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On the approximation of stochastic partial differential equations. I. (English) Zbl 0669.60058
The stochastic abstract partial differential equation $(0)\quad du(t)=(A(t,\omega)u(t)+f(t,\omega))dV(t)+\sum^{d_ 1}_{i=1}(B_ i(t,\omega)u(t)+g_ i(t,\omega))\circ dM^ i(t),\quad u(0)=u_ 0,$ driven by a continuous increasing process V and continuous semimartingales $$M^ i$$ is approximated as $$\delta$$ $$\to 0$$ by (abstract) partial differential equations $(\delta)\quad du_{\delta}(t)=(A_{\delta}(t,\omega)u_{\delta}(t)+f_{\delta}(t,\omega))dV_{\delta}(t) +\sum^{d_ 1}_{i=1}(B_{\delta i}(t,\omega)u_{\delta}(t)+g_{\delta_ i}(t,\omega))dM^ i_{\delta}(t),\quad u_{\delta}(0)=u_{\delta 0},$ driven by continuous, adapted increasing processes $$V_{\delta}$$ and continuous, adapted processes $$M^ i_{\delta}$$ of locally bounded variation. $$((B_ iu(t)+g_ i(t))\circ M^ i(t)$$ denote certain stochastic differentials corresponding to the Stratonovich differentials).
In the special case when $$A_{\delta}=A$$, $$f_{\delta}=f=0$$, $$B_{\delta i}=B_ i$$, $$g_{\delta i}=g_ i=0$$ for every $$\delta >0$$, $$i=1,...,d_ 1$$, and $$B_ i's$$ are time-independent nonrandom linear operators, the main result reads as follows:
Let $$H_ 3\hookrightarrow H_ 2\hookrightarrow H_ 1\hookrightarrow H_ 0$$ be a chain of embedded Hilbert spaces with continuous and dense injections. Assume $\quad | Au|_{\alpha -2}\leq K| u|_{\alpha},\quad | B_ iu|_{\beta -1}\leq K| u|_{\beta},\quad | B^*_ iu|_ 1\leq K| u|_ 2,$ $(1)\quad | [B_ i,B_ j]u|_{\alpha -1}\leq K| u|_{\alpha},\quad | (B_ iu,u)| \leq K| u|^ 2_ 0,\quad | (B_ iu,B_ ju)+(B_ iB_ ju,u)| \leq K| u|^ 2_ 0,$ $| (B_ iAu,u)+(Au,B_ iu)| \leq K| u|^ 2_ 1,\quad (u,Au)+\epsilon | u|^ 2_ 1\leq K| u|^ 2_ 0,$ for every $$u\in H_ 3$$, $$t\in [0,T]$$, $$\omega\in \Omega$$, $$\alpha:=1,2,3$$, $$\beta:=0,1,2,3$$, where $$\epsilon >0$$, $$K\geq 0$$ are constants, (,) denotes the scalar product in $$H_ 0$$, $$| u|_{\alpha}$$ is the norm of u in $$H_{\alpha}$$, $$[B_ i,B_ j]:=B_ iB_ j-B_ jB_ i$$ and $$B^*_ i$$ is the adjoint of $$B_ i$$ with respect to the scalar in $$H_ 0$$; $(2)\quad V_{\delta}(t)\to V(t),\quad M^ i_{\delta}(t)\to M^ i(t),\quad S_{\delta}^{ij}(t)\to 0$ as $$\delta$$ $$\to 0$$, in probability uniformly in t on bounded intervals for every $$i,j:=1,...,d_ 1$$, where $$S_{\delta}^{ij}(t):=\int^{t}_{0}(M^ i-M^ i_{\delta})dM^ j_{\delta}(s)-2^{-1}<M^ i,M^ j>(t).$$
The total variation of $$S_{\delta}^{ij}s$$ over the interval [0,T] is bounded uniformly in $$\delta$$ ; $(3)\quad | u_ 0-u_{\delta 0}|_ 0\to 0\quad in\quad probability.$ Let $$u_{\delta}$$ and u be solutions of equations ($$\delta)$$ and (0), respectively, in the normal triplet $$H_ 1\hookrightarrow H_ 0\equiv H^*_ 0\hookrightarrow H^*_ 1$$ such that $\int^{T}_{0}| u_{\delta}(t)|^ 2_ 2 dV_{\delta}(t)<\infty,\quad \int^{T}_{0}| u(t)|^ 2_ 3 dV(t)<\infty,$ $$u_{\delta}\in C([0,T];H_ 1)$$, $$u\in C([0,T];H_ 2)$$. Then under the above conditions: $$\sup_{t\in [0,T]}| u_{\delta}(t)-u(t)|_ 0\to 0$$ and $$\int^{T}_{0}| u_{\delta}(t)-u(t)|^ 2_ 1 dV_{\delta}(t)\to 0$$ in probability.
[For part II, see the following review, Zbl 0669.60059].
Reviewer: I.Gyöngy

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