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

Abstract

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.

Robust Portfolio Choice and Indifference Valuation

“…We finally need the following announced lemma, which is similar to Lemma 6.2 in Cheridito and Stadje [13]. Then, for every x ∈ R d such that ∂f (t, x) = ∅, there exists Z ∈ P such that f (t, x + y) − f (t, x) ≥ yZ t for all y ∈ R d .…”

The algebra of conditional sets and the concepts of conditional topology and compactness

“…Since Y|a ⊓ W|b = X|0, it must hold c ∧ d = 0 and c ∨ d = 1. 6 It follows from the construction of the supremum ⊔ that a ∨ b = 1 and…”

“…A conditional version of Mazur's lemma for L 0 -modules is shown in Zapata-García [27]. Conditional analysis is successfully applied to dynamic and conditional risk measures and decision theory in Filipović et al [14], Bielecki et al [3], Frittelli and Maggis [17] and Jamneshan and Drapeau [9], to backward stochastic differential equations in Cheridito and Hu [5] and Cheridito and Stadje [6], and to optimization problems in equilibrium and principal-agent models in Horst et al [7] and Horst and Backhoff [1]. This paper is organized as follows.…”

Dynamic Conic Finance via Backward Stochastic Difference Equations

“…As in the existing literature, we define the non-linear expectation, or g-expectation, in terms of the solution of a BS∆E, and we prove a series of standard properties of g-expectations. The results obtained in this section are new, although general ideas of the proofs are similar to those in [Sta09,CE11b,CS13].…”

“…Although the theory of Backward Stochastic Differential Equations is a mature field (cf. [MY99]), there are only few papers [Sta09,CE11b,CE11a,CS13] devoted to their discrete counterpart. Even though the existing results on BS∆Es are quite general, they do not meet our specific needs, which prompted us to establish existence and uniqueness of the solutions for a (large) class of BS∆Es relevant to our needs.…”