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Journal articles on the topic 'Fully coupled forward-backward parabolic equations'

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Published: 5 June 2025

Last updated: 2 August 2025

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1

Ya., Belopolskaya. "Probabilistic Approaches to Nonlinear Parabolic Equations in Jet-Bundles." Global and Stochastic Analysis 1, no. 1 (2014): 1–40. https://doi.org/10.5281/zenodo.7673376.

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We develop two probabilistic approaches that allows to construct a solution of the Cauchy problem for a class of fully nonlinear second order PDEs. Within the first one we reduce an original PDE to a quasilinear PDE in the second order jet-bundle and construct a probabilistic counterpart in terms of forward stochastic equations for the resulting Cauchy problem. The second is based on a reduction to a semilinear  PDE that allows to reduce the original problem to a fully coupled forward-backward BSDE.

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2

Xiao, Lishun, Shengjun Fan, and Dejian Tian. "A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems." ESAIM: Probability and Statistics 24 (2020): 207–26. http://dx.doi.org/10.1051/ps/2019023.

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In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required

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3

Casserini, Matteo, and Gechun Liang. "Fully coupled forward–backward stochastic dynamics and functional differential systems." Stochastics and Dynamics 15, no. 02 (2015): 1550006. http://dx.doi.org/10.1142/s0219493715500069.

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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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4

Wei, Qingmeng. "The Optimal Control Problem with State Constraints for Fully Coupled Forward-Backward Stochastic Systems with Jumps." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/216053.

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We focus on the fully coupled forward-backward stochastic differential equations with jumps and investigate the associated stochastic optimal control problem (with the nonconvex control and the convex state constraint) along with stochastic maximum principle. To derive the necessary condition (i.e., stochastic maximum principle) for the optimal control, first we transform the fully coupled forward-backward stochastic control system into a fully coupled backward one; then, by using the terminal perturbation method, we obtain the stochastic maximum principle. Finally, we study a linear quadratic

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5

Wang, Mingcan, and Xiangjun Wang. "Hybrid Neural Networks for Solving Fully Coupled, High-Dimensional Forward–Backward Stochastic Differential Equations." Mathematics 12, no. 7 (2024): 1081. http://dx.doi.org/10.3390/math12071081.

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The theory of forward–backward stochastic differential equations occupies an important position in stochastic analysis and practical applications. However, the numerical solution of forward–backward stochastic differential equations, especially for high-dimensional cases, has stagnated. The development of deep learning provides ideas for its high-dimensional solution. In this paper, our focus lies on the fully coupled forward–backward stochastic differential equation. We design a neural network structure tailored to the characteristics of the equation and develop a hybrid BiGRU model for solvi

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6

Min, Hui, Ying Peng, and Yongli Qin. "Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/839467.

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We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. Th

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7

Ji, Shaolin, and Shuzhen Yang. "Solutions for functional fully coupled forward–backward stochastic differential equations." Statistics & Probability Letters 99 (April 2015): 70–76. http://dx.doi.org/10.1016/j.spl.2015.01.009.

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8

Lin, X.-Y., Q.-X. Sun, and X.-R. Wang. "The robustness of fully coupled forward-backward stochastic differential equations." Journal of Physics: Conference Series 96 (February 1, 2008): 012207. http://dx.doi.org/10.1088/1742-6596/96/1/012207.

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9

Li, Juan. "FULLY COUPLED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERAL MARTINGALE." Acta Mathematica Scientia 26, no. 3 (2006): 443–50. http://dx.doi.org/10.1016/s0252-9602(06)60068-4.

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10

Al-Hussein, AbdulRahman, and Boulakhras Gherbal. "Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps." Random Operators and Stochastic Equations 28, no. 4 (2020): 253–68. http://dx.doi.org/10.1515/rose-2020-2044.

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AbstractThe 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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11

Zhang, Shuaiqi, and Zhen-Qing Chen. "Fully coupled forward-backward stochastic differential equations driven by sub-diffusions." Journal of Differential Equations 405 (October 2024): 337–58. http://dx.doi.org/10.1016/j.jde.2024.06.003.

