Approximation of partial derivative of a function of stochastic variable

Question

Let $X_t$ be an Ito process $$ dX_t=a(X_t,t)dt + b(X_t,t)dW_t $$ where $W_t$ is a Wiener process.

A numerical approximations of the solution of this equations is proposed by Milstein:

$$ X_T=X_t+a(X_t,t) \Delta t+ b(X_t,t)\Delta W_t+ \frac{1}{2}b(X_t,t) \frac{\partial{b(X_t,t)} }{\partial{x}} \left( \Delta W_t^2 - \Delta t\right) $$

where

$\Delta t = T-t$

$\Delta W_t = W_T-W_t$

According with literature, this can be transformed into a derivative-free scheme via the approximation (known as Platen explicit order 1 strong scheme): $$ b(X_t,t) \frac{\partial{b(X_t,t)} }{\partial{x}} \approx \frac{b(X_t+a(X_t,t) \Delta t+ b(X_t,t)\sqrt{\Delta t},t)-b(X_t,t)}{\sqrt{\Delta t}} $$

(See: 2001, Kloeden, "A brief overview of numerical methods for Stochastic Differential Equations")

Can anybody help understand how this approximation of the partial derivative is obtained?

Thanks

Answer 1

This is discussed Section 11.1 of Kloeden and Platen, "Numerical Solution of Stochastic Differential Equations". There it states:

Using the deterministic Taylor expansion it is easy to show that the ratio $$\frac{1}{\sqrt\Delta}\left\{b(\tau_n,Y_n + a\Delta + b\sqrt\Delta) - b(\tau_n,Y_n)\right\}$$ is a forward difference approximation for $b\frac{\partial{b}}{\partial{x}}$ at $(\tau_n,Y_n)$ if we neglect higher order terms.

Answer 2

A good way to intuit this is that the "average size of a Wiener process step $dW$" is $\sqrt{\Delta t}$. This is because the variance is $\Delta t$. So if you think about the time scale for anything associated with $b$ as $\sqrt{\Delta t}$ the formula follows.

Of course, that's just intution. Kloden's Numerical Solution of Stochastic Differential Equations and Taylor Approximations for Stochastic Partial Differentiations do this more rigorously.