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Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations. Stat Probab Lett 2022. [DOI: 10.1016/j.spl.2022.109608] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Wang H, Yong J, Zhou C. Backward stochastic differential equations and backward stochastic Volterra integral equations with anticipating generators. PROBABILITY, UNCERTAINTY AND QUANTITATIVE RISK 2022. [DOI: 10.3934/puqr.2022018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Tian R, Yu Z. Mean-field type FBSDEs in a domination-monotonicity framework and LQ multi-level Stackelberg games. PROBABILITY, UNCERTAINTY AND QUANTITATIVE RISK 2022. [DOI: 10.3934/puqr.2022014] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
<p style='text-indent:20px;'>Motivated by various mean-field type linear-quadratic (MF-LQ, for short) multi-level Stackelberg games, we propose a kind of multi-level self-similar randomized domination-monotonicity structures. When the coefficients of a class of mean-field type forward-backward stochastic differential equations (MF-FBSDEs, for short) satisfy this kind of structures, we prove the existence, the uniqueness, an estimate and the continuous dependence on the coefficients of solutions. Further, the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games, respectively.</p>
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Affiliation(s)
- Ran Tian
- Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China
| | - Zhiyong Yu
- School of Mathematics, Shandong University, Jinan 250100, China
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Beck C, Hutzenthaler M, Jentzen A. On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. STOCH DYNAM 2021. [DOI: 10.1142/s0219493721500489] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The classical Feynman–Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this paper, we extend the classical Feynman–Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic fixed point equations (SFPEs), which arise when the classical Feynman–Kac identity is formally applied to semilinear Kolmorogov PDEs, as viscosity solutions of the corresponding PDEs. This justifies, in particular, employing full-history recursive multilevel Picard (MLP) approximation algorithms, which have recently been shown to overcome the curse of dimensionality in the numerical approximation of solutions of SFPEs, in the numerical approximation of semilinear Kolmogorov PDEs.
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Affiliation(s)
- Christian Beck
- Applied Mathematics Münster: Institute for Analysis and Numerics, University of Münster, Einsteinstraße 62, 48149 Münster, Germany
- Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland
| | - Martin Hutzenthaler
- Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
| | - Arnulf Jentzen
- Applied Mathematics Münster: Institute for Analysis and Numerics, University of Münster, Einsteinstraße 62, 48149 Münster, Germany
- Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland
- School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, 2001 Longxiang Road, 518172 Shenzhen, China
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Chen Y, Djehiche B, Hamadène S. Mean-field backward–forward stochastic differential equations and nonzero sum stochastic differential games. STOCH DYNAM 2020. [DOI: 10.1142/s0219493721500362] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We study a general class of fully coupled backward–forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coefficients. These strategies are valid for any time horizon of the game.
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Affiliation(s)
- Yinggu Chen
- Department of Mathematics, Shandong University, Jinan 250100, Shandong Province, P. R. China
| | - Boualem Djehiche
- Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden
| | - Said Hamadène
- Le Mans University, LMM, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France
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Al-Hussein A, Gherbal B. Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2020. [DOI: 10.1515/rose-2020-2044] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Abstract
The paper addresses a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. These equations are allowed to live in Euclidean spaces of different dimensions, and the system is Markovian in the sense that the terminal value of the backward equation depends on the terminal value of the solution of the forward one. Under some monotonicity conditions we establish the existence and uniqueness of strong solutions of such equations by using a continuation method.
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Affiliation(s)
- AbdulRahman Al-Hussein
- Department of Mathematics , College of Science , Qassim University , P. O. Box 6644 , Buraydah 51452 , Saudi Arabia
| | - Boulakhras Gherbal
- Laboratory of Mathematical Analysis, Probability and Optimization , University of Mohamed Khider , P. O. Box 145 , Biskra 07000 , Algeria
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Shamarova E, Sá Pereira R. Forward–backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2019.11.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Ankirchner S, Fromm A, Kruse T, Popier A. Optimal position targeting via decoupling fields. ANN APPL PROBAB 2020. [DOI: 10.1214/19-aap1511] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria. Stoch Process Their Appl 2019. [DOI: 10.1016/j.spa.2018.03.002] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Meng Q, Shen Y, Shi P. On the existence of optimal controls for backward stochastic partial differential equations. Stat Probab Lett 2018. [DOI: 10.1016/j.spl.2018.01.013] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Huang J, Wang S, Wu Z. Backward-forward linear-quadratic mean-field games with major and minor agents. PROBABILITY, UNCERTAINTY AND QUANTITATIVE RISK 2016. [DOI: 10.1186/s41546-016-0009-9] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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Yin H. Forward–backward stochastic partial differential equations with non-monotonic coefficients. STOCH DYNAM 2016. [DOI: 10.1142/s0219493716500258] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs) with non-monotonic coefficients. These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of such FBSPDEs by using the method of continuation. Contrary to the common belief, we show that the usual monotonicity assumption can be removed by a change of the diffusion term.
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Affiliation(s)
- Hong Yin
- Department of Mathematics, State University of New York, Brockport, NY 14420, USA
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Abstract
In this paper we study the problem of pricing contingent claims for a large investor (i.e. the coefficients of the price equation can also depend on the wealth process of the hedger) in an incomplete market where the portfolios are constrained. We formulate this problem so as to find the minimal solution of forward-backward stochastic differential equations (FBSDEs) with constraints. We use the penalization method to construct a sequence of FBSDEs without constraints, and we show that the solutions of these equations converge to the minimal solution we are interested in.
