Reflected solutions of backward SDE's, and related obstacle problems for PDE's

Karoui, Nicole El; Kapoudjian, C.; Pardoux, Étienne; Péng, Shigē; Quenez, Marie‐Claire

doi:10.1214/aop/1024404416

Cited by 613 publications

(764 citation statements)

References 10 publications

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“…In this subsection, we prove certain estimates for H and K in terms of the process M . These estimates are similar to those obtained for reflected backward stochastic differential equations in [5]. Proposition 5.9 Let H, K, M be as in (5.4).…”

Section: Estimatessupporting

confidence: 72%

Martingale representation theorem for theG-expectation

Soner

Touzi

Zhang

2011

Stochastic Processes and their Applications

165

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This paper considers the nonlinear theory of G-martingales as introduced by Peng in [16,17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in [20] for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility.

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Section: Estimatessupporting

confidence: 72%

Martingale representation theorem for theG-expectation

Soner

Touzi

Zhang

2011

Stochastic Processes and their Applications

165

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“…To justify the expression of the solution given in Theorem 1, we rely both on the results of the previous section and on the characterizations already provided in Proposition 2.3 and Proposition 5.1 in [EPK97]. In this paper, the authors prove that the solution (Y, Z, K) of a RBSDE with driver f := f (s, y, z), lower obstacle S and terminal condition ξ satisfies…”

Section: Characterization Of the Solution Of The Rbsdementioning

confidence: 97%

“…Some particular RBSDEs are studied, for instance, in [EPK97] in connection with PDE obstacle problems or also in [EQ97], in connection with the problem of pricing an American contingent claim. We consider here, in a brownian setting, a class of reflected BSDEs (those can be viewed as a kind of generalized BSDEs).…”

Section: Motivationmentioning

confidence: 99%

“…Referring to the same arguments in [EPK97], the optimal exercise time τ A is: τ A = inf{t, W A t = U (t, C t )}, and the optimal control u = u A is given in feedback form by: ∀ s, u A s = (U ′ ) −1 (w A s ). Hence, if we rewrite the backward 3 This condition is checked in particular under Novikov's condition E " exp`1 2…”

Section: One Financial Applicationmentioning

confidence: 99%

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Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

Morlais

2012

Stochastics

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In that paper, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver g is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of g-Snell enveloppe. Then, in a second step, we relate this representation to a specific class of dynamic monetary concave functionals already introduced in a discrete time setting. This connection implies that the solution, characterized by means of non linear expectations, has again the time consistency property.

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“…In [12], the authors introduced the penalization method to prove the existence of the solution. For p ∈ N, the penalization equation with respect to the lower barrier L is 5) thanks to the comparison theorem for BSDE, we have…”

Section: Numerical Penalization Schemesmentioning

confidence: 99%