A converse comparison theorem for backward stochastic differential equations with jumps

Scheemaekere, Xavier De

doi:10.1016/j.spl.2010.10.016

Cited by 4 publications

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“…Coquet, Hu, Mémin and Peng [8], Briand, Coquet, Mémin and Peng [2], and Jiang [17] derived converse comparison theorems for BSDEs, with no jumps. De Schemaekere [11], derived a converse comparison theorem for a model with jumps.…”

Section: Introductionmentioning

confidence: 99%

Comparison and converse comparison theorems for backward stochastic differential equations with Markov chain noise

Yang¹,

Ramarimbahoaka²,

Elliott³

2016

Electron. Commun. Probab.

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Comparison and converse comparison theorems are important parts of the research on backward stochastic differential equations. In this paper, we obtain comparison results for one dimensional backward stochastic differential equations with Markov chain noise, extending and generalizing previous work under natural and simplified hypotheses, and establish a converse comparison theorem for the same type of equation after giving the definition and properties of a type of nonlinear expectation: f -expectation. 1 derived comparison theorems for BSDEs with Lipschitz continuous coefficients. Liu and Ren [20] proved a comparison theorem for BSDEs with linear growth and continuous coefficients. Situ [26] obtained a comparison theorem for BSDEs with jumps. Zhang [29] deduced a comparison theorem for BSDEs with two reflecting barriers. Hu and Peng [16] established a comparison theorem for multidimensional BSDEs. Comparison theorems for BSDEs have received much attention because of their importance in applications. For example, the penalization method for reflected BSDEs is based on a comparison theorem (see[10], [12], [18] and [24]). Moreover, research on properties of g-expectations (see, Peng [23]) and the proof of a monotonic limit theorem for BSDEs (see, Peng [22]) both depend on comparison theorems. BSDEs with jumps were also introduced by many. Among others, we mention [1] and [25]. Crepey and Matoussi [9] considered BSDEs with jumps in a more general framework where a Brownian motion is incorporated in the model and a general random measure is used to model the jumps, which in [1] is a Poisson random measure. It is natural to ask whether the converse of the above results holds or not. That is, if we can compare the solutions of two BSDEs with the same terminal conditions, can we compare the driver? Coquet, Hu, Mémin and Peng [8], Briand, Coquet, Mémin and Peng [2], and Jiang [17] derived converse comparison theorems for BSDEs, with no jumps. De Schemaekere [11], derived a converse comparison theorem for a model with jumps. In 2012, van der Hoek and Elliott [27] introduced a market model where uncertainties are modeled by a finite state Markov chain, instead of Brownian motion or related jump diffusions, which are often used when pricing financial derivatives. The Markov chain has a semimartingale representation involving a vector martingale M = {M t ∈ R N , t ≥ 0}. BSDEs in this framework were introduced by Cohen and Elliott [5] asCohen and Elliott [6] and [7] gave some comparison results for multidimensional BSDEs in the Markov Chain model under conditions involving not only the two drivers but also the two solutions. If we consider two onedimensional BSDEs driven by the Markov chain, we extend the comparison result to a situation involving conditions only on the two drivers. Consequently our comparison results are easier to use for the one-dimensional case. Moreover, our result in the Markov chain framework needs less conditions on the drivers compared to those in Crepey and Matoussi [9] which are suitable for more ...

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