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Abstract
In this paper we study the problem of pricing contingent claims for a large investor (i.e. the coefficients of the price equation can also depend on the wealth process of the hedger) in an incomplete market where the portfolios are constrained. We formulate this problem so as to find the minimal solution of forward-backward stochastic differential equations (FBSDEs) with constraints. We use the penalization method to construct a sequence of FBSDEs without constraints, and we show that the solutions of these equations converge to the minimal solution we are interested in.
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Bensoussan A, Yam S, Zhang Z. Well-posedness of mean-field type forward–backward stochastic differential equations. Stoch Process Their Appl 2015. [DOI: 10.1016/j.spa.2015.04.006] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Ma J, Wu Z, Zhang D, Zhang J. On well-posedness of forward–backward SDEs—A unified approach. ANN APPL PROBAB 2015. [DOI: 10.1214/14-aap1046] [Citation(s) in RCA: 67] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Delbaen F, Qiu J, Tang S. Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space. Stoch Process Their Appl 2015. [DOI: 10.1016/j.spa.2015.02.014] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Casserini M, Liang G. Fully coupled forward–backward stochastic dynamics and functional differential systems. STOCH DYNAM 2015. [DOI: 10.1142/s0219493715500069] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
This paper introduces and solves a general class of fully coupled forward–backward stochastic dynamics by investigating the associated system of functional differential equations. As a consequence, we are able to solve many different types of forward–backward stochastic differential equations (FBSDEs) that do not fit in the classical setting. In our approach, the equations are running in the same time direction rather than in a forward and backward way, and the conflicting nature of the structure of FBSDEs is therefore avoided.
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Affiliation(s)
| | - Gechun Liang
- Department of Mathematics, King's College London, UK
- Oxford-Man Institute, University of Oxford, UK
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Ji S, Yang S. Solutions for functional fully coupled forward–backward stochastic differential equations. Stat Probab Lett 2015. [DOI: 10.1016/j.spl.2015.01.009] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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22
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Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations. Stoch Process Their Appl 2014. [DOI: 10.1016/j.spa.2014.07.013] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Cruzeiro AB, Gomes ADO, Zhang L. Asymptotic properties of coupled forward–backward stochastic differential equations. STOCH DYNAM 2014. [DOI: 10.1142/s021949371450004x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we consider coupled forward–backward stochastic differential equations (FBSDEs in short) with parameter ε > 0, of the following type [Formula: see text] We study the asymptotic behavior of its solutions and establish a large deviation principle for the corresponding processes.
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Affiliation(s)
- Ana Bela Cruzeiro
- GFMUL and Dep. de Matemática IST(TUL), Av. Rovisco Pais, 1049-001 Lisboa, Portugal
| | - André de Oliveira Gomes
- GFMUL – Grupo de Física Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
- Institut für Mathematik Humboldt-Universität zu Berlin, Germany
| | - Liangquan Zhang
- School of Mathematics, Shandong University, P. R. China
- Laboratoire de Mathématiques, Université de Bretagne Occidentale, 29285 Brest Cédex, France
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Horst U, Hu Y, Imkeller P, Réveillac A, Zhang J. Forward–backward systems for expected utility maximization. Stoch Process Their Appl 2014. [DOI: 10.1016/j.spa.2014.01.004] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Rozkosz A. Stochastic Representation of Weak Solutions of Viscous Conservation Laws: A BSDE Approach. J THEOR PROBAB 2011. [DOI: 10.1007/s10959-011-0395-y] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Liang G, Lyons T, Qian Z. Backward stochastic dynamics on a filtered probability space. ANN PROBAB 2011. [DOI: 10.1214/10-aop588] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Chen X, Lin L. Nonparametric Estimation for FBSDEs Models with Applications in Finance. COMMUN STAT-THEOR M 2010. [DOI: 10.1080/03610920903046816] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Bahlali S. Necessary and Sufficient Conditions of Optimality for Optimal Control Problems of Forward and Backward Systems. THEORY OF PROBABILITY AND ITS APPLICATIONS 2010. [DOI: 10.1137/s0040585x97984474] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Bahlali K, Mezerdi B, N'zi M, Ouknine Y. Weak solutions and a Yamada–Watanabe theorem for FBSDEs. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2007. [DOI: 10.1515/rose.2007.016] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Guo D, Ji S, Zhao H. On the solvability of infinite horizon forward–backward stochastic differential equations with absorption coefficients. Stat Probab Lett 2006. [DOI: 10.1016/j.spl.2006.05.001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Antonelli F, Hamadène S. Existence of the solutions of backward–forward SDE's with continuous monotone coefficients. Stat Probab Lett 2006. [DOI: 10.1016/j.spl.2006.03.018] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Bahlali S, Labed B. Necessary and sufficient conditions of optimality for optimal control problem with initial and terminal costs. RANDOM OPERATORS AND STOCHASTIC EQUATIONS 2006. [DOI: 10.1515/156939706778239846] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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42
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Yong J. Linear forward-backward stochastic differential equations with random coefficients. Probab Theory Relat Fields 2005. [DOI: 10.1007/s00440-005-0452-5] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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El Karoui N, Peng S, Quenez MC. A dynamic maximum principle for the optimization of recursive utilities under constraints. ANN APPL PROBAB 2001. [DOI: 10.1214/aoap/1015345345] [Citation(s) in RCA: 74] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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