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Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions. (English. French summary) Zbl 1236.60051

Interested in parabolic variational inequalities over the whole Euclidean space with viscosity solution \(V\), and taking into account that the numerical resolution of such inequalities requires to introduce a boundary and artificial boundary conditions, the authors approximate the following variational inequality \[ \left\{\begin{aligned}&\min\Big\{V(t,x)-Lt,x)\mid-\frac{\partial V}{\partial t}(t,x)\\&\quad-\mathcal{A}V(t,x)-f\big(t,x,V(t,x),(\nabla V\sigma)(t,x)\big) \Big\}=0,\quad (t,x)\in[0,T)\times\mathbb{R}^d,\\ &V(T,x)=g(x),\end{aligned}\right. \] where \(\mathcal{A}\) is the infinitesimal generator of a diffusion process, by one over a bounded domain with nonhomogeneous Neumann boundary condition, the viscosity solution of which is denoted by \(v\). In order to estimate \(|V(t,x)-v(t,x)|\), C. Berthelot, M. Bossy and D. Talay [in: J. Akahori (ed.) et al., Stochastic processes and applications to mathematical finance. Proceedings of the Ritsumeikan international symposium, Kusatsu, Shiga, Japan, 2003. River Edge, NJ: World Scientific. 1–25 (2004; Zbl 1191.91052)] used an approach based on the associated stochastic interpretation through a reflected forward stochastic differential equation and a reflected backward stochastic equation. In the present paper, the authors use this stochastic interpretation to provide a stochastic interpretation to the gradient \(\nabla_x v(t,x)\); to this end, they study, in particular, the gradient of the flow of the reflected forward equation mentioned above. Using their stochastic representation of the gradients, the authors estimate \(|\nabla_xV(t,x)-\nabla_xv(t,x0|\).

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K55 Nonlinear parabolic equations

Citations:

Zbl 1191.91052
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References:

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