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Mean square rate of convergence for random walk approximation of forward-backward SDEs. ADV APPL PROBAB 2020. [DOI: 10.1017/apr.2020.17] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractLet (Y,Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk$B^n$from the underlying Brownian motionBby Skorokhod embedding, one can show$L_2$-convergence of the corresponding solutions$(Y^n,Z^n)$to$(Y, Z).$We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in$C^{2,\alpha}$. The proof relies on an approximative representation of$Z^n$and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.
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Ankirchner S, Engelhardt S, Fromm A, dos Reis G. The Skorokhod embedding problem for inhomogeneous diffusions. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2020. [DOI: 10.1214/19-aihp1012] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Geiss C, Labart C, Luoto A. Random walk approximation of BSDEs with Hölder continuous terminal condition. BERNOULLI 2020. [DOI: 10.3150/19-bej1120] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Chassagneux JF, Crisan D, Delarue F. Numerical method for FBSDEs of McKean–Vlasov type. ANN APPL PROBAB 2019. [DOI: 10.1214/18-aap1429] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Delbaen F, Qiu J, Tang S. Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space. Stoch Process Their Appl 2015. [DOI: 10.1016/j.spa.2015.02.014] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Guo W, Zhang J, Zhuo J. A monotone scheme for high-dimensional fully nonlinear PDEs. ANN APPL PROBAB 2015. [DOI: 10.1214/14-aap1030] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Crisan D, Manolarakis K. Second order discretization of backward SDEs and simulation with the cubature method. ANN APPL PROBAB 2014. [DOI: 10.1214/13-aap932] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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dos Reis G, Réveillac A, Zhang J. FBSDEs with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity. Stoch Process Their Appl 2011. [DOI: 10.1016/j.spa.2011.05.002] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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