Numerical Method for FBSDEs of McKean-Vlasov Type

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

doi:10.48550/arxiv.1703.02007

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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 Int...mentioning

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 Int...mentioning

confidence: 99%

“…The first implementation of the Picard iteration, Φ ξ,G , is the tree algorithm presented in Chassagneux, Crisan, Delarue [14]. We now provide a brief presentation of this algorithm.…”

Section: Tree Algorithm For the Global Methodsmentioning

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

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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 Int...mentioning

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 Int...mentioning

confidence: 99%

“…The first implementation of the Picard iteration, Φ ξ,G , is the tree algorithm presented in Chassagneux, Crisan, Delarue [14]. We now provide a brief presentation of this algorithm.…”

Section: Tree Algorithm For the Global Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations