Friends, there were some small typos in my previous post. I have also explained some final equations better in this post. It is a better copy of previous post 1858. A coefficient of .5 was missing in Equation 2 in [$]{\Delta t}^2[$] terms. Please do look at this post especially new equations 7 and 8 that were not in previous original post. This is the right post with some errors removed. Please disregard post 1858.

Suppose we have a general SDE in original (as opposed to Bessel coordinates that we worked with mostly in context of Z-series) coordinates. The SDE is given as

[$] \, dX(t) \, = \, \mu(X(t)) \, dt \,+\, \sigma(X(t)) \, dZ(t) [$] Equation(1)

We know that a second order expansion of the above SDE is given as (using the method we have previously learnt on this forum)

[$] \, X_{t_1} \, = \, X(t_0) \, + \, \mu(X(t_0)) \Delta t \, + \, \sigma(X(t_0)) Z \sqrt{\Delta t} \\ + \, \frac{\partial \mu(X(t_0))}{\partial X} \, \mu(X(t_0)) \, \frac{{\Delta t}^2}{2} \, + \, \frac{\partial \mu(X(t_0))}{\partial X} \, \sigma(X(t_0)) \, (1 \, - \, \frac{1}{\sqrt{3}} ) \, Z \, {\Delta t}^{1.5} \\ +\, .5 \, \frac{\partial^2 \mu(X(t_0))}{\partial X^2} \, {\sigma(X(t_0))}^2 \, \frac{{\Delta t}^2}{2} \, \\ + \, \frac{\partial \sigma(X(t_0))}{\partial X} \, \mu(X(t_0)) \, (\frac{1}{\sqrt{3}} ) \, Z \, {\Delta t}^{1.5} \, + \, \frac{\partial \mu(X(t_0))}{\partial X} \, \sigma(X(t_0)) \, (Z^2\, - \, 1) \, \frac{\Delta t}{2} \\ +\, .5 \, \frac{\partial^2 \sigma(X(t_0))}{\partial X^2} \, {\sigma(X(t_0))}^2 \, \, (\frac{1}{\sqrt{3}} ) \, Z \, {\Delta t}^{1.5} \,[$] Equation(2)

We suppose without any loss of generality that starting distribution is given in the form of a Z-series random variable as

[$]X(t_0)\, = \, f(Z_0)[$] where f(Z_0) is a Z-series representation of [$]X(t_0)[$]. When we are starting from a delta source, this Z-series distribution would be a constant value.

We know that we can represent the functions of any Z-series random variable by a Taylor series. For the type of power exponents we use in drift and volatility functions, which are typically equal to or smaller than one, this Taylor series representation of functions is quite very faithful when seven terms are used in Z-series. We have repeatedly used Taylor series in the past to calculate the functions of Z-series random variables with good results.

Please read this post for introduction:

https://forum.wilmott.com/viewtopic.php?f=4&t=99702&start=1290#p868484
In order for friends to be familiar with this Taylor expansion of functions of Z-series, I would solve a toy problem. Suppose random variable X is represented by a fourth order Z-series (we want to find taylor expansion of its suitable functions). Z-series of X is given as

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

You can verify from various methods (also from the above link and direct series expansion) that a Z-series of a suitable function f(X(Z)) (when expanded around median, Z=0) is given as

[$]f(X(Z)) \, =\, \, f(a_0) \, +\, a_1 f^{'}(a_0) \, Z+\, (a_2 \, f^{'}(a_0)+1/2 {a_1}^2 \, f^{''}(a_0)) \, Z^2 \\ +(a_3 \, f^{'}(a_0)+a_1 \, a_2 \, f^{''}(a_0) \, +1/6 \, {a_1}^3 \, f^{3}(a_0) \, ) \, Z^3 \, \\+ \, (a_4 \, f^{'}(a_0)+1/2 \, ({a_2}^2 \, +\, 2 \, a_1 \, a_3) \, f^{''}(a_0) \, + \, 1/2 \, {a_1}^2 \, a_2 \, f^{3}(a0)+1/24 \, {a_1}^4 \, f^{4}(a_0)) \, Z^4 \, + \, O[Z]^5 [$] Equation(3)

I have expanded the Taylor series of above function to only low order for friends to understand it but in general we will have to write an automated program to expand it to seventh or higher order to keep the expansion faithful to true result. Usually we will almost always expand functions of Z-series using Taylor around the median Z=0.

Now going back to Equation 2 where we expanded the SDE in original coordinates to second order. We suppose that original distribution of the SDE is given by the random variable [$]X(t_0)[$] can be represented by a Z-series to suitable order as

[$]\, X(t_0) \, = \, c_0 \, + \, c_1 \, Z_0 \, + \, c_2 \, {Z_0}^2 \, + \, c_3 \, {Z_0}^3 + \, \ldots \\ =\, g_0(Z_0)[$] Equation(4)

We know that we can expand the functions of initial random variable also as Z-series using Taylor formula around median and therefore we can represent as

[$]\, \mu(X(t_0)) \, = \, g_1(Z_0) [$]

[$]\, \sigma(X(t_0)) \, = \, g_2(Z_0)[$]

[$]\, \frac{\partial \mu(X(t_0))}{\partial X} \, \mu(X(t_0)) \, = \, g_3(Z_0)[$]

