Numerical Resolution of McKean-Vlasov FBSDEs Using Neural Networks

Germain, Maximilien; Mikael, Joseph; Warin, Xavier

doi:10.1007/s11009-022-09946-1

Cited by 19 publications

(19 citation statements)

References 23 publications

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“…Secondly, a suitable numerical procedure is introduced to solve the discrete-time MV-FBSDE, which usually consists of projecting the nested conditional expectations on some trial spaces by least-squares regression (see e.g. [17,23,4,16,18,1,13,15,21,22,31,35,28]).…”

Section: Introductionmentioning

confidence: 99%

“…[4]) or neural networks of fixed sizes (see e.g. [18,21,22]). Hence it is not clear whether the chosen trial space is rich enough to approximate the required conditional expectations up to the desired accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, sharp criteria for convergence of the Picard method are difficult to establish since, on one hand, it is well-known that the Picard theorem only applies to the fully coupled system (1.1) with a sufficiently small maturity T (see e.g. [1,15]), while on the other hand, empirical studies show that the theoretical bound on the maturity to ensure the convergence is usually far too pessimistic (see Remark 1 in [22]).…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, instead of merely estimating Y 0 (the value of Y at the initial time) as in [2], the estimator (1.2) yields upper and lower bounds for the global L 2 -error of the given discrete approximation ( X, Ŷ , Ẑ) over the grid, which subsequently enables us to measure the accuracy of the numerical Nash equilibria and optimal control strategies (see e.g. [1,13,15,22]) or the dynamic risk measures [24] computed over the whole interval.…”

Section: Introductionmentioning

confidence: 99%

“…An important application of our a posteriori error estimates is the convergence analysis of the recently developed neural network-based algorithm (the so-called Deep BSDE Solver) for solving MV-FBSDEs (see e.g. [13,21,22]). The Deep BSDE Solver consists of approximating the process Z by ( Ẑt i ) t i ∈π in a parametric form depending on weights Ξ (for instance a deep neural network), generating ( Xt i , Ŷt i ) t i ∈π by following the explicit forward Euler scheme of (1.1), and then minimizing the objective function E[| ŶT − g( XT , P XT )| 2 ] over the weight Ξ (see (4.21) for more details).…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates for fully coupled McKean-Vlasov forward-backward SDEs

Reisinger¹,

Stockinger²,

Zhang³

2020

Preprint

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Fully coupled McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. This paper proposes an a posteriori error estimator to quantify the L 2 -approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE. A crucial and challenging step in the analysis is the proof of stability of this Euler approximation to the MV-FBSDE, which is of independent interest. We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. In particular, the a posteriori error estimates justify the usage of the Deep BSDE Solver for solving MV-FBSDEs. Numerical experiments on an extended mean field game are presented to illustrate the theoretical results and to demonstrate the practical applicability of the error estimator.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A posteriori error estimates for fully coupled McKean-Vlasov forward-backward SDEs

Reisinger¹,

Stockinger²,

Zhang³

2020

Preprint

View full text Add to dashboard Cite

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Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

Bezemek,

Heldman

2023

Appl Math Optim

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We construct an importance sampling method for computing statistics related to rare events for weakly interacting diffusions. Standard Monte Carlo methods behave exponentially poorly with the number of particles in the system for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) equation on Wasserstein space which arises in the theory of mean-field (McKean-Vlasov) control. We identify conditions under which such a scheme is asymptotically optimal. In the process, we make connections between the large deviations principle for the empirical measure of weakly interacting diffusions, mean-field control, and the HJB equation on Wasserstein space. We also provide evidence, both analytical and numerical, that with sufficient regularity of the HJB equation, our scheme can have vanishingly small relative error in the many particle limit.

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