I want to explain the calculation of variance of dt-integral with a toy example of four time periods.

The gist of analytics below is that variances summation can only be applied on simulation intervals that are orthogonal to each other.

Let us first get oriented with variables and their notations.

We have an SDE for [$]X(t)[$] with four simulation periods on time points [$]t_0[$], [$]t_1[$], [$]t_2[$], [$]t_3[$], [$]t_4[$],

We suppose that SDE is such that its evolution within every time period is orthogonal to all other time periods. This holds for CEV noises and simple SDEs. For more difficult SDEs, you will have to take covariances into account carefully into this analysis.

We denote the evolution within each time period as

[$]\int_{t_0}^{t_1} dX(t) = X(t_1) \, - \, X(t_0) \,=\, X_{0,1} [$]

similarly for other simulation periods for example

[$]\int_{t_1}^{t_2} dX(t) = X(t_2) \, - \, X(t_1) \,=\, X_{1,2} [$]

please note that [$]X_{0,1} [$], [$]X_{1,2} [$], [$]X_{2,3} [$], [$]X_{3,4} [$], are all orthogonal to each other.

Now we write X(t) at end of all simulation periods in terms of above orthogonal increments as

[$] X(t_1) \,= \, X_0 \,+ \, X_{0,1}[$]

[$] X(t_2) \,= \, X_0 \,+ \, X_{0,1}\,+ \, X_{1,2}[$]

[$] X(t_3) \,= \, X_0 \,+ \, X_{0,1}\,+ \, X_{1,2}\,+ \, X_{2,3}[$]

[$] X(t_4) \,= \, X_0 \,+ \, X_{0,1}\,+ \, X_{1,2}\,+ \, X_{2,3}\,+ \, X_{3,4}[$]

Now (this is approximate but I am sure that we can make it exact with proper integration over each time interval)

[$]\, \int_{t_0}^{t_4} \, X(t) \, dt \, \approx \, \sum_{t_n=t_0}^{t_4} \, X(t_n) \, \Delta t \,[$]

[$]=\, \Delta t \, X(t_1) \,+\,\Delta t \, X(t_2) \,+\,\Delta t \, X(t_3) \,+\,\Delta t \, X(t_4) \,[$]

We want to convert above summation into orthogonal intervals so that we could apply summation of variances. Doing that

[$]=\, \Delta t \, [\, X_0 \,+ \, X_{0,1}] \,+\,\Delta t \, [\, X_0 \,+ \, X_{0,1}\,+ \, X_{1,2} ] \,+\, \Delta t \, [\, X_0 \,+ \, X_{0,1}\,+ \, X_{1,2}\,+ \, X_{2,3} ]\,[$]

[$]+\,\Delta t \, [\, X_0 \,+ \, X_{0,1}\,+ \, X_{1,2}\,+ \, X_{2,3}\,+ \, X_{3,4}] [$]

[$]=4 \, \Delta t \, \, X_0 \,+4 \, \Delta t \, X_{0,1}\,+\, 3 \Delta t \, X_{1,2} \,+2\, \Delta t \, X_{2,3} \,+\,\Delta t \, X_{3,4} [$]

So we have

[$]\, \int_{t_0}^{t_4} \, X(t) \, dt \, \approx \, \sum_{t_n=t_0}^{t_4} \, X(t_n) \, \Delta t \,[$]

[$]=4 \, \Delta t \, \, X_0 \,+4 \, \Delta t \, X_{0,1}\,+\, 3 \Delta t \, X_{1,2} \,+2\, \Delta t \, X_{2,3} \,+\,\Delta t \, X_{3,4} [$]

Now matching variances (considering we are evolving from a delta point at [$]X_0[$] which has no variance

[$]\, Var[\int_{t_0}^{t_4} \, X(t) \, dt] \, \approx \, Var[\sum_{t_n=t_0}^{t_4} \, X(t_n) \, \Delta t \,][$]

[$]=16 \, {\Delta t}^2 \, Var[X_{0,1}]\,+\, 9 \, {\Delta t}^2 \, Var[X_{1,2}] \,+4\, {\Delta t}^2 \, Var[X_{2,3}] \,+\, {\Delta t}^2 \, Var[X_{3,4}] [$]

I used the above logic to calculate the variances of dt-integrlas in the graphs for simple CEV SDEs that I attached a few posts earlier. I used simple dt summation in monte carlo with the same simulation period that I used for analytic calculations with hermite polynomials.

