Results 1  10
of
19
Meanfield backward stochastic differential equations and related patial differential equations. Submitted. Available at http://arxiv.org/abs/0711.2167
, 2007
"... Mathematical meanfield approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special meanfield problem in a purely stochastic approa ..."
Abstract

Cited by 66 (8 self)
 Add to MetaCart
Mathematical meanfield approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special meanfield problem in a purely stochastic approach: for the solution (Y,Z) of a meanfield backward stochastic differential equation driven by a forward stochastic differential of McKean–Vlasov type with solution X we study a special approximation by the solution (X N,Y N,Z N) of some decoupled forward–backward equation which coefficients are governed by N independent copies of (X N,Y N,Z N). We show that the convergence speed of this approximation is of order 1 / √ N. Moreover, our special choice of the approximation allows to characterize the limit behavior of √ N(X N − X,Y N − Y,Z N − Z). We prove that this triplet converges in law to the solution of some forward–backward stochastic differential equation of meanfield type, which is not only governed by a Brownian motion but also by an independent Gaussian field. 1. Introduction. Our
A PHASE TRANSITION BEHAVIOR FOR BROWNIAN MOTIONS INTERACTING THROUGH THEIR RANKS
"... Abstract. Consider a timevarying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their s ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Abstract. Consider a timevarying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge ’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a nontrivial PoissonDirichlet distribution. The proof employs, among other things, techniques from Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the PoissonDirichlet law. 1.
Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles: Application to the Burgers Equation
 Annals of Applied Prob
, 1996
"... In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is O(1= p N + p \Deltat) for the L 1 (IR \Theta \Omega\Gamma norm of the error. To obtain that result, we link the PDE and the algorithm to a system ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is O(1= p N + p \Deltat) for the L 1 (IR \Theta \Omega\Gamma norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the difficulty of the analysis comes from the discontinuity of the interaction kernel which is equal to the Heaviside function. In [4], we show how the algorithm and the result extend to the case of non monotone initial conditions for the Burgers equation. We also treat the case of nonlinear PDE's related to particle systems with Lipschitz interaction kernels. Our next objective is to adapt our methodology to the (more difficult) case of the 2D inviscid NavierStokes equation.
Weak approximations. A Malliavin calculus approach
 Math. Comp
, 2001
"... Abstract. We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Abstract. We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions. 1.
Weak rate of convergence for an Euler scheme of nonlinear SDE's
"... We define an Euler type weak approximation for solutions of nonlinear diffusion processes. We find the rate of convergence of this scheme in weak form. As in the diffusion case the weak rate of convergence is better than the strong one. The proof uses the integration by parts formula of Malliavin ca ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
We define an Euler type weak approximation for solutions of nonlinear diffusion processes. We find the rate of convergence of this scheme in weak form. As in the diffusion case the weak rate of convergence is better than the strong one. The proof uses the integration by parts formula of Malliavin calculus.
OPTIMAL RATE OF CONVERGENCE OF A STOCHASTIC PARTICLE METHOD TO SOLUTIONS OF 1D VISCOUS SCALAR CONSERVATION LAWS
"... Abstract. This article presents the analysis of the rate of convergence of a stochastic particle method for 1D viscous scalar conservation laws. The convergence rate result is O(∆t +1 / √ N), where N is the number of numerical particles and ∆t is the time step of the first order Euler scheme applie ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. This article presents the analysis of the rate of convergence of a stochastic particle method for 1D viscous scalar conservation laws. The convergence rate result is O(∆t +1 / √ N), where N is the number of numerical particles and ∆t is the time step of the first order Euler scheme applied to the dynamic of the interacting particles. 1.
Markov transitions and the propagation of chaos
"... The propagation of chaos is a central concept of kinetic theory that serves to relate the equations of Boltzmann and Vlasov to the dynamics of manyparticle systems. Propagation of chaos means that molecular chaos, i.e., the stochastic independence of two random particles in a manyparticle system, ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
The propagation of chaos is a central concept of kinetic theory that serves to relate the equations of Boltzmann and Vlasov to the dynamics of manyparticle systems. Propagation of chaos means that molecular chaos, i.e., the stochastic independence of two random particles in a manyparticle system, persists in time, as the number of particles tends to infinity. We establish a necessary and sufficient condition for a family of general nparticle Markov processes to propagate chaos. This condition is expressed in terms of the Markov transition functions associated to the nparticle processes, and it amounts to saying that chaos of random initial states propagates if it propagates for pure initial states. Our proof of this result relies on the weak convergence approach to the study of chaos due to Sznitman and Tanaka. We assume that the space in which the particles live is homeomorphic to a complete and separable metric space so that we may invoke Prohorov’s theorem in our proof. We also show that, if the particles can be in only finitely many states,
NUMERICAL ALGORITHMS FOR SEMILINEAR PARABOLIC EQUATIONS WITH SMALL PARAMETER BASED ON APPROXIMATION OF STOCHASTIC EQUATIONS
"... Abstract. The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burger ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPPequation with a small parameter. 1.
Numerical Methods For Nonlinear Parabolic Equations With Small Parameter Based On Probability Approach
"... The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equati ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPPequation with a small parameter.
RATE OF CONVERGENCE OF A PARTICLE METHOD TO THE SOLUTION OF THE MC KEAN VLASOV'S EQUATION
"... Acknowledgements: This work was completed while the rst author's visit at Purdue University and he thanks the Mathematics Department for the hospitality. During this period the research of the rst author was partially supported by the CNRNATO grant n. 215.29. The research of the second author was c ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Acknowledgements: This work was completed while the rst author's visit at Purdue University and he thanks the Mathematics Department for the hospitality. During this period the research of the rst author was partially supported by the CNRNATO grant n. 215.29. The research of the second author was completed while visiting the Department of Mathematics of the University ofTokyo. He thanks them for their hospitality and his research was also supported by a DGICYT grant. Typeset by AMSTEX 1 ABSTRACT. This paper studies the rate of convergence of an appropriate discretization scheme of the solution of the Mc Kean Vlasov equation introduce d by Bossy and Talay. More speci cally, we consider approximations of the distribution and of the density of the solution of the stochastic di erential equation associated to the Mc Kean Vlasov equation. The scheme adopted here is a mixed one: Euler/weakly interacting particle system. If n is the number of weakly interacting particles and h is the uniform st ep in the time discretization, we prove that the rate of convergence of the distribution functions of the approximating sequence in the L1 ( 2 R) norm and in the sup norm is of the order of p1 + h, while for the densities is of the o rder h + n 1 p. This result is obtained by carefully nh employing techniques of Malliavin Calculus.