Cubature method to solve BSDEs: Error expansion and complexity control

Chassagneux, Jean-François; Trillos, Camilo A. Garcia

doi:10.1090/mcom/3522

Cited by 14 publications

(10 citation statements)

References 35 publications

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“…Since B n is built using the random walk (15), it can be represented by a recombining binomial tree. Both (Y n t k ) 0≤k≤n and (Z n t k ) 0≤k≤n−1 can then also be represented as a recombining binomial tree.…”

Section: Simulation Of (Y N Z N )mentioning

confidence: 99%

“…However, to be fully implementable, this algorithm requires to have a good approximation of its associated conditional expectation. For this, various methods have been developed (see [24], [19], [15]). Forward methods have also been introduced to approximate (1) : a branching diffusion method (see [26]), a multilevel Picard approximation (see [34]) and Wiener chaos expansion (see [7]).…”

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confidence: 99%

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Random walk approximation of BSDEs with Hölder continuous terminal condition

Geiß¹,

Labart

Luoto³

2020

Bernoulli

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In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L 2 -convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.Keywords : Backward stochastic differential equations, numerical scheme, random walk approximation, speed of convergence MSC codes : 65C30 60H35 60G50 65G99 IntroductionLet (Ω, F, P) be a complete probability space carrying the standard Brownian motion B = (B t ) t≥0 and assume (F t ) t≥0 is the augmented natural filtration. We consider the following backward stochastic differential equation (BSDE for short)where f is Lipschitz continuous and g is a locally α-Hölder continuous and polynomially bounded function (see (3)). In this paper we are interested in the L 2 -convergence of the numerical approximation of (1) by using a random walk. First results dealing with the numerical approximation of BSDEs date back to the late 1990s. Bally (see [2]) was the first to consider this problem by introducing random discretization, namely the jump times of a Poisson process. In his PhD thesis, Chevance (see [17]) proposed the following discretization y k = E(y k+1 + hf (y k+1 )|F n k ), k = n − 1, · · · , 0, n ∈ N * and proved the convergence of (Y n t ) t := (y [t/h] ) t to Y . At the same time, Coquet, Mackevičius and Mémin [18] proved the convergence of Y n by using convergence of filtrations, still in the case of

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Section: Simulation Of (Y N Z N )mentioning

confidence: 99%

mentioning

confidence: 99%

Random walk approximation of BSDEs with Hölder continuous terminal condition

Geiß¹,

Labart

Luoto³

2020

Bernoulli

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“…In order to define the discrete counterpart to (11), we first define the discrete counterpart to (N t s ) s∈ [t,T] given in (12):…”

Section: Definition 23 (Discretized Processes (∇Xmentioning

confidence: 99%

“…But this approach involves conditional expectations. Various methods to approximate these conditional expectations have been developed ( [23], [17], [12]). Also, forward methods have been introduced to approximate (1): a branching diffusion method ( [24]), a multilevel Picard approximation ( [39]), and Wiener chaos expansion ( [8]).…”

Section: Introductionmentioning

confidence: 99%

Mean square rate of convergence for random walk approximation of forward-backward SDEs

Geiß¹,

Labart

Luoto³

2020

Adv. Appl. Probab.

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Let (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 motion B by 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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“…e forward Picard iterations method was designed by [16]. e cubature method was used to solve BSDEs in [14,17]. In [15], authors proposed the BCOS method based on the Fourier cosine series expansions to approximate the solutions of BSDEs.…”

Section: Introductionmentioning

confidence: 99%

Variable Step Size Adams Methods for BSDEs

Qiang

2021

Journal of Mathematics

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For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

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Cubature method to solve BSDEs: Error expansion and complexity control

Cited by 14 publications

References 35 publications

Random walk approximation of BSDEs with Hölder continuous terminal condition

Random walk approximation of BSDEs with Hölder continuous terminal condition

Mean square rate of convergence for random walk approximation of forward-backward SDEs

Variable Step Size Adams Methods for BSDEs

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