I was wondering how to find the solution of the following stochastic integral:
$$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation $$Y_{t}=Y_{0}+\int_{0}^{t}a(W_{s},Y_{s})dW_{s}+\int_{0}^{t}b(W_{s},Y_{s})dZ_{s}$$
where $W_{t}$ and $Z_{t}$ are two independent Wiener processes. Can I approximate this with the Euler scheme? If so, how do I know it will actually converge. If not, is there any way to find it?
Any help would be much appreciated
A nonstandard feature of your SDE is that the driving Brownian motions $W$ appears in the coefficients. This will probably make it more difficult to find convergence results. However, there exists results for a less general case. If youy SDE is given by, say $$ dY_t = \sum_{k=1}^p a_k(Y_t) dW^k_t $$ where the $W^k$ processes are, say, Brownian motions, and the mappings $a_k:\mathbb{R}\to\mathbb{R}$ are Lipschitz continuous, then there exists a unique solution to the SDE, and the Euler scheme converges. This follows from Theorem V.7 and the corollary to Theorem V.16 of P. Protter's "Stochastic integration and differential equations". The proof there is actually stated in the considerably larger generality of a generic semimartingale setup.
My personal guess would be that for your Euler scheme to converge, Lipschitz continuity of the coefficient functions $a, b: \mathbb{R}^2 \to \mathbb{R}$ in your SDE will suffice.