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On the approximation of stochastic partial differential equations. II. (English) Zbl 0669.60059
[For part I see the preceding review, Zbl 0669.60058].
In this second part of our study we apply the results of the first part to the stochastic partial differential equation \[ (0)\quad du(t,\omega,x)=\sum^{d}_{p,q=0}D_ p(a^{pq}D_ qu(t,\omega,x)+f(t,\omega,x)dt+ \] \[ +\sum^{d}_{p=0}\sum^{d_ 1}_{i=1}(b^ p_ i D_ pu(t,\omega,x)+g_ i(t,\omega,x))\circ dM^ i(t),\quad u(0,\omega,x)=u_ 0(\omega,x), \] approximated by the partial differential equations \[ (\delta)\quad du_{\delta}(t,\omega,x)=\sum^{d}_{p,q=0}D_ p(a_{\delta}^{pq} D_ qu_{\delta}(t,\omega,x)+f_{\delta}(t,\omega,x))dt+ \] \[ +\sum^{d}_{p=0}\sum^{d_ 1}_{i=1}(b^ p_{\delta i} D_ pu_{\delta}(t,\omega,x)+g_{\delta i}(t,\omega,x))dM^ i_{\delta}(t),\quad u_{\delta}(0,\omega,x)=u_{\delta 0}(\omega,x), \] as \(\delta\) \(\to 0\), where \(D_ k:=\partial /\partial x^ k\) for \(k:=1,...,d\) and \(D_ 0\) is the identity. As an application we treat the problem of robustness in nonlinear filtering of diffusion processes. In the special case when \(a_{\delta}^{pq}=a^{pq}\), \(f_{\delta}=f=0\), \(b^ p_{\delta i}=b^ p_ i\), \(g_{\delta i}=g_ i=0\) and \(b^ p_ i\) are nonrandom time-independent, the main result can be formulated as follows:
Assume:
(1) The derivates in \(x\in {\mathbb{R}}^ d\) of the coefficients \(a^{pq}\), \(b^ p_ i\) up to the second order are bounded measurable functions (well-measurable in t, \(\omega)\), and the \(d\times d\)-matrix \((a^{k\ell})\) is uniformly elliptic.
(2) \(M^ i_{\delta}\) is an adapted absolutely continuous process for every \(\delta\), i, such that \(M^ i_{\delta}(t)\to M^ i(t)\), \(S_{\delta}^{ij}(t)\to 2^{-1}<M^ i,M^ j>(t)\) in probability, uniformly in \(t\in [0,T]\), and the total variations of \(S_{\delta}^{ij}\) over [0,T] are bounded in probability, where \(S_{\delta}^{ij}(t):=\int^{t}_{0}(M^ i-M^ i_{\delta})dM^ j_{\delta}(s)\). The processes \(M^ i_{\delta}\) are continuous semimartingales whose bounded variation and quadratic variation have bounded derivatives in \(t\in [0,T].\)
(3) \(u_{\delta 0}\) and \(u_ 0\) are \({\mathcal F}_ 0\)-measurable random elements in \(W^ 1_ 2({\mathbb{R}}^ d)\) and in \(W^ 2_ 2({\mathbb{R}}^ d)\), respectively, such that \(| u_{\delta 0}-u_ 0|_ 0\to 0\) in probability, where \(| \cdot |_ 0\) denotes the norm in \(L_ 2({\mathbb{R}}^ d).\)
Then under the above conditions equations (0) and (\(\delta)\) admit a unique generalized solution u and \(u_{\delta}\), respectively, and \[ \sup | u_{\delta}(t)-u(t)|_ 0\to 0,\quad \int^{T}_{0}| u_{\delta}(t)-u(t)|^ 2_ 1dt\to 0\quad in\quad probability, \] as \(\delta\) \(\to 0\), where \(| \cdot |_ 1\) denotes the norm in \(W^ 1_ 2({\mathbb{R}}^ d)\).
Reviewer: I.Gyöngy

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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