[$]\, \frac{\partial \mu(X(t_0))}{\partial X} \, \sigma(X(t_0)) \, = \, g_4(Z_0)[$]

[$]\, \frac{\partial^2 \mu(X(t_0))}{\partial X^2} \, {\sigma(X(t_0))}^2 \, = \, g_5(Z_0)[$]

[$] \, \frac{\partial \sigma(X(t_0))}{\partial X} \, \mu(X(t_0)) \,= \, g_6(Z_0)[$]

[$] \, \frac{\partial \mu(X(t_0))}{\partial X} \, \sigma(X(t_0)) \,=\, g_7(Z_0)[$]

[$]\, \frac{\partial^2 \sigma(X(t_0))}{\partial X^2} \, {\sigma(X(t_0))}^2 \,=\, g_8(Z_0)[$] Equations(5)

Basically all g's above are different Z-series that represent function of original random variable [$]X(t_0)[$] that represents the initial starting distribution of SDE at every evolution step.

We insert above function Z-series expansions in Equation(2) to find the expression as

[$] \, X_{t_1}(Z,Z_0) \, = \, g_0(Z_0) \, + \, g_1(Z_0) \Delta t \, + \, g_2(Z_0) Z \sqrt{\Delta t} + \, g_3(Z_0) \, \frac{{\Delta t}^2}{2} \, + \, g_4(Z_0) \, (1 \, - \, \frac{1}{\sqrt{3}} ) \, Z \, {\Delta t}^{1.5} \\ +\, .5 \, g_5(Z_0) \, \frac{{\Delta t}^2}{2} \, + \, g_6(Z_0) \, (\frac{1}{\sqrt{3}} ) \, Z \, {\Delta t}^{1.5} \, + \, g_7(Z_0) \, (Z^2\, - \, 1) \, \frac{\Delta t}{2} +\, .5 \, g_8(Z_0) \, (\frac{1}{\sqrt{3}} ) \, Z \, {\Delta t}^{1.5} \,[$] Equation(6)

We can simplify the above equation with some algebra in a simpler form as

[$] \, X_{t_1}(Z,Z_0) \, = \, h_0(Z_0) \, + \, h_1(Z_0) Z \, + \, h_2(Z_0) \, (Z^2\, - \, 1) \, \,[$] Equation(7)

In above equation, we have re-arranged and absorbed constant values in coefficients of new re-scaled (after simple algebra) Z-series given as [$]h_0(Z_0)[$], [$]h_1(Z_0)[$], and [$]h_2(Z_0)[$].

Depending upon what is convenient for analytics, we can also re-write a different re-arranged version of above equation as

[$] \, X_{t_1}(Z,Z_0) \, = \, \tilde{h_0}(Z_0) \, + \, \tilde{h_1}(Z_0) Z \, + \, \tilde{h_2}(Z_0) \, Z^2 \, \,[$] Equation(8)

where [$] \tilde{h_0}(Z_0) \,[$], [$] \tilde{h_1}(Z_0) \,[$] and [$] \tilde{h_2}(Z_0) \,[$] are new different but re-arranged versions of previous Z-series given as [$]h_0(Z_0)[$], [$]h_1(Z_0)[$], and [$]h_2(Z_0)[$].

It was actually naiive of me earlier to use cumulant addition for solution of SDEs in Bessel coordinates. We do not necessarily need cumulant addition of previous random variable form of initial distribution the SDE at the start of time step and the SDE itself. We can take moments of the entire expression [$] \, X_{t_1}(Z,Z_0) \, [$] above by expanding the whole expression to its powers by simple algebra and then substituting expected values of powers of Z and [$]Z_0[$] knowing that both Z's are independent in order to calculate value of each moment of [$] \, X_{t_1}(Z,Z_0) \, [$]. We might have to write a good and clever algorithm (for very fast speed) that does it for series expressions embedded as in above equation but it should not be very difficult. Once we have first eight moments of [$] \, X_{t_1}(Z,Z_0) \, [$] calculated from above equation, we can turn them into a Z-series with a single stochastic variable [$]Z_1[$] which is different (but not independent of) from both [$]Z_0[$] and [$] Z[$]. So we have an evolution algorithm for SDEs in original variables in the form of equations 6, 7 and 8 after calculation of its moments and finding an appropriate Z-series for it that matches the calculated moments.

The above method also makes use of SDEs very transparent in the sense that we can acquire all information about conditional variances and higher conditional moments and we can match them to empirical phenomenon much more systematically.

I am thinking of another method that can possibly make use of derivatives of the SDE expression in Z and [$]Z_0[$] to match the Z-series as opposed to using moments to match the Z-series but some ideas are still hazy. If it works out, it would be a lot better algorithm since we would be spared the effort to match the moments for a single final Z-series(which might not possibly work universally all the time and might create some issues from time to time with the present algorithm we have.) If it works out with the derivatives algorithm, I will post the details of derivatives algorithm in next few days here on this forum.

We can also solve for stochastic volatility models in this way but we would need a two dimensional Z-series representation for general stochastic volatility model. We can represent the distribution of asset as one dimensional Z-series in SV models after matching moments but loss of necessary dimensionality would mean that we would not be able to write a proper conditional evolution of the SV SDE conditional on both volatility and and the asset itself. Therefore we would need a two dimensional Z-series for proper evolution of asset in SV system of SDEs. I have calculated most of the details and would present the solution of general SV system of SDEs in a few days.

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