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Mean-Field Backward Stochastic Differential Equations

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Mean-Field Backward Stochastic Differential Equations The Annals of Probability 2009, Vol. 37, No. 4, 1524–1565 DOI: 10.1214/08-AOP442 c Institute of Mathematical Statistics, 2009 MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS: A LIMIT APPROACH By Rainer Buckdahn, Boualem Djehiche, Juan Li1,2 and Shige Peng2 Universit´ede Bretagne Occidentale, Royal Institute of Technology, Shandong University at Weihai and Shandong University Mathematical mean-field 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 The- ory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field 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 (XN ,Y N , ZN ) of some decoupled forward–backward equation which coefficients are governed by N independent copies of (XN ,Y N , ZN ). We show that the convergence speed of this approximation is of order 1/√N. More- over, our special choice of the approximation allows to characterize the limit behavior of √N(XN X,Y N Y, ZN Z). We prove that − − − this triplet converges in law to the solution of some forward–backward stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field. 1. Introduction. Our present work on a stochastic limit approach to a mean-field problem is motivated on the one hand by classical mean-field ap- proaches in Statistical Mechanics and Physics (e.g., the derivation of Boltz- mann or Vlasov equations in the kinetic gas theory), by similar approaches in Quantum Mechanics and Quantum Chemistry (in particular the density arXiv:0711.2162v3 [math.PR] 28 Aug 2009 Received February 2008; revised October 2008. 1Supported in part by the NSF of P. R. China (No. 10701050; 10671112), Shandong Province (No. Q2007A04). 2National Basic Research Program of China (973 Program) (No. 2007CB814904; 2007CB814906). AMS 2000 subject classifications. Primary 60H10; secondary 60B10. Key words and phrases. Backward stochastic differential equation, mean-field ap- proach, McKean–Vlasov equation, mean-field BSDE, tightness, weak convergence. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2009, Vol. 37, No. 4, 1524–1565. This reprint differs from the original in pagination and typographic detail. 1 2 BUCKDAHN, DJEHICHE, LI AND PENG functional models or Hartree and Hartree–Fock type models) but also by a recent series of papers by Lasry and Lions (see [7] and the references therein). On the other hand it stems its motivation from partial differen- tial equations of McKean–Vlasov type, which have found a great interest in recent years and have been studied with the help of stochastic methods by several authors. Lasry and Lions introduced in recent papers a general mathematical mod- eling approach for high-dimensional systems of evolution equations corre- sponding to a large number of “agents” (or “particles”). They extended the field of such mean-field approaches also to problems in Economics and Fi- nance as well as to the game theory: here they studied N-players stochastic differential games [7] and the related problem of the existence of Nash equi- librium points, and by letting N tend to infinity they derived in a periodic setting the mean-field limit equation. On the other hand, in the last years models of large stochastic particle systems with mean-field interaction have been studied by many authors; they have described them by characterizing their asymptotic behavior when the size of the system becomes very large. The reader is referred, for ex- ample, to the works by Bossy [1], Bossy and Talay [2], Chan [4], Kotelenez [6], M´el´eard [8], Overbeck [10], Pra and Hollander [14], Sznitman [15, 16], Talay and Vaillant [17] and all the references therein. They have shown that probabilistic methods allow to study the solution of the linear McKean– Vlasov PDE (see, e.g., M´el´eard [8]). With the objective to give a stochastic interpretation to such semilinear PDEs we introduced in [3] the notion of a mean-field backward stochastic differential equation: backward stochastic differential equations (BSDEs) have been introduced by Pardoux and Peng in their pioneering papers [11] and [12], in which they proved in particular that, in a Markovian framework, the solution of a BSDE describes