In next post I explain how I made the actual variance calculations with hermite polynomials.

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In this post I want to explain how I made the above variance matching calculations for dt-integrals of SDE with hermite polynomials.

I solved the SDE in Bessel coordinates using moment-matching as I have done in some programs that I posted earlier on this forum.

Let us suppose we get a Z-series representation on every time step in Bessel coordinates as

[$] w(t_n) \, = \, a_0(t_n) \, + \, a_1(t_n) \, Z \,+ \, a_2(t_n) \, Z^2 \, + \, a_3(t_n) \, Z^3 \,+ \, a_4(t_n) \, Z^4 \, [$]

I convert the above Z-Series of SDE variable in Bessel coordinates into a Z-Series of SDE variable in original coordinates, X(t) given as

[$] X(t_n) \, = \, c_0(t_n) \, + \, c_1(t_n) \, Z \,+ \, c_2(t_n) \, Z^2 \, + \, c_3(t_n) \, Z^3 \,+ \, c_4(t_n) \, Z^4 \, [$]

Now we convert the above Z-series in original coordinates to a series in Hermite polynomials(It is a bit of algebra and I will share my program so friends can see how I did it.)

Let us suppose that series in Hermite polynomials is given as

[$] X(t_n) \, = \, h_0(t_n) \, + \, h_1(t_n) \, H_1 \,+ \, h_2(t_n) \, H_2 \, + \, h_3(t_n) \, H_3 \,+ \, h_4(t_n) \, H_4 \, [$]

We converted into Hermite basis since it is very suitable for calculation of variances.

Now we calculate the hermite representation of each of the four orthogonal increments(in the context of toy example I presented in previous post) as

[$]X_{0,1} \, = \, h_0(t_1) \, + \, h_1(t_1) \, H_1 \,+ \, h_2(t_1) \, H_2 \, + \, h_3(t_1) \, H_3 \,+ \, h_4(t_1) \, H_4 \, [$]

There is no sum-squared calculations in first increment above since it is starting from delta origin.

[$]X_{1,2} \, = \, h_0(t_2)- \, h_0(t_1) \, + \, \sqrt{h_1(t_2)^2 - h_1(t_1)^2} \, H_1 \,+ \, \sqrt{h_2(t_2)^2 - h_2(t_1)^2} \, H_2 \, + \, \sqrt{h_3(t_2)^2 - h_3(t_1)^2} \, H_3 \,[$]

[$]+ \, \sqrt{h_4(t_2)^2 - h_4(t_1)^2} \, H_4 \, [$]

[$]= \, h_0(t_{1,2}) \, + \, h_1(t_{1,2}) \, H_1 \,+ \, h_2(t_{1,2}) \, H_2 \, + \, h_3(t_{1,2}) \, H_3 \,+ \, h_4(t_{1,2}) \, H_4 \, [$]

similarly

[$]X_{2,3} \, = \, h_0(t_3)- \, h_0(t_2) \, + \, \sqrt{h_1(t_3)^2 - h_1(t_2)^2} \, H_1 \,+ \, \sqrt{h_2(t_3)^2 - h_2(t_2)^2} \, H_2 \, + \, \sqrt{h_3(t_3)^2 - h_3(t_2)^2} \, H_3 \,[$]

[$]+ \, \sqrt{h_4(t_3)^2 - h_4(t_2)^2} \, H_4 \, [$]

[$]= \, h_0(t_{2,3}) \, + \, h_1(t_{2,3}) \, H_1 \,+ \, h_2(t_{2,3}) \, H_2 \, + \, h_3(t_{2,3}) \, H_3 \,+ \, h_4(t_{2,3}) \, H_4 \, [$]

and Similarly

[$]X_{3,4} \, = \, h_0(t_4)- \, h_0(t_3) \, + \, \sqrt{h_1(t_4)^2 - h_1(t_3)^2} \, H_1 \,+ \, \sqrt{h_2(t_4)^2 - h_2(t_3)^2} \, H_2 \, + \, \sqrt{h_3(t_4)^2 - h_3(t_3)^2} \, H_3 \,[$]

