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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?


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