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12

Liu, Ruyi, and Zhen Wu. "Well-Posedness of Fully Coupled Linear Forward-Backward Stochastic Differential Equations." Journal of Systems Science and Complexity 32, no. 3 (2018): 789–802. http://dx.doi.org/10.1007/s11424-018-7424-1.

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13

Kong, Tao, Weidong Zhao, and Tao Zhou. "Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs." Communications in Computational Physics 18, no. 5 (2015): 1482–503. http://dx.doi.org/10.4208/cicp.240515.280815a.

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AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp.

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14

Yin, Hong. "Forward–backward stochastic partial differential equations with non-monotonic coefficients." Stochastics and Dynamics 16, no. 06 (2016): 1650025. http://dx.doi.org/10.1142/s0219493716500258.

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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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15

Chen, Li, Peipei Zhou, and Hua Xiao. "Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations." Mathematics 11, no. 13 (2023): 2898. http://dx.doi.org/10.3390/math11132898.

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In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear–quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward–backward stochastic differential delayed Equation (AFBSDDE), which is composed of an

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16

Wu, Zhen. "Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration." Journal of the Australian Mathematical Society 74, no. 2 (2003): 249–66. http://dx.doi.org/10.1017/s1446788700003281.

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AbstractWe first give the existence and uniqueness result and a comparison theorem for backward stochastic differential equations with Brownian motion and Poisson process as the noise source in stopping time (unbounded) duration. Then we obtain the existence and uniqueness result for fully coupled forward-backward stochastic differential equation with Brownian motion and Poisson process in stopping time (unbounded) duration. We also proved a comparison theorem for this kind of equation.

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17

Peng, Shige, and Zhen Wu. "Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control." SIAM Journal on Control and Optimization 37, no. 3 (1999): 825–43. http://dx.doi.org/10.1137/s0363012996313549.

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18

Wang, Xiangrong, and Hong Huang. "Maximum Principle for Forward-Backward Stochastic Control System Driven by Lévy Process." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/702802.

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We study a stochastic optimal control problem where the controlled system is described by a forward-backward stochastic differential equation driven by Lévy process. In order to get our main result of this paper, the maximum principle, we prove the continuity result depending on parameters about fully coupled forward-backward stochastic differential equations driven by Lévy process. Under some additional convexity conditions, the maximum principle is also proved to be sufficient. Finally, the result is applied to the linear quadratic problem.

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19

Hu, Mingshang, Shaolin Ji, and Xiaole Xue. "Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic controlled systems." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 81. http://dx.doi.org/10.1051/cocv/2020051.

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Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 56 (2018) 4309–4335] and the dynamic programming principle (DPP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 57 (2019) 3911–3938] for a fully coupled forward–backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton–Jacobi–Bellman (HJB) equation combine an algebra equation respectively. With the help of a new decoupling techni

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20

Shi, Jingtao. "Necessary Conditions for Optimal Control of Forward-Backward Stochastic Systems with Random Jumps." International Journal of Stochastic Analysis 2012 (April 2, 2012): 1–50. http://dx.doi.org/10.1155/2012/258674.

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This paper deals with the general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin's type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.

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21

Liu, Ying. "A Fully Discrete Explicit Multistep Scheme for Solving Coupled Forward Backward Stochastic Differential Equations." Advances in Applied Mathematics and Mechanics 12, no. 3 (2020): 643–63. http://dx.doi.org/10.4208/nmtma.oa-2019-0199.

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22

Liu, Ying. "A Fully Discrete Explicit Multistep Scheme for Solving Coupled Forward Backward Stochastic Differential Equations." Advances in Applied Mathematics and Mechanics 12, no. 3 (2020): 643–63. http://dx.doi.org/10.4208/aamm.oa-2019-0079.

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23

Huang, Jianhui, and Jingtao Shi. "Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations." ESAIM: Control, Optimisation and Calculus of Variations 18, no. 4 (2012): 1073–96. http://dx.doi.org/10.1051/cocv/2011204.

