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

  • $\begingroup$ Do you know the functions $\alpha$ and $\beta$? Can you write the process in vector form on $(Y_t,W_t)$? $\endgroup$
    – Kiwiakos
    Jul 25 '14 at 12:08
  • $\begingroup$ $a$ will be something like $Y_{t}*W_{t}$ and $b=1-a$. So you cannot vectorize it. $\endgroup$
    – Math Girl
    Jul 25 '14 at 12:40
  • 1
    $\begingroup$ Does the work of Platen say something about your case? $\int Y_t W_t dW_t$ looks difficult ... $\endgroup$
    – Ric
    Jul 25 '14 at 12:51
  • $\begingroup$ I will check it out. But Euler is a no go? $\endgroup$
    – Math Girl
    Jul 25 '14 at 12:58
  • $\begingroup$ seems legit: en.wikipedia.org/wiki/Euler%E2%80%93Maruyama_method $\endgroup$
    – 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.


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