We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann-Dirichlet boundary condition:where (A t ) t≥0 is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman-Kaç representation formula for the viscosity solution of the PVI problem.
In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as:where η is a stochastic process given by η(t) = η(0) + t 0 σ(s)δB H (s), t ∈ [0, T ], and B H is a fractional Brownian motion with Hurst parameter greater than 1/2. The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng's paper, BDSEs driven by fBm, SIAM J Control Optim. (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equationwhere ∂ϕ is a multivalued operator of subdifferential type associated with the convex function ϕ. (2010): 60H10, 60G22, 47J20, 60H05 (Lucian Maticiuc), nietianyang@163.com (Tianyang Nie) Mathematics Subject ClassificationHere, in our manuscript, we work without such a condition. Let us mention that in [12], it is assumed thatIn our paper, we adopt the formT ], since we will use the proof of Theorem 3.8 from [12], especially the quasi-conditional expectation formulâ. Based on the above-described framework, we prove the existence and the uniqueness for BSDE (1). This approach includes, in particular, first a discussion of the equationAfter, the existence for BSDE (1) is proved by using a fixed point theorem over an appropriate Banach space. Based on our results on BSDE driven by a fBm and on Pardoux and Rȃşcanu [19] on BSVI governed by a standard Brownian motion, we consider the following fractional BSVIwhere ∂ϕ is the subdifferential of a convex lower semicontinuous (l.s.c.) function ϕ : R → (−∞, +∞]. The existence of the solution will be proved. Now, we give the outline of our paper: In Section 2 we recall some definitions and results about fractional stochastic integrals and the related Itô formula. We present the assumptions and some auxiliary results including the Itô formula w.r.t. the divergence-type integral in Section 3. Section 4 is devoted to prove the existence and the uniqueness result for BSDE driven by a fBm. In Section 5, we study the existence for fractional BSVI governed by a fBm. Finally, in the Appendix, we prove a more general Itô formula based on Theorem 8 [3] and an auxiliary lemma. Preliminaries: Fractional stochastic calculusIn this section, we shall recall some important definitions and results concerning the Malliavin calculus, the stochastic integral with respect to a fBm, and Itô's formula. For a deeper discussion, we refer the reader to [2, 3, 7, 9, 10] and [16].Throughout our paper, we assume that the Hurst parameter H always satisfies H > 1/2. Define φ(x) = H(2H − 1)|x| 2H−2 , x ∈ R. Let us denote by |H| the Banach space of measurable functions f : [0, T ] → R such that f 2 |H| := T 0 T 0 φ(u − v)|f (u)||f (v)|dudv < +∞.
The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval: \[\cases{\displaystyle -\mathrm{d}Y_t+\partial_y\Psi (t,Y_t)\,\mathrm{d}Q_t\ni\Phi (t,Y_t,Z_t)\,\mathrm{d}Q_t-Z_t\,\mathrm{d}W_t,\qquad 0\leq t<\tau,\cr \displaystyle{Y_{\tau}=\eta,}}\] where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_y\Psi$ is the subdifferential of the convex lower semicontinuous function $y\longmapsto\Psi (t,y)$. As applications, we obtain from our main results applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.Comment: Published at http://dx.doi.org/10.3150/14-BEJ601 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by solutions of penalized partial differential equations.
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