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24

Zhang, Shuaiqi, and Zhen-Qing Chen. "Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Differential Equations Driven by Subdiffusion." SIAM Journal on Control and Optimization 62, no. 5 (2024): 2433–55. http://dx.doi.org/10.1137/23m1620168.

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25

Xu, Xiaoming. "Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control." Journal of Systems Science and Complexity 33, no. 6 (2020): 1886–902. http://dx.doi.org/10.1007/s11424-020-9027-x.

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26

Naito, Makoto, Taiga Saito, Akihiko Takahashi, and Kohta Takehara. "Asymptotic expansions as control variates for deep solvers to fully-coupled forward-backward stochastic differential equations." PLOS One 20, no. 5 (2025): e0321778. https://doi.org/10.1371/journal.pone.0321778.

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Coupled forward-backward stochastic differential equations (FBSDEs) are closely related to financially important issues such as optimal investment. However, it is well known that obtaining solutions is challenging, even when employing numerical methods. In this paper, we propose new methods that combine an algorithm recently developed for coupled FBSDEs and an asymptotic expansion approach to those FBSDEs as control variates for learning of the neural networks. The proposed method is demonstrated to perform better than the original algorithm in numerical examples, including one with a financia

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27

Herdegen, Martin, Johannes Muhle-Karbe, and Dylan Possamaï. "Equilibrium asset pricing with transaction costs." Finance and Stochastics 25, no. 2 (2021): 231–75. http://dx.doi.org/10.1007/s00780-021-00449-4.

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AbstractWe study risk-sharing economies where heterogeneous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully coupled forward–backward stochastic differential equations. We show that a unique solution exists provided that the agents’ preferences are sufficiently similar. In a benchmark specification with linear state dynamics, the empirically observed illiquidity discounts and liquidity premia correspond to a positive relationship between transaction costs and volatility.

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28

Tang, Maoning. "Stochastic Maximum Principle of Near-Optimal Control of Fully Coupled Forward-Backward Stochastic Differential Equation." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/361259.

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This paper first makes an attempt to investigate the near-optimal control of systems governed by fully nonlinear coupled forward-backward stochastic differential equations (FBSDEs) under the assumption of a convex control domain. By Ekeland’s variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for anyε-near optimal control in a local form with an error order of exactε1/2. Moreover, under additional convexity conditions on Hamiltonian function, we prove that anε-maximum condition in terms of the Hamiltonian in the integ

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29

Hao, Tao, and Juan Li. "Fully coupled forward-backward SDEs involving the value function and associated nonlocal Hamilton−Jacobi−Bellman equations." ESAIM: Control, Optimisation and Calculus of Variations 22, no. 2 (2016): 519–38. http://dx.doi.org/10.1051/cocv/2015016.

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30

Khallout, Rania, and Adel Chala. "A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications." Asian Journal of Control 22, no. 3 (2019): 1360–71. http://dx.doi.org/10.1002/asjc.2020.

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31

WEBB, G. M., A. ZAKHARIAN, M. BRIO, and G. P. ZANK. "Wave interactions in magnetohydrodynamics, and cosmic-ray-modified shocks." Journal of Plasma Physics 61, no. 2 (1999): 295–346. http://dx.doi.org/10.1017/s0022377898007399.

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Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfvén waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled

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32

Tang, Maoning. "A Variational Formula for Nonzero-Sum Stochastic Differential Games of FBSDEs and Applications." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/283418.

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A nonzero-sum stochastic differential game problem is investigated for fully coupled forward-backward stochastic differential equations (FBSDEs in short) where the control domain is not necessarily convex. A variational formula for the cost functional in a given spike perturbation direction of control processes is derived by the Hamiltonian and associated adjoint systems. As an application, a global stochastic maximum principle of Pontryagin’s type for open-loop Nash equilibrium points is established. Finally, an example of a linear quadratic nonzero-sum game problem is presented to illustrate

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33

Liu, Meijuan, Xiangrong Wang, and Hong Huang. "Maximum Principle for Forward-Backward Control System Driven by Itô-Lévy Processes under Initial-Terminal Constraints." Mathematical Problems in Engineering 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/1868560.

