Euler Approximation for Stochastic Differential Equations with Independent Brownian Drivers

Summary

The document asks whether an equation for a process driven by two independent Wiener processes can be approximated with an Euler scheme, and what conditions ensure convergence. The answer reframes the equation as a system of coupled stochastic differential equations, adding a state variable for the process that appears in the coefficients. The diffusion terms are then arranged in a matrix against the two Brownian drivers.

It suggests checking the applicable convergence conditions, mentioning Lipschitz conditions and Platen’s conditions, and gives a particular coefficient form as a case that may be suitable. This is only a brief pointer: it does not state the assumptions precisely, prove convergence, derive an error bound, or describe implementation details. Whether Euler approximation is valid depends on the coefficient regularity and the precise convergence result sought; the example should not be treated as a general guarantee.

Key ideas

• A system with multiple Brownian drivers can be expressed using vector states and a diffusion coefficient matrix.
• Euler approximation requires suitable regularity and growth assumptions on the coefficients.
• The answer points to Lipschitz-type and Platen conditions as criteria to check.
• The supplied coefficient example is tentative and does not establish a general convergence result.
• The document gives no proof, error bound, or detailed numerical procedure.

Tags

# Ito integral approximation by Euler?


# 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

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/14168

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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