[$]+ \, \sqrt{h_4(t_4)^2 - h_4(t_3)^2} \, H_4 \, [$]

[$]= \, h_0(t_{3,4}) \, + \, h_1(t_{3,4}) \, H_1 \,+ \, h_2(t_{3,4}) \, H_2 \, + \, h_3(t_{3,4}) \, H_3 \,+ \, h_4(t_{3,4}) \, H_4 \, [$]

Now hermite representation of the dt-integral as given by variances in the previous post is calculated as

[$]\, \int_{t_0}^{t_4} \, X(t) \, dt \, \approx \, \sum_{t_n=t_0}^{t_4} \, X(t_n) \, \Delta t \,[$]

[$]=4 \, \Delta t \, \, X_0 \,+4 \, \Delta t \, X_{0,1}\,+\, 3 \Delta t \, X_{1,2} \,+2\, \Delta t \, X_{2,3} \,+\,\Delta t \, X_{3,4} [$]

[$] =\, {\Delta t} (4\, X_0 +4\, h_0(t_{0,1})+3\, h_0(t_{1,2})+2\, h_0(t_{2,3})+\, h_0(t_{3,4}))[$]

[$]+\, {\Delta t} \sqrt{(16 \, h_1(t_{0,1})^2+ 9 \, h_1(t_{1,2})^2+4 \, h_1(t_{2,3})^2+\, h_1(t_{3,4})^2)} \, H_1 \,[$]

[$]+\, {\Delta t} \sqrt{(16 \, h_2(t_{0,1})^2+ 9 \, h_2(t_{1,2})^2+4 \, h_2(t_{2,3})^2+\, h_2(t_{3,4})^2)}\, H_2 \,[$]

[$]+\, {\Delta t} \sqrt{(16 \, h_3(t_{0,1})^2+ 9 \, h_3(t_{1,2})^2+4 \, h_3(t_{2,3})^2+\, h_3(t_{3,4})^2)}\, H_3 \,[$]

[$]+\, {\Delta t} \sqrt{(16 \, h_4(t_{0,1})^2+ 9 \, h_4(t_{1,2})^2+4 \, h_4(t_{2,3})^2+\, h_4(t_{3,4})^2)}\, H_4 \,[$]

It is this above formula that I used to calculate graphs of dt-integrals of CEV noises that I posted on last Thursday.

Here are some notes.

1. Conversion of Z-series in Bessel coordinates to Z_series in original Coordinates is not exactly faithful (in extreme tails) when volatility is very high or when time is large especially for CEV exponents close to one (close to lognormal) and close to CEV exponent=.5 even though conversion to original coordinates on a grid is usually very good.

2. I tried to improve the accuracy of original coordinates by increasing the number of terms in Z-series of Bessel coordinates from 5 to 7 i.e. taking the calculations to eighth moment and many times it greatly helps to increase the accuracy of converted Z-series in original coordinates but sometimes including

seventh moments or eighth moments causes errors which I believe are due to lack of precision in 15-digit accuracy we mostly have. I think if a program is written in C++ employing float-128 high precision non-standard type, we can get a more accurate representation of Z-series of original coordinates but I have not done it myself yet so I cannot be perfectly sure.
3. Even though I could use variances matching to get dt-integrals quite exact, when I tried to solve for the evolution for densities of SDE itself by matching variances (sum of squared coefficients of hermite polynomials), it did not work at high volatilities but seemed to work at low volatilities. Friends are encouraged to experiment with it since I might be doing something wrong posssibly.
4. In calculations of square-root of sum of squared hermite coefficients, you have to be careful with signs, two negatives should convert to a larger negative and a large negative plus a smaller positive should sum-square to a smaller negative etc.

I will share my simple dt-integrals calculations program that I used to make previously posted graphs sometime tomorrow.

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