A numerical scheme for BSDEs

By Jianfeng Zhang. 2004

421 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/7d90294dd99149d5f37fcb0e6a0900848a466772

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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”
Discrete-time approximation for continuously and discretely reflected BSDEs (2008, background): “In the non-re ected case, such approximations have been studied by [3] and [16], see also [2] and [6] for BSDEs with jumps.”
Least-Squares Monte Carlo for Backward SDEs (2012, background): “While these ideas can be traced back to the papers by Bally (1997) and Chevance (1997), a detailed analysis of the corresponding time discretization scheme under quite general assumptions was first given by Zhang (2004) and Bouchard and Touzi (2004).”
TRUE UPPER BOUNDS FOR BERMUDAN PRODUCTS VIA NON‐NESTED MONTE CARLO (2009, background): “…= XTj + ∫ t Tj a(s, Xs) ds + ∫ t Tj b(s, Xs) dW s Ȳt = u(Tj+1, XTj+1 ) − ∫ Tj+1 t Zt dW t. Due to the Lipschitz continuity of u(Tj+1, ·) results on L2-regularity obtained for the control part of FBSDEs in more general situations by Zhang (2004) and Bender and Zhang (2008) can be applied.”
Numerical simulation of quadratic BSDEs (2013, positive): “tionisnecessaryon˙. 1.1 KnownresultsontheapproximationofquadraticBSDEs ThedesignofeﬃcientalgorithmstosolveBSDEsinanyreasonabledimensionhasbeen intensivelystudiedsincetheﬁrstworkofChevance[15],seee.g. [43,5,23,11,10]and the references therein. In all these articles, the driver f of the BSDE is a Lips”
Numerical simulation of BSDEs with drivers of quadratic growth (2010, methodology): “Actually, the design of effici ent algorithms which are able to solve BSDEs in any reasonable dimension has been intensively studied si nce the first work of Chevance [6], see for instance [19, 1, 11].”
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].”
Time discretization and Markovian iteration for coupled FBSDEs (2008, methodology): “Since the PDE approach requires existence of a classical solution to (1.3), there is typically need for some smoothness, boundedness and regularity conditions, such as uniform ellipticity of the differential operator.”
A regression-based Monte Carlo method to solve backward stochastic differential equations (2005, methodology): “They follow the approach by Zhang [29] by approximating (1)–(2) by a discrete time FBSDE with N time steps [see (5)–(6) below], with an L2-error of order N −1/2.”
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”
Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition (2011, positive): “2) The study of the error ∑n−1 k=0 E [ ∫ tk+1 tk ∣ Zt − Z̄tk ∣ ∣ 2 dt ] was done by Zhang in [29].”
Second order discretization of backward SDEs and simulation with the cubature method (2014, methodology): “However, it was the work of Bouchard and Touzi [5] and, in parallel, Zhang [26] [essentially scheme (6) below] that first suggested a method for the discretization of a fairly general class of BSDEs and paved the way for algorithms of probabilistic flavor.”
Error Criteria for Numerical Solutions of Backward SDEs (2010, methodology): “Referring to Zhang [34] and Gobet et al.”
Error estimates of the $\theta$-scheme for backward stochastic differentialequations (2009, background): “In [31], Zhang proposed a numerical scheme and studied its convergence rate under weak regularity assumptions.”
A PRIMAL–DUAL ALGORITHM FOR BSDES (2013, background): “0. Z⊤ t σ −1 t corresponds to the vector of amounts of money invested in the risky assets at time t. Discretizing time and taking conditional expectations gives a recursion of the form (1) for Y, see Zhang (2004); Bouchard and Touzi (2004). In this European case Si ≡ −∞ for i < n and Sn ≡ h(XT ).”