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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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