zbMATH — the first resource for mathematics

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)
Full Text: