The Experts below are selected from a list of 24273 Experts worldwide ranked by ideXlab platform
Magdalena Kobylanski - One of the best experts on this subject based on the ideXlab platform.
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backward stochastic differential equations and partial differential equations with quadratic growth
Annals of Probability, 2000Co-Authors: Magdalena KobylanskiAbstract:We provide existence, comparison and stability results for one-dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) F(t, Y, Z) is continuous and has a quadratic growth in Z and the Terminal Condition is bounded. We also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations.
Alexandre Popier - One of the best experts on this subject based on the ideXlab platform.
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asymptotic approach for backward stochastic differential equation with singular Terminal Condition
Stochastic Processes and their Applications, 2021Co-Authors: Paulwin Graewe, Alexandre PopierAbstract:Abstract In this paper, we provide a one-to-one correspondence between the solution Y of a BSDE with singular Terminal Condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behaviour of Y close to the final time and enlarges the uniqueness result to a wider class of generators.
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Asymptotic approach for backward stochastic differential equation with singular Terminal Condition *
2020Co-Authors: Paulwin Graewe, Alexandre PopierAbstract:In this paper, we provide a one-to-one correspondence between the solution Y of a BSDE with singular Terminal Condition and the solution H of a BSDE with singular generator. This result provides the precise asymptotic behavior of Y close to the final time and enlarges the uniqueness result to a wider class of generators.
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Minimal supersolutions for BSDEs with singular Terminal Condition and application to optimal position targeting
Stochastic Processes and their Applications, 2016Co-Authors: Thomas Kruse, Alexandre PopierAbstract:We study the existence of a minimal supersolution for backward stochastic differential equations when the Terminal data can take the value +∞ with positive probability. We deal with equations on a general filtered probability space and with generators satisfying a general monotonicity assumption. With this minimal supersolution we then solve an optimal stochastic control problem related to portfolio liquidation problems. We generalize the existing results in three directions: firstly there is no assumption on the underlying filtration (except completeness and quasi-left continuity), secondly we relax the Terminal liquidation constraint and finally the time horizon can be random.
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Backward stochastic differential equations with singular Terminal Condition
Stochastic Processes and their Applications, 2006Co-Authors: Alexandre PopierAbstract:In this paper, we are concerned with backward stochastic differential equations (BSDE for short) of the following type:
San J Martin - One of the best experts on this subject based on the ideXlab platform.
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backward stochastic differential equations with continuous coefficient
Statistics & Probability Letters, 1997Co-Authors: Jeanpierre Lepeltier, San J MartinAbstract:We prove the existence of a solution for "one dimensional" backward stochastic differential equations where the coefficient is continuous, it has a linear growth, and the Terminal Condition is squared integrable. We also obtain the existence of a minimal solution.
E Weinan - One of the best experts on this subject based on the ideXlab platform.
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deep learning based numerical methods for high dimensional parabolic partial differential equations and backward stochastic differential equations
Communications in Mathematics and Statistics, 2017Co-Authors: E Weinan, Jiequn Han, Arnulf JentzenAbstract:We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed Terminal Condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation, the Hamilton–Jacobi–Bellman equation, and a nonlinear pricing model for financial derivatives.
Khaled Bahlali - One of the best experts on this subject based on the ideXlab platform.
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quadratic bsde with mathbb l 2 Terminal data krylov s estimate ito krylov s formula and existence results
Annals of Probability, 2017Co-Authors: Khaled Bahlali, M'hamed Eddahbi, Youssef OuknineAbstract:We establish a Krylov-type estimate and an Ito–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary Terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square integrable Terminal Condition and sometimes a merely measurable generator. It turns out that neither the existence of exponential moments of the Terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions. We also establish a comparison theorem for solutions of a particular class of QBSDEs with measurable generator. As a byproduct, we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations (QPDEs) with a square integrable Terminal datum.
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quadratic bsdes with l 2 Terminal data existence results krylov s estimate and ito krylov s formula
arXiv: Probability, 2014Co-Authors: Khaled Bahlali, M'hamed Eddahbi, Youssef OuknineAbstract:In a first step, we establish the existence (and sometimes the uniqueness) of solutions for a large class of quadratic backward stochastic differential equations (QBSDEs) with continuous generator and a merely square integrable Terminal Condition. Our approach is different from those existing in the literature. Although we are focused on QBSDEs, our existence result also covers the BSDEs with linear growth, keepingsquare integrable in both cases. As byproduct, the existence of viscosity solutions is established for a class of quadratic partial differential equations (QPDEs) with a square integrable Terminal datum. In a second step, we consider QBSDEs with measurable generator for which we establish a Krylov's type a priori estimate for the solutions. We then deduce an Ito-Krylov's change of variable formula. This allows us to establish various existence and uniqueness results for classes of QBSDEs with square integrable Terminal Condition and sometimes a merely measurable generator. Our results show, in particular, that neither the existence of expo- nential moments of the Terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions for quadratic BSDEs. Some comparison theorems are also established for solutions of a class of QBSDEs.
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Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs
Comptes Rendus Mathematique, 2010Co-Authors: Khaled Bahlali, El Hassan Essaky, M. HassaniAbstract:Abstract We establish the existence and uniqueness as well as the stability of p-integrable solutions to multidimensional backward stochastic differential equations (BSDEs) with super-linear growth coefficient and a p-integrable Terminal Condition ( p > 1 ) . The generator could neither be locally monotone in the variable y nor locally Lipschitz in the variable z. As application, we establish the existence and uniqueness of weak (Sobolev) solutions to the associated systems of semilinear parabolic PDEs. The uniform ellipticity of the diffusion matrix is not required. Our result covers, for instance, certain systems of PDEs with logarithmic nonlinearities which arise in physics.