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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
Optimal Consumption and Portfolio Selection with Stochastic Differential Utility
, 1999
"... We develop the utility gradient (or martingale) approach for computing portfolio and consumption plans that maximize stochastic differential utility (SDU), a continuoustime version of recursive utility due to D. Duffie and L. Epstein (1992, Econometrica 60, 353 394). We characterize the firstorder ..."
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Cited by 88 (4 self)
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We develop the utility gradient (or martingale) approach for computing portfolio and consumption plans that maximize stochastic differential utility (SDU), a continuoustime version of recursive utility due to D. Duffie and L. Epstein (1992, Econometrica 60, 353 394). We characterize the firstorder conditions of optimality as a system of forward backward SDEs, which, in the Markovian case, reduces to a system of PDEs and forward only SDEs that is amenable to numerical computation. Another contribution is a proof of existence, uniqueness, and basic properties for a parametric class of homothetic SDUs that can be thought of as a continuoustime version of the CES Kreps Porteus utilities studied by L. Epstein and A. Zin (1989, Econometrica 57, 937 969). For this class, we derive closedform solutions in terms of a single backward SDE (without imposing a Markovian structure). We conclude with several tractable concrete examples involving the type of ``affine'' state price dynamics that are familiar from the term structure literature.
Solution of forward–backward stochastic di erential equations.
 Probab. Theory Related Fields 103,
, 1995
"... Abstract Solvability of forwardbackward stochastic di erential equations with nonsmooth coe cients is considered using the FourStep Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion ..."
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Cited by 64 (2 self)
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Abstract Solvability of forwardbackward stochastic di erential equations with nonsmooth coe cients is considered using the FourStep Scheme and some approximation arguments. For the onedimensional case, the existence of an adapted solution is established for the equation which allows the di usion in the forward equation to be degenerate. As a byproduct, we obtain the existence of a viscosity solution to a onedimensional nonsmooth degenerate quasilinear parabolic partial di erential equation.
A regressionbased Monte Carlo method to solve backward stochastic differential equations
, 2005
"... We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experime ..."
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Cited by 57 (6 self)
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We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.
ForwardBackward Stochastic Differential Equations and quasilinear parabolic PDEs
 PROBAB. THEORY RELATED FIELDS
, 1999
"... This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We us ..."
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Cited by 52 (2 self)
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This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate.
Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions
, 2000
"... In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of Backward SDE's and forwardbackward SDE's. The controlled process (X #Y ) takes values in IR d \Theta IR and agiven initial data for X (0). Then, the ..."
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Cited by 43 (11 self)
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In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of Backward SDE's and forwardbackward SDE's. The controlled process (X #Y ) takes values in IR d \Theta IR and agiven initial data for X (0). Then, the control problem is to find the minimal initial data for Y so that it reaches a stochastic target at a specified terminal time T . The main application is from financial mathematics in which the process X is related to stockprice,Y is the wealth process, and is the portfolio. Weintroduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. The boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense. This provides a unique characterization of the value function which is the minimal initial data for ...
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.
Hedging of Defaultable Claims
, 2004
"... Contents 1 Replication of Defaultable Claims 7 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 DefaultFree Market . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Random Time . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Defaultable Claims . . . . . . . ..."
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Cited by 38 (12 self)
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Contents 1 Replication of Defaultable Claims 7 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 DefaultFree Market . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Random Time . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Defaultable Claims . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Default Time . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 RiskNeutral Valuation . . . . . . . . . . . . . . . . . . . 13 1.2.3 Defaultable Term Structure . . . . . . . . . . . . . . . . . 15 1.3 Properties of Trading Strategies . . . . . . . . . . . . . . . . . . . 16 1.3.1 DefaultFree Primary Assets . . . . . . . . . . . . . . . . 17 1.3.2 Defaultable and DefaultFree Primary Assets . . . . . . . 23 1.4 Replication of Defaultable Claims . . . . . . . . . . . . . . . . . . 30 1.4.1 Replication of a Promised Payo# . . . . . . . . . . . . . . 30 1.4.2 Replication of a Recovery Payo# . . . . . . . . . . . . . . 34 1.4.3 Replication
Stochastic linear quadratic optimal control problems
 Appl. Math. Optim
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