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Convergence of a spatial semidiscretization for a backward semilinear stochastic parabolic equation. (English) Zbl 1517.49017

The authors consider the transposition solution \( (p, z)\in D_{\mathbb{F}}([0, T]; L^2(\Omega; H))\times L^2_{\mathbb{F}}(0, T; H)\) of the following backward semilinear stochastic equation defined on a bounded convex polygonal domain \( O\subset \mathbb{R}^d, \, d=1,2,3\): \[ \begin{cases} dp(t) = -(\Delta p(t) + f(t, p(t), z(t)))dt+ z(t) dW(t), \quad 0\leq t\leq T, \\ p(T) = p_T, \end{cases} \tag{1} \] where \(H:= L^2(O)\), \( \mathbb{F}:= (\mathcal{F}_t)_{t\geq 0}\) is a normal filtration on a complete probability space \( (\Omega, \mathcal{F}, P)\), \( W(.)\) is a \( \mathbb{F}-\)adapted real valued Brownian motion, \(\Delta\) is the Laplace operator with the homogeneous Dirichlet boundary condition, \( f(., p, z)\in L^2_{\mathbb{F}}(0, T; H)\) satisfying \(P-a.s.\) for almost every \(t\in [0, T]\) a Lipschitz condition for the couple \((p, z)\in H\), \( D_{\mathbb{F}}([0, T]; L^2(\Omega; H))\) is the space of \(H-\)valued \(\mathbb{F}-\)adapted Cadlag processes in \(L^2(\Omega; H)\), \( p_T \in L^2(\Omega, \mathcal{F}_T, P; \dot{H}^1)\), with \(\dot{H}^k\) being the Sobolev space of order \(k\). The first result of the paper concerns the regularity of the transposition solution. The authors prove that \( p\in L^2_{\mathbb{F}}(0, T; \dot{H}^2)\), \( z\in L^2_{\mathbb{F}}(0, T; \dot{H}^1)\) and \(p\) admits a modification in \(D_{\mathbb{F}}([0, T]; L^2(\Omega; \dot{H}^1))\). The second result of the paper concerns the approximation of the transposition solution. To this aim, the authors introduce the following discretized scheme: \[ \begin{cases} dp_h(t) = -(\Delta_h p_h(t) + Q_hf(t, p_h(t), z_h(t)))dt+ z_h(t) dW(t), \quad 0\leq t\leq T, \\ p_h(T) = Q_hp_T, \end{cases} \tag{2} \] where \(h\) is the maximum diameter of a quasi triangulation \(\mathcal{K}_h\) of \( O\), \(\Delta_h\) is the discrete Laplace operator, \( Q_h\) is the orthogonal projection onto the finite element space \( \mathcal{V}_h\). They prove that it exists \(c>0\), s.t.: \[ \sup_{t\in [0,T]}|p(t)-p_h(t)|_H+ |p-p_h|_{L^2(0, T; \dot{H}^1)}+ |z-z_h|_{L^2(0, T; H)}\leq ch, \] where \((p_h, z_h)\) is the unique transposition solution of (2). An application of this result with numerical implementation for a stochastic linear quadratic control problem is also provided.

MSC:

49M25 Discrete approximations in optimal control
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35K58 Semilinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60J65 Brownian motion
49N10 Linear-quadratic optimal control problems
49J55 Existence of optimal solutions to problems involving randomness
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