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57
A Finite Horizon Optimal Multiple Switching Problem Preprint Université du
, 2007
"... We consider the problem of optimal multiple switching in finite horizon, when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and completely solved using probabilistic tools such as the ..."
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Cited by 28 (5 self)
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We consider the problem of optimal multiple switching in finite horizon, when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and completely solved using probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. Finally, when the state of the system is a Markov diffusion process, we show that the vector of value functions of the optimal problem is a viscosity solution to a system of variational inequalities with interconnected obstacles.
Error expansion for the discretization of Backward Stochastic Differential Equations
, 2006
"... We study the error induced by the time discretization of a decoupled forwardbackward stochastic differential equations (X,Y,Z). The forward component X is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme X N with N time steps. The backward component i ..."
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Cited by 17 (4 self)
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We study the error induced by the time discretization of a decoupled forwardbackward stochastic differential equations (X,Y,Z). The forward component X is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme X N with N time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (Y N −Y,Z N −Z) measured in the strong Lpsense (p ≥ 1) are of order N −1/2 (this generalizes the results by Zhang [20]). Secondly, an error expansion is derived: surprisingly, the first term is proportional to X N − X while residual terms are of order N −1.
Malliavin calculus for backward stochastic differential equations and application to numerical solutions. To appears in Annals of Applied Probability
, 2011
"... In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, eithe ..."
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Cited by 13 (2 self)
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In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the L pHölder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained L pHölder continuity results. The main tool is the Malliavin calculus.
Time discretization and Markovian iteration for coupled FBSDEs
 Ann. Appl. Probab
, 2006
"... In this paper we lay the foundation for a numerical algorithm to simulate highdimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular, we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for ..."
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Cited by 12 (2 self)
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In this paper we lay the foundation for a numerical algorithm to simulate highdimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular, we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the number of iterations. This feature seems to be indispensable for an efficient iterative scheme from a numerical point of view. We finally suggest a fully explicit numerical algorithm and present some numerical examples with up to 10dimensional state space. 1. Introduction. Motivated
2012, Perturbative expansion techniques for nonlinear FBSDEs with interacting particle method, working paper
"... Discussion Papers are a series of manuscripts in their draft form. They are not intended for circulation or distribution except as indicated by the author. For that reason Discussion Papers may not be reproduced or distributed without the written consent of the author. CIRJEF954 ..."
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Cited by 10 (4 self)
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Discussion Papers are a series of manuscripts in their draft form. They are not intended for circulation or distribution except as indicated by the author. For that reason Discussion Papers may not be reproduced or distributed without the written consent of the author. CIRJEF954
Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions
 Available on http://hal.archivesouvertes.fr/hal00642685, 2013. EJP
, 2015
"... Abstract. We design a numerical scheme for solving the Multi stepforward Dynamic Programming (MDP) equation arising from the timediscretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the ..."
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Cited by 9 (5 self)
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Abstract. We design a numerical scheme for solving the Multi stepforward Dynamic Programming (MDP) equation arising from the timediscretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical leastsquares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy.
A GENERALIZED θSCHEME FOR SOLVING BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
"... (Communicated by Qiang Du) Abstract. In this paper we propose a new type of θscheme with four parameters ({θi} 4 i=1) for solving the backward stochastic differential equation −dyt = f(t, yt, zt)dt − ztdWt. We rigorously prove some error estimates for the proposed scheme, and in particular, we show ..."
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Cited by 7 (3 self)
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(Communicated by Qiang Du) Abstract. In this paper we propose a new type of θscheme with four parameters ({θi} 4 i=1) for solving the backward stochastic differential equation −dyt = f(t, yt, zt)dt − ztdWt. We rigorously prove some error estimates for the proposed scheme, and in particular, we show that accuracy of the scheme can be high by choosing proper parameters. Various numerical examples are also presented to verify the theoretical results. 1. Introduction. Let (Ω
Results on numerics for FBSDE with drivers of quadratic growth
, 2009
"... We consider the problem of numerical approximation for forwardbackward stochastic differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial derivative using correlated ass ..."
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Cited by 7 (2 self)
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We consider the problem of numerical approximation for forwardbackward stochastic differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial derivative using correlated assets. For the convergence of numerical approximation schemes for such systems of stochastic equations, path regularity of the solution processes is instrumental. We present a method based on the truncation of the driver, and explicitly exhibit error estimates as functions of the truncation height. We discuss a reduction method to FBSDE with globally Lipschitz continuous drivers, by using the ColeHopf exponential transformation. We finally illustrate our numerical approximation methods by giving simulations for prices and optimal hedges of simple insurance derivatives.
A monotone scheme for high dimensional fully nonlinear PDEs, preprint
, 2013
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