A regression-based Monte Carlo method to solve backward stochastic differential equations

By Emmanuel Gobet, Jean-Philippe Lemor, Xavier Warin. 2005

423 citations recorded by the source corpus.

We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.

Subfields: Demography, Finance

View Paper: https://www.semanticscholar.org/paper/b4c388bd5c6c4670b7ecde874d55571468fde417

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Citations in the corpus

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks (2019, methodology): “, [2, 6, 7, 10, 12, 13, 17, 18, 19, 20, 25, 26, 27, 28, 38, 39, 31, 32, 40, 41, 42, 43, 44, 45, 46, 57, 69, 70, 71, 72, 73, 74, 78, 79, 82, 83, 84, 85, 89, 90, 91, 96, 102, 103, 104] for probabilistic approximation methods for PDEs using discretizations of the associated backward stochastic differen”
On some recent aspects of stochastic control and their applications (2005, methodology): “We mention for BSDE the Markov chain approximation in [64], regression methods in [ 43 ], quantization methods in [4], the Monte-Carlo Malliavin method in [18] , and for FBSDE the four-step scheme in [29], and quantization method in [28].”
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations (2017, methodology): “, [12, 2, 3, 5, 8, 11] where this example has been used as a test example for numerical methods for BSDEs).”
Pricing Asset Scheduling Flexibility using Optimal Switching (2008, methodology): “Another possible alternative (see, for instance, Carmona and Durrleman, 2003) is to discard (Pt) and (Gt) and directly work with a one-dimensional (Xt) that models the spark-spread.”
RUNGE-KUTTA SCHEMES FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS (2014, background): “Under weak regularity assumption on the coefficient g, the order 12 can be retrieved; see, for example, [2, 5, 7, 8, 17].”
On the Monte Carlo simulation of BSDEs: An improvement on the Malliavin weights (2010, neutral): “The simulation could rely on a Monte Carlo method as is the case in [6,15], or on an evaluation on a tree as in [11].”
Numerical simulation of quadratic BSDEs (2013, positive): “The design of efficient algorithms to solve BSDEs in any reasonable dimension has been intensively studied since the first work of Chevance [15]; see, for example, [5, 10, 11, 23, 43] and the references therein.”
BSDE, path-dependent PDE and nonlinear Feynman-Kac formula (2011, methodology): “This nonlinear Feynman-Kac formula also provides a nonlinear MonteCarlo method via the BSDE ([3], [11]) to solve numerically the PDE.”
Numerical simulation of BSDEs with drivers of quadratic growth (2010, background): “he conditional expectation given Ft k . Let us notice that the classical dynamic programming equation do not use a projection in (4.3): it is the only difference with our time approximation, see e.g. [11] for the classical case. This projection comes directly from the estimate of Z in Proposition 4.”
Malliavin calculus for backward stochastic differential equations and application to numerical solutions (2011, methodology): “In the work by Zhang [16], the L2-regularity of Z is obtained, which allows one to use deterministic time partitions as well as to obtain the rate estimate (see Bender and Denk [2], Gobet, Lemor and Warin [8] and Zhang [16] for different algorithms).”
Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition (2011, methodology): “Besides the fact that this variation gives a strong insight into the quantitative behavior of the BSDE, in particular var2(ξ, f, τ) was used to describe the error in adapted backward Euler schemes for ξ = g(XT ) with g being a Lipschitz function; see [5, 23] for implicit schemes and [15, 16] for exp”
Time discretization and Markovian iteration for coupled FBSDEs (2008, neutral): “Xn,m0 , x, Xn,mi+1 ,X n,m i + b(ti,X n,m i , u n,m−1 i (X n,m i ))h + σ(ti,X n,m i , u n,m−1 i (X n,m i ))∆Wi+1, Y n,mn , g(X n,m n ), Ẑn,mi , 1 h Eti{Y n,mi+1 ∆Wi+1}, Y n,mi ,Eti{Y n,m i+1 + f(ti,X n,m i , Y n,m i+1 , Ẑ n,m i )h}, un,mi (X n,m i ) = Y n,m i .”
Path regularity and explicit convergence rate for BSDE with truncated quadratic growth (2009, background): “Much has been done in recent years to create schemes for BSDE with Lipschitz continuous drivers (see [2], [7] or [5] and the references therein).”