arXiv subjects

Alexander Steinicke

Publications and source records attributed to Alexander Steinicke.

23 records · Page 2Linked to original sources

arXiv2019

Existence, Uniqueness and Malliavin Differentiability of Lévy-driven BSDEs with locally Lipschitz Driver

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.

math.PR↗

arXiv2019

Existence, Uniqueness and Comparison Results for BSDEs with Lévy Jumps in an Extended Monotonic Generator Setting

We show existence of a unique solution and a comparison theorem for a one-dimensional backward stochastic differential equation with jumps that emerge from a Lévy process. The considered generators obey a time-dependent extended monotonicity condition in the y-variable and have linear time-dependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we consider BSDEs where the Poisson random measure admits only jumps of size larger than $1/n$. We show convergence of their solutions to those of the original BSDE, as $n \to \infty.$ The proofs only rely on Itô's formula and the Bihari-LaSalle inequality and do not use Girsanov transforms.

math.PR↗

arXiv2016

Malliavin derivative of random functions and applications to Lévy driven BSDEs

We consider measurable $F: Ω\times \mathbb{R}^d \to \mathbb{R}$ where $F(\cdot, x)$ belongs for any $x$ to the Malliavin Sobolev space $\mathbb{D}_{1,2}$ (with respect to a Lévy process) and provide sufficient conditions on $F$ and $G_1,\ldots,G_d \in \mathbb{D}_{1,2}$ such that $F(\cdot, G_1,\ldots,G_d) \in \mathbb{D}_{1,2}.$ The above result is applied to show Malliavin differentiability of solutions to BSDEs (backward stochastic differential equations) driven by Lévy noise where the generator is given by a progressively measurable function $f(ω,t,y,z).$

math.PR↗

arXiv2016

$L_2$-variation of Lévy driven BSDEs with non-smooth terminal conditions

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 $(X_t)_{t\in[0,T]}$. 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.

math.PR↗

arXiv2014

Functionals of a Lévy Process on Canonical and Generic Probability Spaces

We develop an approach to Malliavin calculus for Lévy processes from the perspective of expressing a random variable $Y$ by a functional $F$ mapping from the Skorohod space of càdlàg functions to $\mathbb{R}$, such that $Y=F(X)$ where $X$ denotes the Lévy process. We also present a chain-rule-type application for random variables of the form $f(ω,Y(ω))$. An important tool for these results is a technique which allows us to transfer identities proved on the canonical probability space (in the sense of Solé et al.) associated to a Lévy process with triplet $(γ,σ,ν)$ to an arbitrary probability space $(Ω,\mathcal{F},\mathbb{P})$ which carries a Lévy process with the same triplet.

math.PR↗