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

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**April 17th, 2016, 12:24 pm**I will try to give wilmott friends an idea about my new research on stochastic differential equations and their simulation. My formal research paper will be ready in a week but I will present some preview details about the research for the friends for discussion.Let us write a simple stochastic differential equation in the general form with a finite step here as[$]x(t)=\int _0^t\mu (x(t)) \text{dt}+\int _0^t\sigma (x(t)) \text{dz}(t)\text{ }\text{Eq (1)}[$]When we simulate the above stochastic differential equation, the simplest method is to freeze the drift and volatility coefficients at their initial value, and multiply with dt and Normal(sqrt(t)) as[$]x(t)=\mu (x(0)) \text{dt}+\sigma (x(0)) \text{sqrt}(t)z.[$]Though many better techniques exist to simulate the stochastic differential equations for a finite step or to find their density, most of them lack generalization and have to be applied in special settings. We derive a generic new method that can simulate even non-linear stochastic differential equations to arbitrary accuracy. The method is extremely simple and is based on the simple understanding of how the non-linear coefficient functions are changing in time. I have tried to give some intuition about it in the exposition below.Before, we embark on a full-fledged scheme for the above general SDE, we would like to give the reader some crucial insights. To give the proper insights, we would like to start by taking a normal diffusion so that we could proceed in a step by step manner [$]\text{dx}(s)=\sigma \text{dz}(s)[$] Eq(2)or[$]x(t)=x(0)+\int _0^t\sigma \text{dz}(s)[$] Eq(3)We would like to start by learning how to calculate the following time integral with extreme precision and arbitrary accuracy[$]I(t)=\int _0^t\mu (x(s)) \text{ds}[$] Eq(4)We would also like to explain how to calculate the following stochastic integral with extreme precision and arbitrary accuracy[$]S(t)=\int _0^t\sigma (x(s)) \text{dz}(s)[$] Eq(5)Since we want to proceed step by step, we will first calculate these integrals for a normal brownian motion as given in Eq (2) and after gaining intuition , we later generalize the SDE from Eq(2) to Eq(1) and calculate I(t) and S(t) with the most general dynamics as given by Eq(1).We start by calculating I(t) as given by Eq(4) and Eq(2).[$]\int _0^t\mu (x(s))\text{ds}=t \mu (x(t)) -\int _0^ts d[\mu (x(s))][$] Eq(6)Expanding the second term on RHS[$]\int _0^ts d[\mu (x(s))]=\int _0^t \frac{\partial \mu (x(s))}{\partial x}s\text{$\sigma $dz}(s)+.5\int _0^t \frac{\partial ^2\mu (x(s))}{\partial x^2}s\sigma ^2 \text{ds}[$] Eq(7)We know that [$]\frac{\partial \mu (x(s))}{\partial x}[$] in first term of Eq(7) is a function of x(s) and we can apply Ito Change of variable formula on this as [$]\frac{\partial \mu (x(s))}{\partial x}=\frac{\partial \mu (x(0))}{\partial x}+\int _0^s \frac{\partial ^2\mu (x(u))}{\partial x^2}\text{$\sigma $dz}(u)+.5\int _0^s \frac{\partial ^3\mu (x(u))}{\partial x^3}\sigma ^2 \text{du}[$] Eq(8)Now we come to the most important step towards understanding of this expansion. We expand the first noise term on RHS of Eq(7) by substituting Eq(8) in it. [$]\int _0^t \frac{\partial \mu (x(s))}{\partial x}s\text{$\sigma $dz}(s)=\int _0^t \frac{\partial \mu (x(0))}{\partial x}s\text{$\sigma $dz}(s)+\int _0^t s \text{$\sigma $dz}(s)\int _0^s \frac{\partial ^2\mu (x(u))}{\partial x^2}\text{$\sigma $dz}(u) +\int _0^t s\text{$\sigma $dz}(s)\int _0^s .5 \frac{\partial ^3\mu (x(u))}{\partial x^3}\sigma ^2 \text{du}\text{ }[$] Eq(9)So we have a few nested integrals and we can easily deal with them. We could continue the expansions of [$]\frac{\partial ^2\mu (x(u))}{\partial x^2}[$] and [$]\frac{\partial ^3\mu (x(u))}{\partial x^3}[$] to further order just by following Eq(8) but we stop the order here for initial simplicity and explanation of the basic example. We freeze [$]\frac{\partial ^2\mu (x(u))}{\partial x^2}[$] and [$]\frac{\partial ^3\mu (x(u))}{\partial x^3}[$] in the above equation to their initial time zero values (we could have continued to expand them to desired accuracy if we wished) and write Eq(9) as[$]\int _0^t \frac{\partial \mu (x(s))}{\partial x}s\text{$\sigma $dz}(s)=\frac{\partial \mu (x(0))}{\partial x}\int _0^t s\text{$\sigma $dz}(s)+\frac{\partial ^2\mu (x(0))}{\partial x^2}\int _0^t s \text{$\sigma $dz}(s)\int _0^s \text{$\sigma $dz}(u) +.5 \frac{\partial ^3\mu (x(0))}{\partial x^3}\int _0^t s\text{$\sigma $dz}(s)\int _0^s \sigma ^2 \text{du}\text{ }[$] Eq(10)Now we are left to evaluation of intgrals[$]\int _0^t \text{s$\sigma $dz}(s)[$] Eq(11)and double nested integrals[$]\int _0^t s \text{$\sigma $dz}(s)\int _0^s \text{$\sigma $dz}(u)=\int _0^t s \sigma ^2\text{ }(z(s)-z(0))\text{dz}(s)[$][$]\int _0^t s\text{$\sigma $dz}(s)\int _0^s \sigma ^2 \text{du}=\int _0^t s\sigma \left(\left.\sigma ^2 s\right)\text{dz}(s)\right.[$]None of the above nested integrals vanishes and their continuations do not go to zero. I have devised schemes in my research that can calculate all different types of above integrals with arbitrary complexity of nesting to perfect precision using very simple integral calculus in a form that is a function of only time t, and a unit gaussian.We would similarly expand the second term of Eq(7) on RHS using Ito expansion of [$]\frac{\partial ^2\mu (x(s))}{\partial x^2}[$] and substituting it in Eq(7) . We will freeze the coefficients of above expansion at time zero like we did to get Eq(10) and get different integrals like Eq(11-13). All these stochastic integrals can be trivially solved by scheme I have devised in my research.To be continued tomorrow. Sorry, tired of writing more at the moment.