A Forward-Backward Stochastic Algorithm For Quasi-Linear PDEs

By Franccois Delarue, Stéphane Menozzi. 2006

154 citations recorded by the source corpus.

We propose a time-space discretization scheme for quasi-linear PDEs. The algorithm relies on the theory of fully coupled Forward-Backward SDEs, which provides an efficient probabilistic representation of this type of equations. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1cc2014853163b4c0f13ce9381135caae8ac6f2

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Citations in the corpus

Time discretization and Markovian iteration for coupled FBSDEs (2008, background): “Xt = x+ ∫ t 0 b(s,Xs, Ys)ds+ ∫ t 0 σ(s,Xs, Ys)dWs, Yt = g(XT ) + ∫ T t f(s,Xs, Ys,Zs)ds− ∫ T t Zs dWs, (1.1) where b, σ, f, g are deterministic and Lipschitz continuous functions of linear growth which are additionally supposed to satisfy some weak coupling or monotonicity condition.”
Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations (2018, background): “It is shown in [14] and [15] (see also [3, 16, 17]) that coupled forward-backward stochastic diﬀerential equations (1) are related to quasi-linear partial diﬀerential equations of the form with terminal condition u ( T, x ) = g ( x ), where u ( t, x ) is the unknown solution and Here, Du and D 2 u…”
Error expansion for the discretization of backward stochastic differential equations (2006, result): “The first result of this Proposition corresponds to Theorem 2.1 of Delarue and Menozzi [ 4 ].”
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”
Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications (2011, background): “There are actually very active researches on numerical analysis and calculations of BSDE, see Douglas, Ma and Protter (1996), Ma and Zhang (2002), Zhanng (2004), Bouchard and Touzi (2004), Peng and Xu (2003), Gobet, Lemor and Warin (2005), Zhao et al (2006), Delarue and Menozzi (2006).”
Weak existence and uniqueness for forward–backward SDEs (2006, background): “Buckdahn and Hu [6], and numerical approximation, see Delarue and Menozzi [10].”
Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs (2005, methodology): “But they have been overcome, and now an impressive theory is available; see for instance, Bismut [7, 8], Arkin and Saksonov [2], Pardoux and Peng [39, 40, 41], Peng [43, 44, 45, 46, 47, 48], Antonelli [1], Ma et al. [36, 37, 38], Douglas et al. [22], Cvitani¶c et al. [17, 18, 19], Chevance [15], Par”
Least-Squares Monte Carlo for Backward SDEs (2012, methodology): “• Quantization methods, see Bally and Pagès (2003) for reflected BSDEs and Delarue and Menozzi (2006) for coupled FBSDEs.”
Numerical simulation of quadratic BSDEs (2013, unknown): “We refer to [26, 37] for general results about quantization and [2] for application to American options pricing and to [17] for application to coupled forward– backward SDEs.”
Numerical method for FBSDEs of McKean–Vlasov type (2017, methodology): “Then the estimate (12) follows from standard results on non-linear PDEs, see e.g. Theorem 2.1 in [21] .”
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2017, methodology): “tations based on extensions of the classical Feynman-Kac formula (cf., e.g., [34, 58, 84]). We refer, e.g., to [23, 30, 65, 69, 95, 103, 104, 106] for deterministic approximation methods for PDEs, to [3, 6, 7, 13, 18, 19, 20, 21, 25, 26, 27, 28, 31, 32, 43, 44, 45, 46, 47, 48, 56, 64, 71, 72, 73, 74”
Error estimates of the $\theta$-scheme for backward stochastic differentialequations (2009, methodology): “Delarue and Menozzi proposed a time-space discretization scheme for solving quasi-linear parabolic PDEs in [8] and then improved this scheme through a special interpolation procedure [9].”
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 ].”
Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition (2011, methodology): “s a speciﬁc form 1, it is possible to get around the problem by using an exponential transformation method (see [19]) or by using results on fully coupled forward-backward differential equations (see [11]). Papers [18, 26] only study the case of a bounded terminal condition: The ﬁrst one investigate”