Mean-field backward stochastic differential equations and related partial differential equations

By Rainer Buckdahn, Juan Li, Shige Peng. 2007

377 citations recorded by the source corpus.

In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Ann. Probab. (2007) (in press). Available online: http://www.imstat.org/aop/future_papers.htm] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of "particles" (or "agents"). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we study a mean-field BSDE in a Markovian framework, associated with a McKean-Vlasov forward equation. By combining classical BSDE methods, in particular that of "backward semigroups" introduced by Peng [S. Peng, J. Yan, S. Peng, S. Fang, L. Wu (Eds.), in: BSDE and Stochastic Optimizations; Topics in Stochastic Analysis, Science Press, Beijing (1997) (Chapter 2) (in Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions.

Subfields: Finance, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/35d6fbcc9b191c388123c17bdd5ef17aa0eb2b99

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. P. J. Graber
2. Jiongmin Yong
3. Jingrui Sun
4. Juan Li
5. Jin Ma
6. Huyen Pham
7. Xunjing Li
8. Shige Peng

In Mathematical Physics

1. Juan Li
2. P. J. Graber
3. X. Mao
4. Jiongmin Yong
5. Boualem Djehiche
6. Rainer Buckdahn
7. Shige Peng
8. Jingrui Sun

In Modeling and Simulation

1. Juan Li
2. P. J. Graber
3. X. Mao
4. Boualem Djehiche
5. Rainer Buckdahn
6. Jiongmin Yong
7. Jin Ma
8. Shige Peng

Citations in the corpus

• 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”
• Probabilistic Analysis of Mean-Field Games (2012, neutral): “See for example [3, 4].”
• 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.”
• 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].”
• 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 Maximum Principle for SDEs of Mean-Field Type (2011, result): “Under the assumptions (A.1)–(A.2), this is a linear mean-field backward SDE with bounded coefficients and it follows from [ 5 ], Theorem 3.1, that it has a unique adapted solution such that”
• A mean-field stochastic control problem with partial observations (2017, result): “The main idea is similar to earlier works of the first two authors [4, 16], with some significant modifications.”
• 7 – Backward Stochastic Differential Equations (2011, positive): “[14] to the BSDE driven by a Brownian motion and a compensated Poisson process.”
• 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].”
• 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.”
• BSDEs with mean reflection (2016, neutral): “Observe first the the solution (Y , Z) is well and uniquely defined, according to the results of [5] up to slight modifications discussed for example in [7].”
• Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability (2016, neutral): “See, for examples, McKean [16], Buckdahn–Djehiche–Li–Peng [5], Buckdahn–Li–Peng [6], Andersson–Djenhiche [2], Buckdahn–Djehiche–Li [4], Meyer-Brandis–Øksendal–Zhou [17], Yong [25], Elliott–Li–Ni [10], Cui–Li–Li [9], Huang–Li–Wang [11], Huang–Li–Yong [12].”
• Mean-field stochastic differential equations and associated PDEs (2014, result): “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”
• Mean-field backward stochastic differential equations: A limit approach (2007, methodology): “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 [1”
• Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations --- Time-Consistent Solutions (2013, background): “A main motivation of that is a kind of conditional/dynamic mean-variance problem (see [1, 4]) See also [5, 16, 7] for relevant results.”
• Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, methodology): “= 0 Now by [11, 12] we have a solution to the McKean-Vlasov type BSDE (2.”
• A General Stochastic Maximum Principle for SDEs of Mean-field Type (2011, methodology): “in Buckdahn, Li and Peng [ 6 ], under the Condition 2.1 ,( 2.7) admits a unique F-adapted solution (p, q) such that”