Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

Henry-Labordère, Pierre; Oudjane, Nadia; Tan, Xiaolu; Touzi, Nizar; Warin, Xavier

doi:10.48550/arxiv.1603.01727

Cited by 15 publications

(44 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…That being said, one can derive stochastic representations for more general PDEs using so-called Forward Backward SDEs (FBSDEs), but FBSDEs are computationally expensive and difficult to handle (see Section 4). However, [56], [41], [43] and [42] have further developed the "branching diffusions" methodology as an efficient method to solve wider classes of nonlinear PDEs than the originally proposed by McKean.…”

Section: Branching Diffusions and The Kpp Equationmentioning

confidence: 99%

“…The idea of marked branching diffusions was developed by [41] as a solution to pricing credit valuation adjustments (CVAs) (see Section 3.2.2), but since then has been generalized by [43] and [42]; similar ideas appeared in [56]. Following [43], we consider a terminal value PDE of the form (recall the time-inversion argument in Section 2.2)…”

Section: Marked Branching Diffusionsmentioning

confidence: 99%

“…Although the set up here has been carried out using exponential random variables as the lifetime of particles, more recent work suggest that other distributions yield smaller variances, see [42], [25], [60] and [19].…”

Section: Marked Branching Diffusionsmentioning

confidence: 99%

“…PDD is currently well understood for the linear case, although recent advances in stochastic representations have opened the door for using PDD with nonlinear PDEs. A class of nonlinear PDEs strongly amenable to PDD are those whose solution can be represented by the so-called branching processes, as introduced by [61], [57], [51] and recently extended by [43] and [42]. This methodology avoids backward regressions, the so-called "Monte Carlo of Monte Carlo simulation" problem, see Section 4 below or [41,Section 3.1].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hybrid PDE solver for data-driven problems and modern branching

Bernal¹,

Reis²,

Smith³

2017

Eur. J. Appl. Math

View full text Add to dashboard Cite

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD.We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.

show abstract

Section: Branching Diffusions and The Kpp Equationmentioning

confidence: 99%

Section: Marked Branching Diffusionsmentioning

confidence: 99%

Section: Marked Branching Diffusionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hybrid PDE solver for data-driven problems and modern branching

Bernal¹,

Reis²,

Smith³

2017

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

“…For this issue, Weinan E et al propose the deep learning algorithm which can deal with 100-dimensional nonlinear PDEs (see [3,13,14,21,25]). Also, the branching diffusion method does not suffer from the curse of dimensionality (see [22]) and this method is extended to the non-Markovian case and the non-linearities case (see [24] and [23] respectively).…”

Section: Introductionmentioning

confidence: 99%

Novel multi-step predictor-corrector schemes for backward stochastic differential equations

Qiang¹,

Ji²

2021

Preprint

View full text Add to dashboard Cite

Novel multi-step predictor-corrector numerical schemes have been derived for approximating decoupled forward-backward stochastic differential equations (FBSDEs). The stability and high order rate of convergence of the schemes are rigorously proved. We also present a sufficient and necessary condition for the stability of the schemes. Numerical experiments are given to illustrate the stability and convergence rates of the proposed methods.

show abstract

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

2019

View full text Add to dashboard Cite

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black-Scholes-Barenblatt equation, 1 arXiv:1709.05963v1 [math.NA] 18 Sep 2017 a 100-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

show abstract

Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

Cited by 15 publications

References 0 publications

Hybrid PDE solver for data-driven problems and modern branching

Hybrid PDE solver for data-driven problems and modern branching

Novel multi-step predictor-corrector schemes for backward stochastic differential equations

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

Contact Info

Product

Resources

About