Numerical method for FBSDEs of McKean–Vlasov type

Chassagneux, Jean-François; Crisan, Dan; Delarue, François

doi:10.1214/18-aap1429

Cited by 49 publications

(67 citation statements)

References 27 publications

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“…But in practice, this is not always the case: It may happen that the FBSDE system is uniquely solvable, but that we are unable to prove that the Picard sequence converges. In order to overcome this issue, we follow the approach introduced in Chassagneux, Crisan, Delarue [14]. Basically, the point is to divide the time interval into smaller intervals, called levels, and to apply a Picard solver recursively between the levels.…”

Section: Continuation In Time Of the Global Methods For Arbitrary Intementioning

confidence: 99%

“…We will see in Section 4 that the continuation in time successfully achieves this goal for our benchmark examples. We refer to [14] for its theoretical analysis.…”

Section: Continuation In Time Of the Global Methods For Arbitrary Intementioning

confidence: 99%

“…In this regard, we implement (and compare) two different algorithms. The first algorithm, which is based on the paper of Chassagneux, Crisan, and Delarue [14], relies on a tree structure to represent the pathwise law of the solution. The second algorithm, and main contribution of this report, takes the algorithm presented in the paper of Delarue and Menozzi [18] for solving FBSDEs and extends it to the mean field framework in hand.…”

Section: Introductionmentioning

confidence: 99%

“…For sure, this limitation should persist in the mean field setting for the global Picard method; as exemplified below, it turns out that it persists as well when the Picard scheme is applied to the marginal laws. One of our main contribution in this report is to apply the time continuation approach presented in [14] to the grid algorithm and to compare the results with the tree algorithm for which the time continuation approach was originally designed in [14]. In brief, the time continuation permits to extend, by a continuation argument, the time interval on which the Picard scheme converges.…”

Section: Introductionmentioning

confidence: 99%

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Cemracs 2017: numerical probabilistic approach to MFG

Angiuli

Graves

et al. 2019

ESAIM: ProcS

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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 [27], and by Huang, Caines, and Malhamé, see [24]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

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Section: Continuation In Time Of the Global Methods For Arbitrary Intementioning

confidence: 99%

“…We will see in Section 4 that the continuation in time successfully achieves this goal for our benchmark examples. We refer to [14] for its theoretical analysis.…”

Section: Continuation In Time Of the Global Methods For Arbitrary Intementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cemracs 2017: numerical probabilistic approach to MFG

Angiuli

Graves

et al. 2019

ESAIM: ProcS

Self Cite

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“…Forward methods have also been introduced to approximate (1) : a branching diffusion method (see [26]), a multilevel Picard approximation (see [34]) and Wiener chaos expansion (see [7]). Many extensions of (1) have also been considered : high order schemes (see [11], [10]), schemes for reflected BSDEs (see [3], [14]), for fully-coupled BSDEs (see [21], [9]), for quadratic BSDEs (see [13]), for BSDEs with jumps (see [23]) and for McKean-Vlasov BSDEs (see [1], [16], [12]). From a numerical point of view, the random walk is of course not competitive with recent methods listed above.…”

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confidence: 99%

Random walk approximation of BSDEs with Hölder continuous terminal condition

Geiß¹,

Labart

Luoto³

2020

Bernoulli

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In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L 2 -convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.Keywords : Backward stochastic differential equations, numerical scheme, random walk approximation, speed of convergence MSC codes : 65C30 60H35 60G50 65G99 IntroductionLet (Ω, F, P) be a complete probability space carrying the standard Brownian motion B = (B t ) t≥0 and assume (F t ) t≥0 is the augmented natural filtration. We consider the following backward stochastic differential equation (BSDE for short)where f is Lipschitz continuous and g is a locally α-Hölder continuous and polynomially bounded function (see (3)). In this paper we are interested in the L 2 -convergence of the numerical approximation of (1) by using a random walk. First results dealing with the numerical approximation of BSDEs date back to the late 1990s. Bally (see [2]) was the first to consider this problem by introducing random discretization, namely the jump times of a Poisson process. In his PhD thesis, Chevance (see [17]) proposed the following discretization y k = E(y k+1 + hf (y k+1 )|F n k ), k = n − 1, · · · , 0, n ∈ N * and proved the convergence of (Y n t ) t := (y [t/h] ) t to Y . At the same time, Coquet, Mackevičius and Mémin [18] proved the convergence of Y n by using convergence of filtrations, still in the case of

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Towards control of dam and reservoir systems with forward–backward stochastic differential equations driven by clustered jumps

Yoshioka

2022

Adv Control Appl

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We deal with a new maximum principle‐based stochastic control model for river management through operating a dam and reservoir system. The model is based on coupled forward–backward stochastic differential equations (FBSDEs) derived from jump‐driven streamflow dynamics and reservoir water balance. A continuous‐time branching process with immigration driven by a tempered stable subordinator efficiently describes clustered inflow streamflow dynamics. This is a completely new attempt in hydrology and control engineering. Applying a stochastic maximum principle to the dynamics based on an objective functional for designing cost‐efficient control of dam and reservoir systems leads to the FBSDEs as a system of optimality equations. The FBSDEs under a linear‐quadratic ansatz lead to a tractable model, while they are solved numerically in the other cases using a least‐squares Monte‐Carlo method. Optimal controls are found in the former, while only sub‐optimal ones are computable in the latter due to a hard state constraint. Model parameters are successfully identified from a real data of a river in Japan having a dam and reservoir system. We also show that the linear‐quadratic case can capture the real operation data of the system with underestimation of the outflow discharge. More complex cases with a realistic time horizon are analyzed numerically to investigate impacts of considering the environmental flows and seasonal operational purposes. Key challenges towards more sophisticated modeling and analysis with jump‐driven FBSDEs are discussed as well.

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Numerical method for FBSDEs of McKean–Vlasov type

Cited by 49 publications

References 27 publications

Cemracs 2017: numerical probabilistic approach to MFG

Cemracs 2017: numerical probabilistic approach to MFG

Random walk approximation of BSDEs with Hölder continuous terminal condition

Towards control of dam and reservoir systems with forward–backward stochastic differential equations driven by clustered jumps

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