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This paper investigates a stochastic optimal control problem where the control system is driven by Itô-Lévy process. We prove the necessary condition about existence of optimal control for stochastic system by using traditional variational technique under the assumption that control domain is convex. We require that forward-backward stochastic differential equations (FBSDE) be fully coupled, and the control variable is allowed to enter both diffusion and jump coefficient. Moreover, we also require that the initial-terminal state be constrained. Finally, as an application to finance, we show an

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34

Wang, Guangchen, and Hua Xiao. "Arrow Sufficient Conditions for Optimality of Fully Coupled Forward–Backward Stochastic Differential Equations with Applications to Finance." Journal of Optimization Theory and Applications 165, no. 2 (2014): 639–56. http://dx.doi.org/10.1007/s10957-014-0625-4.

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35

Drapeau, Samuel, Peng Luo, Alexander Schied, and Dewen Xiong. "An FBSDE approach to market impact games with stochastic parameters." Probability, Uncertainty and Quantitative Risk 6, no. 3 (2021): 237. http://dx.doi.org/10.3934/puqr.2021012.

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<p style='text-indent:20px;'>In this study, we have analyzed a market impact game between <i>n</i> risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting

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36

Aurell, Alexander, René Carmona, Gökçe Dayanıklı, and Mathieu Laurière. "Finite State Graphon Games with Applications to Epidemics." Dynamic Games and Applications 12, no. 1 (2022): 49–81. http://dx.doi.org/10.1007/s13235-021-00410-2.

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AbstractWe consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented with a graphon, which can be viewed as the limit of a dense random graph. A player’s transition rates between the states depend on their control and the strength of interaction with the other players. We develop a rigorous mathematical framework for the game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential

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37

Angiuli, Andrea, Christy V. Graves, Houzhi Li, Jean-François Chassagneux, François Delarue, and René Carmona. "Cemracs 2017: numerical probabilistic approach to MFG." ESAIM: Proceedings and Surveys 65 (2019): 84–113. http://dx.doi.org/10.1051/proc/201965084.

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This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhamé, see [26]. Generally speaki

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38

Hao, Tao, and Qingfeng Zhu. "General fully coupled FBSDES involving the value function and related nonlocal HJB equations combined with algebraic equations." Stochastics and Dynamics, October 24, 2020, 2150032. http://dx.doi.org/10.1142/s0219493721500325.

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Recently, Hao and Li [Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton–Jacobi–Bellman equations, ESAIM: Control Optim, Calc. Var. 22(2016) 519–538] studied a new kind of forward-backward stochastic differential equations (FBSDEs), namely the fully coupled FBSDEs involving the value function in the case where the diffusion coefficient [Formula: see text] in forward stochastic differential equations depends on control, but does not depend on [Formula: see text]. In our paper, we generalize their work to the case where [Formula: see text] depends on both control

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39

Ji, Shaolin, Haodong Liu, and Xinling Xiao. "Fully coupled forward-backward stochastic differential equations on Markov chains." Advances in Difference Equations 2016, no. 1 (2016). http://dx.doi.org/10.1186/s13662-016-0859-6.

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Oukdach, Omar, Said Boulite, Abdellatif Elgrou, and Lahcen Maniar. "Stackelberg–Nash Null Controllability for Stochastic Parabolic Equations." Mathematical Methods in the Applied Sciences, May 26, 2025. https://doi.org/10.1002/mma.11092.

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ABSTRACTWe study a hierarchical control problem for stochastic parabolic equations that involve a gradient term in the drift part. We employ the Stackelberg–Nash strategy with two leaders and two followers. The leaders are responsible for selecting the policy targeting null controllability, while the followers solve a bi‐objective optimal control problem which consists of maintaining the solution process close to prefixed targets. Once the Nash equilibrium is determined, the problem reduces to achieving null controllability of a coupled forward‐backward stochastic system. To solve this problem

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41

Han, Jiequn, and Jihao Long. "Convergence of the deep BSDE method for coupled FBSDEs." Probability, Uncertainty and Quantitative Risk 5, no. 1 (2020). http://dx.doi.org/10.1186/s41546-020-00047-w.

