Backward stochastic differential equations and integral-partial differential equations

Barles, Guy; Buckdahn, Rainer; Pardoux, Étienne

doi:10.1080/17442509708834099

Cited by 453 publications

(750 citation statements)

References 6 publications

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“…In a Markovian setting, g-expectations correspond to viscosity solutions to semi-linear parabolic PDEs (or PIDEs in the case of jumps); see, for example, El Karoui, Peng and Quenez [23] in a Brownian setting and Barles, Buckdahn and Pardoux [3] in the case of jumps. In particular, our results on BSDEs also give rise to results for viscosity solutions to the corresponding PIDEs in the case of infinite jump activity.…”

Section: Bsdesmentioning

confidence: 99%

Robust Portfolio Choice and Indifference Valuation

Laeven

Stadje²

2014

Mathematics of OR

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We solve, theoretically and numerically, the problems of optimal portfolio choice and indifference valuation in a general continuous-time setting. The setting features (i) ambiguity and ambiguity averse preferences, (ii) discontinuities in the asset price processes, with a general and possibly infinite activity jump part next to a continuous diffusion part, and (iii) general and possibly non-convex trading constraints. We characterize our solutions as solutions to Backward Stochastic Differential Equations (BSDEs). We prove existence and uniqueness of the solution to the general class of BSDEs with jumps having a drift (or driver) that grows at most quadratically, encompassing the solutions to our portfolio choice and valuation problems as special cases. We provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium, and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the jump part. We further compute our solutions in a few examples by numerically solving the corresponding BSDEs using regression techniques.

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Section: Bsdesmentioning

confidence: 99%

Robust Portfolio Choice and Indifference Valuation

Laeven

Stadje²

2014

Mathematics of OR

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“…These authors obtained the existence of a unique solution to a BSDEJ with a Lipschitz generator and square-integrable terminal data. Then Barles, Buckdahn and Pardoux [17,7] showed that the wellposedness of BSDEJs gives rise to a viscosity solution of a semilinear parabolic partial integro-differential equation (PIDE) and thus provides a probabilistic interpretation of such a PIDE. Later, Pardoux [63] relaxed the Lipschitz condition of the generator on variable y by assuming a monotonicity condition on variable y instead.…”

Section: Introductionmentioning

confidence: 99%

Lp solutions of backward stochastic differential equations with jumps

Yao

2017

Stochastic Processes and their Applications

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Given p ∈ (1, 2), we study L p solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in (y, z)−variables. We show that such a BSDEJ with p−integrable terminal data admits a unique L p solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.

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“…We shall take any other conditions necessary for uniqueness (for example, square-integrability of the solution) as implicit. (2) degenerates into the tautology Y t = Q, which clearly has a unique solution up to indistinguishability. Property 3 is simply due to a rearrangement of (2).…”

Section: Bsdes With Arbitrary Martingalesmentioning

confidence: 99%

“…(2) degenerates into the tautology Y t = Q, which clearly has a unique solution up to indistinguishability. Property 3 is simply due to a rearrangement of (2). Property 4 is due to the assumed integrability condition, evaluated for Y t = Q.…”

Section: Bsdes With Arbitrary Martingalesmentioning

confidence: 99%

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A general comparison theorem for backward stochastic differential equations

2010

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A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

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Backward stochastic differential equations and integral-partial differential equations

Cited by 453 publications

References 6 publications

Robust Portfolio Choice and Indifference Valuation

Robust Portfolio Choice and Indifference Valuation

Lp solutions of backward stochastic differential equations with jumps

A general comparison theorem for backward stochastic differential equations

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