Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting

Abstract

We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116: 1358–1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346: 345–358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growth- and monotonicity-condition for the generator to be random and time-dependent. In the L 2 -case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88: 491–539, 2016). For the proof of the comparison result, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/ n .

Existence, uniqueness and Malliavin differentiability of Lévy-driven BSDEs with locally Lipschitz driver

“…If a triple (Y, Z, U ) ∈ S 2 × L 2 (W ) × L 2 (Ñ ) satisfies (7) it is called a solution to the BSDE (7). We will recall first the existence and uniqueness result from [17]. (H 2) There are nonnegative, progressively measurable processes K 1 , K 2 and F with…”

“…The quadratic integrability with respect to (r, v) also follows from [17,Proposition 4.2] since ξ ∈ D 1,2 .…”

“…which shows that the standard procedure for Lipschitz generators (e.g. the one from [17,Proposition 4.2]) is applicable. The uniqueness of the solution then follows.…”

Correction to: “Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting”

“…The proof of (Geiss and Steinicke (2018), Theorem 3.5) needs an extra step addressing the problem that our conditions on the generator are not sufficient to guarantee the existence of the considered optional projection:…”

“…Since lim x→0 h(a, b, x) = 0, one derives that lim K →∞ Y t − Y K t = 0, and in the same way it follows lim K →∞ Y t − Y K t = 0. Moreover, Theorem 3.4 and Lemma 5.1 in Geiss and Steinicke (2018) are only valid, if f n in Definition 3.3 exists. For the proof of Theorem 3.5 this does not cause a problem since we need these results for f K n only.…”