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Myshkin
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Reference request: Wong-Zakai smooth approximation in probabilty for stochastic differential equations

Added top-level tags; fixed punktuation.
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Stefan Kohl
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I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to aan $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) with $\sigma$ and $\mu$ smooth coefficients, and denote by $X_\epsilon$ the solution to the random ODE $$dX_\epsilon = \mu(X_\epsilon)dt + \sigma(X_\epsilon)dB_\epsilon$$$$dX_\epsilon = \mu(X_\epsilon)dt + \sigma(X_\epsilon)dB_\epsilon.$$

Then $X_\epsilon$ converges in probability, as $\epsilon \to 0$, so the solution of the SDE $$dX=\mu(X)dt+\sigma(X)\circ dB$$$$dX=\mu(X)dt+\sigma(X)\circ dB.$$

In particular I'm seeking results with no ellipticity requirement on $\sigma$. Many thanks.

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to a $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) with $\sigma$ and $\mu$ smooth coefficients and denote by $X_\epsilon$ the solution to the random ODE $$dX_\epsilon = \mu(X_\epsilon)dt + \sigma(X_\epsilon)dB_\epsilon$$

Then $X_\epsilon$ converges in probability, as $\epsilon \to 0$, so the solution of the SDE $$dX=\mu(X)dt+\sigma(X)\circ dB$$

In particular I'm seeking results with no ellipticity requirement on $\sigma$. Many thanks.

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) with $\sigma$ and $\mu$ smooth coefficients, and denote by $X_\epsilon$ the solution to the random ODE $$dX_\epsilon = \mu(X_\epsilon)dt + \sigma(X_\epsilon)dB_\epsilon.$$

Then $X_\epsilon$ converges in probability, as $\epsilon \to 0$, so the solution of the SDE $$dX=\mu(X)dt+\sigma(X)\circ dB.$$

In particular I'm seeking results with no ellipticity requirement on $\sigma$. Many thanks.

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Reference request: Wong-Zakai smooth approximation in probabilty for stochastic differential equations

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to a $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) with $\sigma$ and $\mu$ smooth coefficients and denote by $X_\epsilon$ the solution to the random ODE $$dX_\epsilon = \mu(X_\epsilon)dt + \sigma(X_\epsilon)dB_\epsilon$$

Then $X_\epsilon$ converges in probability, as $\epsilon \to 0$, so the solution of the SDE $$dX=\mu(X)dt+\sigma(X)\circ dB$$

In particular I'm seeking results with no ellipticity requirement on $\sigma$. Many thanks.