Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

Bally, Vlad; Pardoux, Étienne; Stoica, Lucreţiu

doi:10.1007/s11118-003-6457-8

Cited by 31 publications

(66 citation statements)

References 9 publications

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“…Our main theorem says that under standard, in the L 2 -theory of PDEs, assumptions on the data of the Cauchy problem there exists a unique solution of the associated BSDE and it is represented in terms of the analytical solution in a similar way as in the case of usual BSDEs and viscosity solutions. This strenghtens the corresponding results from [2,3,9].…”

Section: Introductionsupporting

confidence: 86%

“…Let us note here that some results on solutions to BSDEs with driving process being a time-homogeneous diffusion corresponding to second order divergence form operator are given in [2,3,9]. In particular, in [3] it is proved that (1.4) holds under the assumption that a is regular, so that the driving process is in fact an Itô process.…”

Section: X U(t X) (σ ∇U)(t X))mentioning

confidence: 99%

“…But let us remark, that the most interesting results of [3] concern degenerate diffusions, which are not considered here. In [2,9] no regularity assumption on a is imposed. In [2], where as in [3] the Cauchy problem is considered and uniform ellipticity of a is not required, (1.4) is proved for quasi-every starting point.…”

Section: X U(t X) (σ ∇U)(t X))mentioning

confidence: 99%

“…In [2,9] no regularity assumption on a is imposed. In [2], where as in [3] the Cauchy problem is considered and uniform ellipticity of a is not required, (1.4) is proved for quasi-every starting point. In [9] Cauchy-Dirichlet problem in bounded domain is considered.…”

Section: X U(t X) (σ ∇U)(t X))mentioning

confidence: 99%

“…Therefore the classical definition of solutions of BSDEs is not applicable. Following [2,9] we propose slightly more general definition which make use of Fukushima's decomposition of additive functionals of time-inhomogeneous diffusions with divergence form generators (see [11,16]) and seems to be well adjusted to problems with nonsmooth data. Our main theorem says that under standard, in the L 2 -theory of PDEs, assumptions on the data of the Cauchy problem there exists a unique solution of the associated BSDE and it is represented in terms of the analytical solution in a similar way as in the case of usual BSDEs and viscosity solutions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Backward SDEs and Cauchy problem for semilinear equations in divergence form

Rozkosz¹

2003

Probab Theory Relat Fields

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We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces.

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Section: Introductionsupporting

confidence: 86%

Section: X U(t X) (σ ∇U)(t X))mentioning

confidence: 99%

Section: X U(t X) (σ ∇U)(t X))mentioning

confidence: 99%

Section: X U(t X) (σ ∇U)(t X))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Backward SDEs and Cauchy problem for semilinear equations in divergence form

Rozkosz¹

2003

Probab Theory Relat Fields

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Necessary and Sufficient Conditions of Optimalcontrol for Infinite Dimensional SDEs

Al-Hussein

2016

Springer Proceedings in Mathematics &Amp; Statistics

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A general maximum principle (necessary and sufficient conditions) for an optimal control problem governed by a stochastic differential equation driven by an infinite dimensional martingale is established. The solution of this equation takes its values in a separable Hilbert space and the control domain need not be convex. The result is obtained by using the adjoint backward stochastic differential equation.MSC 2010: 60H10, 60G44.

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Martingale Representations for Diffusion Processes and Backward Stochastic Differential Equations

Qian

Ying

2012

Lecture Notes in Mathematics

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In this paper we explain that the natural filtration of a continuous Hunt process is continuous, and show that martingales over such a filtration are continuous. We further establish a martingale representation theorem for a class of continuous Hunt processes under certain technical conditions. In particular we establish the martingale representation theorem for the martingale parts of (reflecting) symmetric diffusions in a bounded domain with a continuous boundary. Together with an approach put forward in [21], our martingale representation theorem is then applied to the study of initial and boundary problems for quasilinear parabolic equations by using solutions to backward stochastic differential equations over the filtered probability space determined by reflecting diffusions in a bounded domain with only continuous boundary. 0 AMS Classification. 60H10, 60H30, 60J45

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Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

Cited by 31 publications

References 9 publications

Backward SDEs and Cauchy problem for semilinear equations in divergence form

Backward SDEs and Cauchy problem for semilinear equations in divergence form

Necessary and Sufficient Conditions of Optimalcontrol for Infinite Dimensional SDEs

Martingale Representations for Diffusion Processes and Backward Stochastic Differential Equations

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