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Christel Geiss

Publications and source records attributed to Christel Geiss.

14 recordsLinked to original sources
arXiv2023

Product formulas for multiple stochastic integrals associated with L\'evy processes

In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a L\'evy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider L\'evy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

math.PR↗
arXiv2020

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

Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk B n from the underlying Brownian motion B by Skorohod embedding, one can show L 2 convergence of the corresponding solutions (Y n , Z n) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function in C 2,$α$. 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 PDE associated to the FBSDE as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by stochastic methods.

math.PR↗
arXiv2019

Donsker-Type Theorem for BSDEs: Rate of Convergence

In this paper, we study in the Markovian case the rate of convergence in the Wasserstein distance of an approximation of the solution to a BSDE given by a BSDE which is driven by a scaled random walk as introduced in Briand, Delyon and M{é}min (Electron. Comm. Probab. 6(2001),1-14).

math.PR↗
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↗
arXiv2018

Random walk approximation of BSDEs with H{ö}lder continuous terminal condition

In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally H{ö}lder continuous function of the Brownian motion. We state the rate of the L 2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.

math.PR↗
arXiv2018

Product and Moment Formulas for Iterated Stochastic Integrals (associated with Lévy Processes)

In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Lévy process. We propose a new approach applying the theory of compensated-covariation stable families of martingales. Our main tool is a representation formula for products of elements of a compensated-covariation stable family, which enables to consider Lévy processes, with both jumps and Gaussian part.

math.PR↗
arXiv2017

On first exit times and their means for Brownian bridges

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

math.PR↗
arXiv2017

Simulation of BSDEs with jumps by Wiener Chaos Expansion

We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.

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↗
arXiv2012

A note on Malliavin fractional smoothness for Lévy processes and approximation

Assume a Lévy process $X$ on the time interval $[0,1]$ that is an $L_2$-martingale and let $Y$ be either its stochastic exponential or $X$ itself. We consider Riemann-approximations of certain stochastic integrals driven by $Y$ and relate the $L_2$-approximation rates to the Malliavin fractional smoothness of the integral to be approximated. The Malliavin fractional smoothness is described by Besov spaces generated with the real interpolation method.

math.PR↗
arXiv2008

Denseness of certain smooth Lévy functionals in $\DD_{1,2}$

The Malliavin derivative for a Lévy process $(X_t)$ can be defined on the space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\Om) \supseteq \mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space $\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\DD_{1,2}$ into $\DD_{1,2}.$

math.PR↗