A general comparison theorem for backward stochastic differential equations

Cohen, Samuel N.; Elliott, Robert J.; Pearce, Charles E. M.

doi:10.1017/s0001867800050485

Cited by 15 publications

(13 citation statements)

References 18 publications

(23 reference statements)

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“…Lately, [12,62] removed the quasileft continuity assumption from the filtration so that the quadratic variation of the driving martingale does not need to be absolutely continuous. On the other hand, based on a general martingale representation result due to Davis and Varaiya [28], Cohen and Elliott [22,23] discussed the case where the driving martingales are not a priori chosen but imposed by the filtration; see Hassani and Ouknine [38] for a similar approach on a BSDE in form of a generic map from a space of semimartingales to the spaces of martingales and those of finite-variation processes. Also, Mania and Tevzadze [57] and Jeanblanc et al [40] studied BSDEs for semimartingales and their applications to mean-variance hedging.…”

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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Section: Introductionmentioning

confidence: 99%

Lp solutions of backward stochastic differential equations with jumps

Yao

2017

Stochastic Processes and their Applications

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“…is a triplet of processes that satisfies the BSDE (ξ, f ) with ξ ∈ L 2 and (A 1), (A 2), then (Y, Z, U ) is a solution to (7), i.e., (Y, Z, U ) ∈ S 2 × L 2 (W ) × L 2 (Ñ ). In particular, there exists a constant C 1 > 0 such that…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting

Geiß

Steinicke

2018

Probab Uncertain Quant Risk

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We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116: 1358–1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346: 345–358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growth- and monotonicity-condition for the generator to be random and time-dependent. In the L 2 -case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88: 491–539, 2016). For the proof of the comparison result, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/ n .

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“…For such an equation, the proof of existence and uniqueness is a difficult task, and counterexamples can be obtained even in simple cases, see [14]. Only recently, some results in the unconstrained case have been obtained in this context, see [13], [12], [2]. In order to have an existence and uniqueness result for our BSDE, we have to impose the following additional assumption on p * .…”

Section: The Control Problemmentioning

confidence: 99%

“…The jump mechanism from the boundary plays a fundamental role as it leads, among other things, to the study of BSDEs driven by a non quasi-left-continuous random measure. Only recently, some results have been obtained on this subject, see [13], [12], [2]; in particular, in [2] well-posedness is proved for unconstrained BSDEs in a general non-diffusive framework, under a specific condition involving the Lipschitz constants of the BSDE generator and the size of the predictable jumps. In the present paper we extend the results in [2] to our class of constrained BSDEs.…”

Section: Introductionmentioning

confidence: 99%

Constrained BSDEs Driven by a Non-Quasi-Left-Continuous Random Measure and Optimal Control of PDMPs on Bounded Domains

Bandini¹

2019

SIAM J. Control Optim.

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We consider an optimal control problem for piecewise deterministic Markov processes (PDMPs) on a bounded state space. The control problem under study is very general: a pair of controls acts continuously on the deterministic flow and on the two transition measures (in the interior and from the boundary of the domain) describing the jump dynamics of the process. For this class of control problems, the value function can be characterized as the unique viscosity solution to the corresponding fully-nonlinear Hamilton-Jacobi-Bellman equation with a non-local type boundary condition.By means of the recent control randomization method, we are able to provide a probabilistic representation for the value function in terms of a constrained backward stochastic differential equation (BSDE), known as nonlinear Feynman-Kac formula. This result considerably extends the existing literature, where only the case with no jumps from the boundary is considered. The additional boundary jump mechanism is described in terms of a non quasi-left-continuous random measure and induces predictable jumps in the PDMP's dynamics. The existence and uniqueness results for BSDEs driven by such a random measure are non trivial, even in the unconstrained case, as emphasized in the recent work [2].Keywords: Backward stochastic differential equations, optimal control problems, piecewise deterministic Markov processes, non quasi-left-continuous random measure, randomization of controls.MSC 2010: 93E20, 60H10, 60J25.

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A general comparison theorem for backward stochastic differential equations

Cited by 15 publications

References 18 publications

Lp solutions of backward stochastic differential equations with jumps

Lp solutions of backward stochastic differential equations with jumps

Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting

Constrained BSDEs Driven by a Non-Quasi-Left-Continuous Random Measure and Optimal Control of PDMPs on Bounded Domains

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