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:

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


$\Delta t = T-t$

$\Delta W_t = W_T-W_t$

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?



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.


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.


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