during that time for more than six months in 2013-2014. I would have to go to remote parts of the city, go to villages around the city and sometimes even go to other cities and only then I would be able to get some good water after repeated attempts and my only worry during the day would be to somehow get good food and water.
At this point I like to tell friends that a lot of food and beverages that are drugged with mind control chemicals would be eaten or taken by most of the population that is not on mind control and they would never notice any anomaly in taste or any other effect since these mind control chemicals are inactive and inert and become chemically active only when targeted remotely with precise electromagnetic wave frequencies so only target victims are affected and this is a very cruel method to target intelligent people by drugging the entire supply of food and beverages and then focusing waves on the targeted individuals to activate the mind control chemicals. Pakistan army and mind control agencies would drug the entire public water supply and beverages knowing that the victim would have to drink something somewhere. Ground water is also so thoroughly drugged with mind control chemicals that entire public water supply of the city and several kilometers out of the city would be completely drugged with mind control chemicals. Pakistan army routinely does this in Lahore city on behest of American mind control agencies.
Ten years ago, army soldiers and agents would go to markets to drug the targeted food and beverages with mind control chemicals which they still. Large scale movements of soldiers and agents raise suspicions in general public and therefore mind control chemicals are supplied to beverage and food manufacturing companies in Pakistan, and they simply add the chemicals in food and beverages during the manufacturing process.
In Lahore city, there are more drugged beverages at any time than good beverages in small stores or large supermarkets alike since most of the beverages are drugged at manufacturing source by coca-cola, pepsi-cola and Nestle.
Coffee stimulates the brain and the brain connects in new ways after taking coffee and good coffee creates problems with keeping victims on mind control since their brain starts connecting in many new ways after taking good coffee. All coffee and I repeat that entire coffee in the country present in the stores is drugged with mind control chemicals. Most of the coffee on good stores is imported from foreign countries. I do not know how it works but importer is probably told by army to drug the imported coffee before distributing it in the market. It is even very difficult or rather impossible to get good coffee beans. Two or three years ago it was possible to find some brand of coffee that was good by trying all new brands but it has not worked for past two years now for me despite that I keep trying. I even know of instances that when I was able to buy good coffee of a certain brand from very large and posh stores, they took entire coffee brand off the shelf from all their branches within two to three days. People know that Corrupt Army in Pakistan that ensures mind control is more powerful than any political government here.
Tea used to be good but for past one year most mainstream Pakistani brands of black tea are drugged with mind control drugs. This has still not caught up thoroughly like coffee is thoroughly drugged and it is still possible to find some good foreign brands tea that is good.
All stimulant/energy drinks including red bull are well drugged at manufacturing source. Red bull was better till last year but then it became drugged everywhere. I even tried it at remote places in several cities other than Lahore and it has always been drugged lately.

Most of the soaps, shampoos, creams, toothpastes are drugged with mind control chemicals. I do not use shampoos or creams but whatever rare I tried them once in a while they were drugged and they charged my skin. All of these have mind control chemicals that charge the skin and make mind control extremely effective. Many people would be amazed when I tell them that even though I would be in my full senses before washing my head and face with a soap but after washing my head and face with a drugged soap, my eyesight would deteriorate and my conception of reality would completely change and sometimes I would face extreme anxiety. It is hard to emphasize that even a small thing like this can completely change the victim's state of mind and consciousness. Getting a good soap was very difficult for me till last year and I would drive for several hours in different parts of the city especially looking at all the pharmacies on the way to find some unknown brand or some soap that would have been manufactured several years ago when soaps were not getting completely drugged. I tried all the soaps including beauty soaps that are specially made for women, even many unknown brands, and specially imported soaps(which were all universally drugged thoroughly) and it was very very difficult to find a good soap. After finding a good soap, I would keep it in my pocket all the time. This is a small thing but you cannot live without it. Since our mouth is kept charged and some charges continue to go into victim's body from one's mouth. Therefore toothpastes are also added with mind control chemicals to charge inside of our mouth. You can imagine that after buying five or six tooth pastes of different remote brands, I would get a good tooth paste. I have recently been using toothpaste that is imported from abroad and is good. Toothpastes manufactured in Pakistan are all drugged with mind control chemicals.
Another big problem is that pharmaceutical companies add mind control chemicals to their drugs(pills etc). Most of the mind control victims are targeted due to their talent and intelligence in mind many times might require special minerals that are rare in our food.  Most of the mineral and vitamin formulas are also drugged with mind control chemicals. Most mainstream mineral and vitamin pills in the market are thoroughly drugged with mind control chemicals. I never like to take names but I would like to tell friends that greatest crook of all times is Glaxosmithkline( I would invite them to sue me for writing this on Wilmott). GSK's mineral and vitamin formulas, effervescent calcium tablets, and toothpastes are all drugged with mind control chemicals. There is a tablet by the name Kemadrin that has to be taken with antipsychotics and GSK manufactures it and it is also drugged with mind control chemicals. My experience with GSK is so bad that If I have to buy any medicine and if it is made by GSK, I would try my utmost to avoid taking it for the fear of its being drugged with mind control chemicals and look for alternatives.
Injection fluanxol and clopixol are also drugged with mind control chemicals.
Again all this thorough infrastructure took more than fifteen years to develop in Pakistan and Pakistan army continued to work hard to develop it over the years. There are at least five hundred brilliant Pakistanis most of whom are computer scientists who are on active mind control to thwart them to contribute to society in any good way.
All this infrastructure exists in Pakistan and was developed because neocon generals wanted to please bad but rich jews and American mind control agencies continue to perfect it over last twenty years. Again, mind control in Pakistan is ensured by top Pakistani army generals since they are given several hundred million dollars by US army
I have not talked about special infrastructure that is used to project EM waves on the target and this infrastructure is present on all major roads and public places in big cities and major highways and motorways in the country.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal

Amin
Topic Author
Posts: 2411
Joined: July 14th, 2002, 3:00 am

### Re: Breakthrough in the theory of stochastic differential equations and their simulation

My Letter To HRW About My Human Rights Abuse

Above is a very incomplete account of how various psychiatrists continued to play in hands of Pakistan army to retard me. It also does not cover all the time till year 2022 and the account stops several years earlier.
Pakistan Army has no personal animosity with me. Chief of Pakistan Army staff is given more than 50 billion rupees every year come rain or shine so that army continues American Mind control of intelligent people in Pakistan. Even when the relations between both countries are extremely bad, this bribe is always given to Pakistan army generals in time.

Before writing this post, I want to apologize to good jews but want to tell them there are enough evil jews who are doing everything to tarnish the great goodwill jews have so rightly earned through generations of hard work and being nice and kind to other people in the society. I want to request good jews to Please force these evil(jewish) people to end the bad practices that are starting to take root in American and other societies on their behest and before it is too late. I also really hope that my mind control would finally end(as I have been begging for past twenty years) and I would never have to write a post like this again.