Showing papers by "Benjamin Jourdain published in 2007"
Posted Content•
18 Jul 2007
TL;DR: In this paper, the authors studied general nonlinear stochastic differential equa- tions, where the usual Brownian motion is replaced by a Levy process, and they proved that the time-marginals of the solutions are abso- lutely continuous with respect to the Lebesgue measure.
Abstract: In this paper we study general nonlinear stochastic differential equa- tions, where the usual Brownian motion is replaced by a Levy process. Moreover, we do not suppose that the coefficient multiplying the increments of this process is linear in the time-marginals of the solution as is the case in the classical McKean- Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Levy process with a smooth but unbounded Levy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are abso- lutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian.
109 citations
Posted Content•
TL;DR: In this article, a stratified sampling algorithm is proposed in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum.
Abstract: In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm.
52 citations
Posted Content•
TL;DR: In this article, the authors studied general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\'evy process, and they proved that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure.
Abstract: In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\'evy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump L\'evy process with a smooth but unbounded L\'evy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian.
43 citations
TL;DR: On a simple one-dimensional example, this paper proves the convergence of the Diffusion Monte Carlo method for a fixed number of reconfigurations when the number of walkers tends to +1 while the timestep tends to 0.
Abstract: The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a Stochastic Differential Equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to +1 while the timestep tends to 0. We confirm our theoretical rates of convergence by numerical experiments. Various resampling algorithms are investigated, both theoretically and numerically.
29 citations
TL;DR: In this article, an exact simulation based technique for pricing continuous arithmetic average Asian options in the Black and Scholes framework is presented, which is no longer prone to the discretization bias resulting from the approximation of continuous time processes through discrete sampling.
Abstract: Taking advantage of the recent litterature on exact simulation algorithms (Beskos, Papaspiliopoulos and Roberts) and unbiased estimation of the expectation of certain fonctional integrals (Wagner, Beskos et al. and Fearnhead et al.), we apply an exact simulation based technique for pricing continuous arithmetic average Asian options in the Black and Scholes framework. Unlike existing Monte Carlo methods, we are no longer prone to the discretization bias resulting from the approximation of continuous time processes through discrete sampling. Numerical results of simulation studies are presented and variance reduction problems are considered.
27 citations
TL;DR: In this paper, the long time behavior of the nonlinear process associated to the one-dimensional viscous scalar conservation law is studied. But the authors only consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this non-linear process by the empirical cdf.
Abstract: In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincare inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.
22 citations
TL;DR: In this paper, the equivalent probabilistic formulation of Dupire's PDE is the put-call duality equality in local volatility models including exponential Levy jumps, and the authors give a direct proof for this result based on stochastic flow arguments.
Abstract: The equivalent probabilistic formulation of Dupire’s PDE is the put-call duality equality. In local volatility models including exponential Levy jumps, we give a direct probabilistic proof for this result based on stochastic flow arguments. This approach also enables us to check the equivalent probabilistic formulation of various generalizations of Dupire’s PDE recently obtained by Pironneau [C. R. Acad. Sci. Paris Ser. I 344(2) 127–133 (2007)] by the adjoint equation technique in the case of complex options.
17 citations
Posted Content•
TL;DR: In this article, the authors focus on implementations of diffusion Monte Carlo which consist in exploring the configuration space with a fixed number of random walkers evolving according to a Stochastic Differential Equation discretized in time.
Abstract: The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a {\bf fixed} number of random walkers evolving according to a Stochastic Differential Equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to $+\infty$ while the timestep tends to 0. We confirm our theoretical rates of convergence by numerical experiments. Various resampling algorithms are investigated, both theoretically and numerically
3 citations