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Abstract The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy o

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42

Liu, Ruyi, Zhen Wu, and Detao Zhang. "Two equivalent families of linear fully coupled forward backward stochastic differential equations." ESAIM: Control, Optimisation and Calculus of Variations, November 8, 2022. http://dx.doi.org/10.1051/cocv/2022073.

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In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs are proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structur

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43

Si, Yu, and Jingtao Shi. "Decentralized Strategies for Backward Linear‐Quadratic Mean Field Games and Teams." Optimal Control Applications and Methods, May 31, 2025. https://doi.org/10.1002/oca.3320.

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ABSTRACTThis paper studies a new class of linear‐quadratic mean field games and teams problem, where the large‐population system satisfies a class of weakly coupled linear backward stochastic differential equations (BSDEs), and (a part of solution of BSDE) enter the state equations and cost functionals. Inspired by the stochastic control problem with partial information, we adopt the backward separation approach to solve the problem. By virtue of the stochastic maximum principle and optimal filter technique, we obtain a Hamiltonian system first, which is a fully coupled forward‐backward stocha

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Ji, Shaolin, Haodong Liu, and Xinling Xiao. "Erratum to: Fully coupled forward-backward stochastic differential equations on Markov chains." Advances in Difference Equations 2016, no. 1 (2016). http://dx.doi.org/10.1186/s13662-016-0870-y.

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45

Sun, Yabing, and Weidong Zhao. "Numerical schemes for fully coupled mean-field forward backward stochastic differential equations." Discrete and Continuous Dynamical Systems - S, 2023, 0. http://dx.doi.org/10.3934/dcdss.2023024.

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46

Al-Hussein, AbdulRahman. "Forward-backward doubly stochastic differential equations with Poisson jumps in infinite dimensions." Random Operators and Stochastic Equations, May 22, 2025. https://doi.org/10.1515/rose-2025-2020.

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Abstract In this paper, we investigate the existence and uniqueness of the solution for a system of nonlinear, fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our study is conducted within the framework of a separable real Hilbert space, and we employ a continuation method to establish the proof.

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47

Saranya, G., P. Muthukumar, and Mokhtar Hafayed. "Maximum Principle for Optimal Control of Fully Coupled Mean‐Field Forward‐Backward Stochastic Differential Equations With Teugels Martingales Under Partial Observation." Optimal Control Applications and Methods, December 4, 2024. https://doi.org/10.1002/oca.3228.

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ABSTRACTThe necessary conditions for the optimal control of partially observed, fully coupled forward‐backward mean‐field stochastic differential equations driven by Teugels martingales are discussed in this paper. In this context, we make the assumption that the forward diffusion coefficient and the martingale coefficient are independent of the control variable, and the control domain may not necessarily be convex. For this class of optimal control problems, we derive the stochastic maximum principle based on the classical method of spike variations and the filtering techniques. The adjoint p

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CHEN, Yanbo, and Tianyang NIE. "Well-posedness of fully coupled McKean-Vlasov FBSDE and application to Stackelberg games." ESAIM: Control, Optimisation and Calculus of Variations, April 28, 2025. https://doi.org/10.1051/cocv/2025041.

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Abstract:

In this paper, motivated by the study of linear-quadratic (LQ) Stackelberg differential games of McKean-Vlasov type, we investigate the solvability for the McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Inspired by the work of Tian and Yu (2023), we propose a type of domination-monotonicity conditions. Under these conditions and Lipschitz condition, we prove the well-posedness of such MV-FBSDEs by the method of continuation. As an application, we consider the LQ Stackelberg differential games of McKean-Vlasov type. The related Stackelberg solutions are given expl

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Tsuchiya, Takahiro. "Fully coupled drift-less forward and backward stochastic differential equations in a degenerate case." Japan Journal of Industrial and Applied Mathematics, February 2, 2023. http://dx.doi.org/10.1007/s13160-023-00566-x.

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50

Song, Teng. "Solvability of general fully coupled forward–backward stochastic difference equations with delay and applications." Optimal Control Applications and Methods, June 28, 2023. http://dx.doi.org/10.1002/oca.3023.

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