the vis- cosity solution of the associated semilinear PDE. The reader interested in more details is also referred to the paper by El Karoui, Peng and Quenez [5] and the references therein. However, in order to generalize this stochastic interpretation to semilinear McKean–Vlasov PDEs we have had to study a new type of BSDE, which takes into account the specific, nonlocal struc- ture of these PDEs; we have called this new type of backward equations mean-field BSDE (MFBSDE). The objective of the present paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a MF- BSDE driven by a forward SDE of McKean–Vlasov type with solution X we study a special approximation by the solution (XN ,Y N ,ZN ) of some decoupled forward–backward equation which coefficients are governed by N independent copies of (XN ,Y N ,ZN ). 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(XN − MFBSDES: A LIMIT APPROACH 3 X,Y N Y, ZN Z). We prove that this triplet converges in law to the so- lution of− some forward–backward− SDE of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field. To be more precise, for a given finite time horizon T > 0, a d-dimensional Brownian motion W = (Wt)t [0,T ] and a driving d-dimensional adapted ∈ stochastic process X = (Xt)t [0,T ] we consider the BSDE of mean-field type ∈ dYt = E[f(t, λ, Λt)] λ=Λt dt + Zt dWt, t [0, T ], (1.1) − | ∈ YT = E[Φ(x, XT )]x=XT with Λt = (Xt,Yt,Zt), where (Y, Z) = (Yt,Zt)t [0,T ] denotes the solution of the above equation. This equation also can be∈ written in the form dYt = f(t, Λt, λ)PΛt (dλ) dt + Zt dWt, t [0, T ], − Rd R Rd ∈ Z × × YT = Φ(XT ,x)]PXT (dx). Rd Z Such type of BSDEs has been studied recently by Buckdahn, Li and Peng [3] in a more general framework. In [3] for the “Markovian-like” case in which the process X is the solution of a forward equation of McKean–Vlasov type (of course, X is not a Markov process) it has been shown that such a BSDE gives a stochastic interpretation to the associated nonlocal partial differential equations. In the present work, given an arbitrary sequence of N N N 2 adapted processes X = (Xt )t [0,T ] such that E[supt [0,T ] Xt Xt ] 0, we study the approximation of the∈ above MFBSDE by∈ a backward| − equation| → of the form N N 1 N N,j N dYt = f(t, Λt , Λt ) dt + Zt dWt, t [0, T ], −N j=1 ∈ (1.2) X 1 N Y N = Φ(XN , XN,j). T N T T jX=1 This approximating BSDE is driven by the i.i.d. sequence ΛN,j = (XN,j,Y N,j, ZN,j), 1 j N, of triplets of stochastic processes, which are supposed to obey the≤ same≤ law as ΛN = (XN ,Y N ,ZN ) and to be independent of the Brownian motion W . In Section 3 we will introduce a natural framework in which such BSDEs can be solved easily, and we will show that the solution (Y N ,ZN ) converges to that of (1.1) (Theorem 3.1). For the study of the speed of the convergence we assume that the process X is the solution of the stochastic differential equation (SDE) of McKean–Vlasov type dXt = E[σ(t,x,Xt)] x=Xt dWt + E[b(t,x,Xt)] x=Xt dt, (1.3) | | X = x Rm, t [0, T ], 0 0 ∈ ∈ 4 BUCKDAHN, DJEHICHE, LI AND PENG and we approximate X in the same spirit as (Y, Z) by the solution XN of the forward equation N N N 1 N N,j 1 N N,j dXt = σ(t, Xt , Xt ) dWt + b(t, Xt , Xt ) dt, N j=1 N j=1 (1.4) X X XN = x , t [0, T ]. 0 0 ∈ We point out the conceptional difference between our approximation and the standard approximation for the McKean–Vlasov SDE which considers N N,1 as X the dynamics X′ of the first element of a system of N particles which dynamics is described by the system of SDEs N N N,i 1 N,i N,j i 1 N,i N,j dXt′ = σ(t, Xt′ , Xt′ ) dWt + b(t, Xt′ , Xt′ ) dt, N j=1 N j=1 (1.5) X X XN,i = x , t [0, T ], 1 i N, 0 0 ∈ ≤ ≤ governed by the independent Brownian motions W 1,...,W N (see, e.g., Bossy [1], Bossy and Talay [2], Chan [4], M´el´eard [8], Overbeck [10], Pra and Hollander [14], Sznitman [15, 16], Talay and Vaillant [17]). By proving that the speed of the convergence of ΛN = (XN ,Y N ,ZN ) to Λ = (X, Y, Z) is of order 1/√N (see Proposition 3.2 and Theorem 3.2) we know that the sequence √N(XN X,Y N Y, ZN Z), N 1, is bounded, − − − ≥ T N 2 N 2 N 2 sup E sup ( Xt Xt + Yt Yt )+ Zt Zt dt < + , N 1 t [0,T ] | − | | − | 0 | − | ∞ ≥ ∈ Z so that the question of the limit behavior of the sequence √N(XN X,Y N Y, ZN Z), N 1, arises.
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