Mean-field backward stochastic differential equations: A limit approach

By Rainer Buckdahn, Boualem Djehiche, Juan Li, Shige Peng. 2007

341 citations recorded by the source corpus.

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 Theory. 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 (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/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root 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 mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73bd79fa3327407b6ce32ca583b656525f6ea900

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. René A. Carmona
2. Daniel Lacker
3. Jiongmin Yong
4. François Delarue
5. Juan Li
6. Jingrui Sun
7. Jin Ma
8. Shige Peng

In Mathematical Physics

1. Juan Li
2. Jiongmin Yong
3. Rainer Buckdahn
4. Shige Peng
5. Jingrui Sun
6. Jin Ma
7. Xunjing Li
8. René A. Carmona

Citations in the corpus

• A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations in Infinite Horizon (2012, neutral): “For the relevant works of recent years, see, for examples, Veretennikov [34], Huang– Malhamé–Caines [22], Buckdahn-Djehiche-Li-Peng [12], Buckdahn-Li-Peng [13], Borkar–Kumar [9], Crisan– Xiong [16], Kotelenez–Kurtz [25], and so on.”
• Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs (2017, methodology): “On the other hand, from Proposition 4.1, I3,2(x, y, Pξ ;x ′, y′, Pξ′) ≤ CpMp ( E [(∫ T t (|Πt,x,Pξr −Π t,x′,Pξ′ r |2 +W2(PΠt,ξr , PΠt,ξ′r ) 2 + |Πt,y,Pξr −Π t,y′,Pξ′ r |2)dr )p]) 1 2 + CpE [(∫ T t |∂xΠt,y,Pξr |2I {|∂xΠ t,y,Pξ r |2≥M} dr )p] ≤ CpMp ( |x− x′|p +W2(Pξ , Pξ′)p + |y − y′|p ) + ρM,p(t, y,”
• A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations (2011, background): “In recent years, related topics and problems have attracted more and more researchers’ attentions, see, for examples, Veretennikov [32], Huang–Malhamé– Caines [17], Mahmudov–McKibben [23], Buckdahn-Djehiche-Li-Peng [8], Buckdahn-Li-Peng [9], Borkar– Kumar [6], Crisan–Xiong [12], Kotelenez–Kurtz [20”
• Linear Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations (2016, background): “Note that H1 allows the coefficients A(·) andC(·) to be unbounded, which is a little different from the standard case [6,7].”
• Linear-Quadratic Mean Field Games (2014, background): “For more details about the existence and convergence rate of the related mean field backward stochastic differential equations, one can refer to [48] and [49].”
• Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications (2011, background): “We also refer to stochastic differential maximization and games with recursive or other utilities (see Peng (1997a), Pham (2004), Buckdahn and Li (2008)), Mean-field BSDE (see Buckdahn et al (2009)).”
• A mean-field stochastic control problem with partial observations (2017, background): “, φ·∧t = φt), then the problem becomes a stochastic control problem with McKean-Vlasov dynamics and/or a Mean-field game (see, for example, [7, 8, 9] in its “forward” form, and [2, 3, 4] in its “backward” form).”
• Mean-field stochastic differential equations and associated PDEs (2014, background): “In [5], deepening the investigation of mean-field SDEs and associated mean-field BSDEs, Buckdahn, Li and Peng generalized their previous results on mean-field BSDEs, and in a “Markovian” framework in which the initial data (t, x) were frozen in the law variable of the coefficients; they investigated”
• Probabilistic Analysis of Mean-Field Games (2012, neutral): “See for example [3, 4].”
• Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Open-Loop Solvabilities (2015, neutral): “For recent development of MF-SDEs, readers may refer to Huang–Malhamé–Caines [17], Veretennikov [27], Mahmudov–McKibben [20], Buckdahn–Djehiche–Li–Peng [7], Buckdahn–Li–Peng [8], Borkar–Kumar [5], Crisan–Xiong [11], Kotelenez–Kurtz [19] and the references cited therein.”
• Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics (2013, background): “The main lines of the proof are given in [5], Proposition 3.”