Functionals of a Lévy Process on Canonical and Generic Probability Spaces

Probab Uncertain Quant Risk

2018

Self Cite

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 .

Section: Proof Of Theorem 34mentioning

confidence: 99%

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

Geiß

Probab Uncertain Quant Risk

2018

Self Cite

“…• For a jointly measurable and adapted generator f : Ω×[0, T ]×R×R×L 2 (ν) → R we have by [26,Theorem 3.4] that there exists a jointly measurable g f :…”

Section: Existence Result Bounds and Malliavin Differentiability Formentioning

confidence: 99%

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

Geiß

2019

Stochastics

Self Cite

We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a locally Lipschitz condition in the Z and U variable. This includes settings of linear, quadratic and exponential growths in those variables.Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value ξ and its Malliavin derivative Dξ.Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in U. BSDEs of the latter type find use in exponential utility maximization.

“…We have that up to indistinguishability. (For details on this representation, see [ 27 ].) Denoting the Lévy process with cutoff Lévy measure again by (see Sect.…”

Section: Appendixmentioning

confidence: 99%

$${{\varvec{L}}}^{{\varvec{p}}}$$-Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

Kremsner

2020

J Theor Probab

Self Cite

We present a unified approach to $$L^p$$ L p -solutions ($$p > 1$$ p > 1 ) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.