arXiv subjects

Jerzy Zabczyk

Publications and source records attributed to Jerzy Zabczyk.

18 recordsLinked to original sources

arXiv2025

Mild solutions of HJB equations associated with cylindrical stable L\'evy noise in infinite dimensions

We study the optimal control of an infinite-dimensional stochastic system governed by an SDE in a separable Hilbert space driven by cylindrical stable noise. We establish the existence and uniqueness of a mild solution to the associated HJB equation. This result forms the basis for the proof of the Verification Theorem, which is the subject of ongoing research and will provide a sufficient condition for optimality.

math.PR↗

arXiv2020

Ergodicity of Burgers' system

We consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a unique invariant measure is investigated. The paper generalizes the results of Da Prato, Debussche and Temam, and Da Prato and Gatarek, dealing with one equation describing the turbulent motion only.

math.PR↗

arXiv2019

On generalized CIR equations

The paper is concerned with stochastic equations for the short rate process $R$ $$ dR(t)=F(R(t))dt+G(R(t-))dZ(t), $$ in the affine model of the bond prices. The equation is driven by a L\'evy martingale $Z$. It is shown that the discounted bond prices are local martingales if either $Z$ is a stable process of index $\alpha\in(1,2]$,\,$F(x)= ax +b, b\geq 0$, $G(x)=cx^{1/\alpha}, c>0$ or $Z$ must be a L\'evy martingale with positive jumps and trajectories of bounded variation, $F(x)= ax +b, b\geq 0$ and G is a constant. The result generalizes the well known Cox-Ingersoll-Ross result and extends the Vasicek result to non-negative short rates.

math.PR↗

arXiv2016

Completeness of bond market driven by Lévy process

The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a countable, dense subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the Lévy measure consists of a finite number of points. Explicit constructions of contingent claims which can not be replicated are provided.

math.PR↗

arXiv2016

On incompleteness of bond markets with infinite number of random factors

The completeness of a bond market model with infinite number of sources of randomness on a finite time interval in the Heath-Jarrow-Morton framework is studied. It is proved that the market is not complete. A construction of a bounded contingent claim, which can not be replicated, is provided.

q-fin.CP↗

arXiv2015

Forward rate models with linear volatilities

Existence of solutions to the Heath-Jarrow-Morton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions the Lévy process should necessarily be without the Gaussian part and without negative jumps. If this is the case then necessary and sufficient conditions for the existence are formulated either in terms of the behavior of the Lévy measure of the noise near the origin or the behavior of the Laplace exponent of the noise at infinity.

q-fin.MF↗

arXiv2015

Heath-Jarrow-Morton-Musiela equation with Lévy perturbation

The paper studies the Heath-Jarrow-Morton-Musiela equation of the bond market. The equation is analyzed in weighted spaces of functions defined on $[0,+\infty)$. Sufficient conditions for local and global existence are obtained . For equation with the linear diffusion term the conditions for global existence are close to the necessary ones.

q-fin.MF↗

arXiv2015

Regularity of Ornstein-Uhlenbeck processes driven by a L{é}vy white noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise "obtained by subordination of a Gaussian white noise". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.

math.PR↗

arXiv2013

Gauss-Markov processes on Hilbert spaces

K. Itô characterised in \cite{ito} zero-mean stationary Gauss Markov-processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of Itô in the case of Hilbert spaces: Gauss-Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coefficients. Choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup, the generator and its core thus providing an infinite-dimensional extension of the classical result of Courrège \cite{courrege} in the case of Gauss-Markov semigroups.

math.PR↗

arXiv2011

Structural properties of semilinear SPDEs driven by cylindrical stable processes

We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities.

math.AP↗

arXiv2011

Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal "energy" needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ ("energy" of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.

math.OC↗

arXiv2010

CDO term structure modelling with Levy processes and the relation to market models

This paper considers the modelling of collateralized debt obligations (CDOs). We propose a top-down model via forward rates generalizing Filipović, Overbeck and Schmidt (2009) to the case where the forward rates are driven by a finite dimensional Lévy process. The contribution of this work is twofold: we provide conditions for absence of arbitrage in this generalized framework. Furthermore, we study the relation to market models by embedding them in the forward rate framework in spirit of Brace, Gatarek and Musiela (1997).

q-fin.PR↗

arXiv2010

Heath-Jarrow-Morton-Musiela equation with linear volatility

The paper is concerned with the problem of existence of solutions for the Heath-Jarrow-Morton equation with linear volatility. Necessary conditions and sufficient conditions for the existence of weak solutions and strong solutions are provided. It is shown that the key role is played by the logarithmic growth conditions of the Laplace exponent.

math.PR↗

arXiv2010

Exponential mixing for some SPDEs with Lévy noise

We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical $α$-stable noises, $α\in (0,2)$, introduced in Priola-Zabczyk "Structural properties of semilinear SPDEs driven by cylindrical stable processes" (PTRF to appear). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.

math.AP↗

arXiv2010

Large deviations for stochastic PDE with Levy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Levy noise. We use general large deviations theorems of Varadhan and Bryc, viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Levy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.

math.PR↗

arXiv2009

Bonds with volatilities proportional to forward rates

The problem of existence of solution for the Heath-Jarrow-Morton equation with linear volatility and purely jump random factor is studied. Sufficient conditions for existence and non-existence of the solution in the class of bounded fields are formulated. It is shown that if the first derivative of the Levy-Khinchin exponent grows slower then logarithmic function then the answer is positive and if it is bounded from below by a fractional power function of any positive order then the answer is negative. Numerous examples including models with Levy measures of stable type are presented.

q-fin.CP↗

arXiv2009

On linear evolution equations with cylindrical Lévy noise

We study an infinite-dimensional Ornstein-Uhlenbeck process $(X_t)$ in a given Hilbert space $H$. This is driven by a cylindrical symmetric Lévy process without a Gaussian component and taking values in a Hilbert space $U$ which usually contains $H$. We give if and only if conditions under which $X_t$ takes values in $H$ for some $t>0$ or for all $t>0$. Moreover, we prove irreducibility for $(X_t)$.

math.AP↗

arXiv2008

Densities for Ornstein-Uhlenbeck processes with jumps

We consider an Ornstein-Uhlenbeck process with values in R^n driven by a Lévy process (Z_t) taking values in R^d with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the Lévy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the Lévy process is of $α$-stable type, $α\in (0,2)$, we show that such density is a $C^{\infty}$-function.

math.PR↗