Malliavin calculus for backward stochastic differential equations and application to numerical solutions

Hu, Yaozhong; Nualart, David; Song, Xiaoming

doi:10.1214/11-aap762

Cited by 64 publications

(81 citation statements)

References 15 publications

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“…In this sense, our paper extends the work done in [16] to an FBSDE with jumps and terminal value g(X ε T ). In this sense, our paper extends the work done in [16] to an FBSDE with jumps and terminal value g(X ε T ).…”

Section: Introductionmentioning

confidence: 55%

“…The second numerical scheme is inspired by the work of Hu et al [16], who studied a backward stochastic differential equation driven by a Brownian motion with general terminal variable ξ . The second numerical scheme is inspired by the work of Hu et al [16], who studied a backward stochastic differential equation driven by a Brownian motion with general terminal variable ξ .…”

Section: Introductionmentioning

confidence: 99%

“…The importance of the above scheme is that it can be adapted to the case where the terminal value is not given by the forward diffusion equation X ε , as is the case in [16]. The importance of the above scheme is that it can be adapted to the case where the terminal value is not given by the forward diffusion equation X ε , as is the case in [16].…”

Section: Introductionmentioning

confidence: 99%

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Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes

Aazizi

2013

Adv. Appl. Probab.

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We present a new algorithm to discretize a decoupled forward-backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forwardbackward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain the L p -Hölder continuity of the solution of the FBSDEBP and we construct two numerical schemes for this FBSDEBP. Based on the L p -Hölder estimate, we prove the convergence of the scheme when the number of time steps n goes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.FBSDEs driven by pure jump Lévy processes 793 approximated X ε by its Euler schemeX n,ε and (Y ε , Z ε , ε ) by the discrete-time process (Ȳ n,ε t ,Z n,ε t ,¯ n,ε t ), i.e.

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Section: Introductionmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes

Aazizi

2013

Adv. Appl. Probab.

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“…Second, instead of the representation of Z t i or Z i using integration by parts formula, we could take advantage of the BSDE-type equation satisfied by (Z t ) t (see [21] for a recent account on the subject). However, these equations involve "the Z of the Z", i.e.…”

Section: Digression On the Driver Not Depending On Zmentioning

confidence: 99%

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

Gobet

Turkedjiev²

2017

Stochastic Processes and their Applications

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Please cite this article as: E. Gobet, P. Turkedjiev, Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations, Stochastic Processes and their Applications (2016), http://dx.doi.org/10. 1016/j.spa.2016.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe design an importance sampling scheme for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in least-squares MonteCarlo (LSMC) algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore it is computed adaptively within the LSMC procedure. To allow robust error estimates w.r.t. the unknown change of measure, we properly randomize the initial value of the forward process. We introduce novel methods to analyze the error: firstly, we establish norm stability results due to the random initialization; secondly, we develop refined concentration-of-measure techniques to capture the variance reduction. Our theoretical results are supported by numerical experiments.

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“…, X r m ), and satisfies a path-dependent Malliavin fractional smoothness condition which is weaker than the Lipschitz condition on g. Using these results and following the ideas of [11], one can show that the convergence rate of the error between the discretizations (Y τ , Z τ ) and the solution (Y, Z) is of order 1 2 , that is provided that the time grid for the discretization is chosen in an appropriate way (like in [14]), and the discretized terminal condition converges in this order. Without any assumptions on the dependence of the terminal condition ξ on a forward process X, Hu, Nualart and Song [22] have shown the convergence rate 1 2 supposing Malliavin differentiability properties of ξ (and of the generator).…”

Section: -Variation Properties Of the Exact Solution (Y Z)mentioning

confidence: 99%

$L_{2}$-variation of Lévy driven BSDEs with non-smooth terminal conditions

Geiß¹,

Steinicke²

2016

Bernoulli

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We consider the L 2 -regularity of solutions to backward stochastic differential equations (BSDEs) with Lipschitz generators driven by a Brownian motion and a Poisson random measure associated with a Lévy process (X t ) t∈ [0,T ] . The terminal condition may be a Borel function of finitely many increments of the Lévy process which is not necessarily Lipschitz but only satisfies a fractional smoothness condition. The results are obtained by investigating how the special structure appearing in the chaos expansion of the terminal condition is inherited by the solution to the BSDE.

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Malliavin calculus for backward stochastic differential equations and application to numerical solutions

Cited by 64 publications

References 15 publications

Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes

Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes

Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations

$L_{2}$-variation of Lévy driven BSDEs with non-smooth terminal conditions

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