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On Wong-Zakai approximations with $$\delta$$-martingales. (English) Zbl 1055.60056
The authors study approximations of the solutions of the following Stratonovich SDE $dx(t)= b(x(t))\,dt+ \sum^n_{j=1} \sigma_j(x(t))\circ dW^i(t),\quad x(0)= \xi\in\mathbb{R}^d,\tag{$$*$$}$ driven by an $$n$$-dimensional Wiener process $$(W^1(t),\dots,W^n(t))$$. Starting point of their investigation are known approximations of the driving Wiener process (say, by smoothing by convolution or polynomial interpolation) through absolutely continuous processes $$B_\delta(t)$$ in such a way that $$\sup_{0\leq t\leq T}\mathbb{E}| W(t)- B_\delta(t)|^2\leq C_T\delta^\varepsilon$$ (‘approximation with exponent $$\varepsilon$$’). The main result is the problem whether this pushes through to the solution $$x_\delta(t)$$ of the SDE $$(*)$$ with $$dB_\delta(t)$$ replacing $$dW(t)$$ as driving noise. The answer is affirmative if, say, $$B_\delta(t)$$ is a good approximation with exponent $$\varepsilon$$ and if $$b$$, $$\sigma$$ and $$\nabla\sigma$$ are bounded and globally Lipschitz continuous. In order to be able to treat whole classes of approximating noise terms (and not just the examples mentioned above) the authors coin the notion of approximating $$\delta$$-martingale which is essentially an $${\mathcal F}_{t+\delta}$$-adapted semimartingale $$m_\delta(t)$$ with locally bounded variation which satisfies for every $${\mathcal F}_t$$-adapted process $$f(t)$$ the additional requirement that $$| E\int^t_0 f(s)\,dm_\delta(s)|$$ is bounded by the $$L^2(P\times dt)$$-norm of $$f(t)$$ and the oscillations of $$f$$ (taken in $$L^1(P)\otimes L^\infty(dt)$$-norm). (More manageable criteria for a process to be an approximating $$\delta$$-martingale are also given.) Most of the ‘classical’ approximations of the Wiener process satisfy these criteria.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J65 Brownian motion 60G15 Gaussian processes
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