discretization of a transformation of a stochastic process

Question

Let :

$$dS_t = \lambda \phi(S_t)dW_t$$

a better way of discretizing this SDE is to perform this transformation :

$$y(S)=\int \frac{dS}{\phi(S)}$$

and discretize this instead :

$$dy = \lambda dW_t$$

and recover $S$ as $S=y^{-1}(.)$

Now considering the disretization in the $S-$ domain , $\{ S_n\} ^{m+1}_{n=0}$ with $y_n=y(S_n), n=0 , ..., m+1$ we have :

$$y_n=y(S_0)+\frac{n}{m+1}(y(S_{m+1})-y(S_0))$$

Where this last equation come from?

Answer 1

With $Y_t=\psi(S_t)$, $ψ'(S)=\frac1{ϕ(S)}$ and thus $ψ''(S)=-\frac{ϕ'(S)}{ϕ(S)^2}$ the Ito formula gives $$ dY_t=ψ'(S_t)dS_t+\frac12ψ''(S_t)d[S]_t=λdW_t-\frac12λ^2ϕ'(S_t)dt $$ As you see, your implicit idea that the SDE reduces to a simple integration is not true. Still it might be that this form is easier to integrate numerically as the Euler-Maruyama method coincides with the slightly better Milshtein method, that is, you get the better result using the simpler method.