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Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order
 in Stochastic Analysis and Related Topics VI: The Geilo Workshop
, 1996
"... The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have ..."
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Cited by 260 (15 self)
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The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have appeared long time ago, both as the equations for the adjoint process in
Secondorder elliptic integrodifferential equations: viscosity solutions’ theory revisited
 Ann. Inst. H. Poincaré Anal. Non Linéaire
"... Abstract. The aim of this work is to revisit viscosity solutions ’ theory for secondorder elliptic integrodifferential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new JensenIshii’s Lemma for integrod ..."
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Cited by 94 (8 self)
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Abstract. The aim of this work is to revisit viscosity solutions ’ theory for secondorder elliptic integrodifferential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new JensenIshii’s Lemma for integrodifferential equations, which is stated for solutions with no restriction on their growth at infinity. The proof of this result, which is of course a key ingredient to prove comparison principles, relies on a new definition of viscosity solution for integrodifferential equation (equivalent to the two classical ones) which combines the approach with testfunctions and subsuperjets.
A finite difference scheme for option pricing in jump diffusion and exponential Lévy models
, 2003
"... We present a finite difference method for solving parabolic partial integrodierential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a timeinhomogeneous jumpdiffusio ..."
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Cited by 66 (2 self)
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We present a finite difference method for solving parabolic partial integrodierential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a timeinhomogeneous jumpdiffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicitimplicit finite dierence scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Our convergence analysis requires neither the smoothness of the solution nor the nondegeneracy of coefficients and applies to European and barrier options in jumpdiffusion and pure jump models used in the literature. Numerical tests are performed with smooth and nonsmooth initial conditions.
Stochastic Differential Games and Viscosity Solutions of HamiltonJacobiBellmanIsaacs Equations
, 2007
"... In this paper we study zerosum twoplayer stochastic differential games with the help of theory of Backward Stochastic Differential Equations (BSDEs). At the one hand we generalize the results of the pioneer work of Fleming and Souganidis [8] by considering cost functionals defined by controlled BS ..."
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Cited by 48 (12 self)
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In this paper we study zerosum twoplayer stochastic differential games with the help of theory of Backward Stochastic Differential Equations (BSDEs). At the one hand we generalize the results of the pioneer work of Fleming and Souganidis [8] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game (which implies that the cost functionals become random variables), on the other hand the application of BSDE methods, in particular that of the notion of stochastic “backward semigroups ” introduced by Peng [14] allows to prove a dynamic programming principle for the upper and the lower value functions of the game in a straightforward way, without passing by additional approximations. The upper and the lower value functions are proved to be the unique viscosity solutions of the upper and the lower HamiltonJacobiBellmanIsaacs equations, respectively. For this Peng’s BSDE method (Peng [14]) is translated from the framework of stochastic control theory into that of stochastic differential games.
Numerical Method for Backward Stochastic Differential Equations
 Ann. Appl. Probab
, 2002
"... Abstract. We propose a method for numerical approximation of Backward Stochastic Differential Equations. Our method allows the final condition of the equation to be quite general and simple to implement. It relies on an approximation of Brownian Motion by simple random walk. 1. ..."
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Cited by 42 (3 self)
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Abstract. We propose a method for numerical approximation of Backward Stochastic Differential Equations. Our method allows the final condition of the equation to be quite general and simple to implement. It relies on an approximation of Brownian Motion by simple random walk. 1.
A “maximum principle for semicontinuous functions” applicable to integropartial differential equations
, 2006
"... We formulate and prove a nonlocal “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integropartial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain co ..."
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Cited by 42 (9 self)
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We formulate and prove a nonlocal “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integropartial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain comparison/uniqueness results for integropartial differential equations, but proofs have so far been lacking.
Multidimensional BSDE with Oblique Reflection and Optimal Switching
, 2007
"... In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The e ..."
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Cited by 38 (1 self)
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In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.
Stochastic linear quadratic optimal control problems
 Appl. Math. Optim
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Solvability of Backward Stochastic Differential Equations with quadratic growth
, 2007
"... We prove the existence of the unique solution of a general Backward Stochastic Differential Equation with quadratic growth driven by martingales. Some kind of comparison theorem is also proved. ..."
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Cited by 34 (2 self)
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We prove the existence of the unique solution of a general Backward Stochastic Differential Equation with quadratic growth driven by martingales. Some kind of comparison theorem is also proved.
Continuous dependence estimates for viscosity solutions of integroPDEs
 J. DIFFERENTIAL EQUATIONS
, 2004
"... We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integroPDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lév ..."
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Cited by 34 (12 self)
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We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integroPDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integroPDEs arising in stochastic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integroPDEs arising when attempting to price European/American options in an incomplete stock market driven by a geometric Lévy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an