 Amin
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Posts: 2212
Joined: July 14th, 2002, 3:00 am

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

Sorry, I was too sleepy last night to express myself clearly and I was being very incoherent. But the true idea that I had yesterday was that we should freeze the expectations made at earlier times and simply forward transport them along the Z-standard deviation grid associated with standard normal probability mass. By Z-standard deviation grid, I mean an expanding or contracting grid where grid cells are divided so that probability mass within each cell remains constant over time. So if there are payoffs at t1, t2, t3 and t4. We could freeze the expected payoffs at t1, on expanding/contracting standard deviations grid that preserves the probability mass in them and simply transport the expectation at t1 to respective cells on standard deviations grid at time t2 if we need to analyze the probability distribution at time t2 otherwise we could simply transport the frozen expectations to corresponding standard deviations grid at terminal time t4.
This could work for many types of "payoffs". But it still remains to be seen if the idea would work for all "payoffs". But I am sure exploring it will help solve the most general problem.
I am trying to superimpose a standard deviations grid on regular fixed grid and trying to learn "calculus" to convert expectations across two grids. Amin
Topic Author
Posts: 2212
Joined: July 14th, 2002, 3:00 am

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

Here is why the expectation on transition probabilities grid maintains its mean and its variance changes as the probability density evolves. First of all, let us suppose we have evolved the expectation using appropriate diffusion till time $t_{1}$ or alternatively calculated it at $t_1$ particular point in time using the grid, this expectation is given as $E[f(Y(t_{1}))]=f(Y(t_{1}))P(Y(t_{1}))$. This expectation stops changing even after it continues to evolve to step $t_2$ after we do not add an appropriate diffusion generator for the next step. At time $t_{2}$, this expectation changes to
$E[f(Y_1(t_{2}))]=\frac{E[f(Y(t_{1}))]}{P(Y(t_{2}))} P(Y(t_{2})) =\frac{f(Y(t_{1}))P(Y(t_{1}))}{P(Y(t_{2}))} P(Y(t_{2}))=f(Y(t_{1}))P(Y(t_{1}))$
Here I have used the term $E[f(Y_1(t_{2}))]$ for an expectation that was calculated at time $t_1$ and was diffused/evolved to time $t_2$ without its appropriate Ito-generator where we needed to calculate the expectation with probability $P(Y(t_{2}))$. As we saw in the above calculating the expectation with a different probability did not change its expectation. However repeating the above step for variance gives us the new equations as (supposing the mean is zero)
$Var[f(Y_1(t_{2}))]={[\frac{E[f(Y(t_{1}))]}{P(Y(t_{2}))}]}^2 P(Y(t_{2}))$
$={\frac{[f(Y(t_{1}))P(Y(t_{1}))}{P(Y(t_{2}))}]}^2 P(Y(t_{2}))=f(Y(t_{1}))P(Y(t_{1}))$
$=\frac{[P(Y(t_{1}))}{P(Y(t_{2}))} [{f(Y(t_{1}))]}^2 P(Y(t_{1}))]$
$=K(t_1,t_2)*Var(Y(t_1))$

where $K(t_1,t_2)=\frac{P(Y(t_{1}))}{P(Y(t_{2}))}$ is the one step variance multiplier and  true variance is given as $Var(Y(t_1))={f(Y(t_{1}))}^2 P(Y(t_{1}))$
So the variance of a $t_1$ expectation calculated with time $t_2$ probabilities is given as
$Var[f(Y_1(t_{2}))]=K(t_1,t_2)*Var(Y(t_1))$ Amin
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Joined: July 14th, 2002, 3:00 am

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

Friends, there is more progress on the work regarding densities of arithmetic path integrals of arbitrary SDEs. In our ito-taylor expnasion of the volatility, some terms are of order sqrt(dt) other are of order dt and even other are of order dt^1.5. The noise of order sqrt(dt) is transported forward on step by multipliplying the noise with sqrt(t2/t1)=sqrt((t1+dt)/t1). while noise terms of order dt are transported forward by multiplication with (t2/t1)=((t1+dt)/t1) and noise terms of order dt^1.5 are transported forward by multiplication with (t2/t1)^1.5=((t1+dt)^1.5/(t1^1.5); When noises of different order are separately evolved, it is very easy to forward transport them by appropriate multiplication ratio for relevant "time order" of the noise. This was a very needed discovery to forward transport the "payoffs" or "data" on our transition probabilities grid for non-linear SDEs. Having done this, I am very confident that I will very soon be able to write and upload a complete program for the densities of arithmetic path integrals of functions of arbitrary SDEs. If you try this on your own, please know that sign on mean-reverting noise terms is negative. Amin
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Posts: 2212
Joined: July 14th, 2002, 3:00 am

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

I have been able to concentrate on my research for past few days. While working on the arithmetic path integrals in transition probabilities framework project, I had some ideas and I was able to find a precise hermite polynomial representation for a general (existing) density in bessel coordinates. I found an algorithm using which we can easily convert any general bessel density(that has been arbitrarily evolved in time already) into a hermite polynomial representation. This was earlier not possible. When I found this algorithm, I digressed from the arithmetic path integrals project for a bit and decided to switch for a very few days towards the density evolution algorithm using SD fractions that expand or contract with time. Ok, here is the good news that I was able to refine my earlier algorithm and make it quite sharply precise for the evolution of densities of the SDEs in SD fractions grid(moving and expanding grid so as to keep probability mass in each SD fraction constant) framework. Here are the features of the new algorithm.
1. It is far more accurate than the earlier version. The accuracy is excellent everywhere.

2. It is very accurate for mean reverting SDEs  with smaller values of gamma when SDE mean is very close to zero where the earlier versions were being very inaccurate. For example with gamma=.75 or gamma=.65, the density was far too peaked as compared to true density when SDE mean was close to zero. This problem has been perfectly corrected in the new version since we are using an exact hermite representation of the bessel density.

3. The same density evolution algorithm works for mean reverting, partially mean-reverting,  non-mean reverting, zero mean and all other types of SDE densities. There are no arbitrary multiplications with parameters like gamma or kappa like I had in the previous version. I am sure that algorithm would work for all possible SDEs that are well behaved whether mean-reverting or partially mean-reverting or otherwise.

I am testing the algorithm for all possible errors and will be posting it in another two to three days. This new program became possible only because I was able to find a hermite representation of the arbitrary SDE density in bessel coordinates. I will be psoting the code here with explanations very soon.  