# Ito integral approximation by Euler?

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

• Do you know the functions $\alpha$ and $\beta$? Can you write the process in vector form on $(Y_t,W_t)$? – Kiwiakos Jul 25 '14 at 12:08
• $a$ will be something like $Y_{t}*W_{t}$ and $b=1-a$. So you cannot vectorize it. – Math Girl Jul 25 '14 at 12:40
• Does the work of Platen say something about your case? $\int Y_t W_t dW_t$ looks difficult ... – Ric Jul 25 '14 at 12:51
• I will check it out. But Euler is a no go? – Math Girl Jul 25 '14 at 12:58
• – LazyCat Jul 25 '14 at 13:24

You can write it as $$\left(\begin{array}{c}dY_t\\ dX_t\end{array} \right) = \left(\begin{array}{cc}\alpha(X_t, Y_t)& \beta(X_t,Y_t)\\ 1 & 0\end{array} \right)\cdot \left(\begin{array}{c}dW_t\\ dZ_t\end{array} \right)$$ and check Platen's conditions (Lipschitz?) as Richard pointed out on the matrix perhaps?
If it is $$\left(\begin{array}{c}dY_t\\ dX_t\end{array} \right) = \left(\begin{array}{cc}X_t Y_t& 1-X_t Y_t\\ 1 & 0\end{array} \right)\cdot \left(\begin{array}{c}dW_t\\ dZ_t\end{array} \right)$$ I think that it should be fine.