Corrigendum to “Stability of solutions of BSDEs with random terminal time”

Toldo, Sandrine

doi:10.1051/ps:2007025

ESAIM: PS

2007

DOI: 10.1051/ps:2007025

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Corrigendum to “Stability of solutions of BSDEs with random terminal time”

Sandrine Toldo

Abstract: Abstract. This paper is a corrigendum to paper Toldo, ESAIM, P&S 10 (2006) 141-163 where we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time.Mathematics Subject Classification. 60H10, 60Fxx, 60G40.

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

Geiß¹,

Labart

Luoto³

2020

Adv. Appl. Probab.

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BS$\Delta$Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

Cheridito¹,

Stadje²

2013

Bernoulli

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We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS∆Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS∆Es and BSDEs are governed by drivers f N (t, ω, y, z) and f (t, ω, y, z), respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS∆Es are based on d-dimensional random walks W N approximating the d-dimensional Brownian motion W underlying the BSDE and that f N converges to f . Conditions are given under which for any bounded terminal condition ξ for the BSDE, there exist bounded terminal conditions ξ N for the sequence of BS∆Es converging to ξ, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when f N and f are convex in z. We show that in this situation, the solutions of the BS∆Es converge to the solution of the BSDE for every uniformly bounded sequence ξ N converging to ξ. As a consequence, one obtains that the BSDE is robust in the sense that if (W N , ξ N ) is close to (W, ξ) in distribution, then the solution of the N th BS∆E is close to the solution of the BSDE in distribution too.

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Corrigendum to “Stability of solutions of BSDEs with random terminal time”

Cited by 2 publications

References 1 publication

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

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

BS$\Delta$Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

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