Frank Proske

White noise analysis for Lévy processes

We construct a white noise theory for Levy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark–Haussmann–Ocone formula for Levy processes F(ω)=E[F]+∑m⩾1∫0TE[Dt(m)F|Ft]♢Y•t(m)dt. Here E[F] is the generalized expectation, the operators Dt(m)F,m⩾1 are (generalized) Malliavin derivatives, ♢ is the Wick product and for all m⩾1Y•t(m) is the white noise of power jump processes Yt(m). In particular, Y•t(1) is the white noise of the Levy process. The formula holds for all F∈G∗⊃L2(μ), where G∗ is a space of stochastic distributions and μ is a white noise probability measure. Finally, we give an application of this formula to partial observation minimal variance hedging problems in financial markets driven by Levy processes.

MALLIAVIN CALCULUS AND ANTICIPATIVE ITÔ FORMULAE FOR LÉVY PROCESSES

We introduce the forward integral with respect to a pure jump Levy process and prove an Ito formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Ito formula for the Skorohod integral.

Stochastic Partial Differential Equations driven by Lévy Space-Time White Noise

In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Levy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Levy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d\leq 3, then this solution can be represented as a classical random field in L2(\mu ), where \mu is the probability law of the Levy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Levy Hermite transform.

Optimal portfolio for an insider in a market driven by Levy processes

We consider a financial market driven by a Lévy process with filtration . An insider in this market is an agent who has access to more information than an honest trader. Mathematically, this is modelled by allowing a strategy of an insider to be adapted to a bigger filtration . The corresponding anticipating stochastic differential equation of the wealth is interpreted in the sense of forward integrals. In this framework, we study the optimal portfolio problem of an insider with logarithmic utility function. Explicit results are given in the case where the jumps are generated by a Poisson process. §Dedicated to the memory of Axel Grorud.

Central limit theorems for generalized set-valued random variables

Abstract We give central limit theorems for generalized set-valued random variables whose level sets are compact both in R d or in a Banach space under milder conditions than those obtained recently by the latter two authors.

Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations (SPDE) with jumps. This is a type of equations, which appear as adjoint equations in the maximum principle approach to optimal control of systems described by SPDE driven by Lévy processes.

On explicit strong solution of Itô-SDE's and the Donsker delta function of a diffusion.

We determine a new explicit representation of strong solutions of Ito-diffusions and elicit its correspondence to the general stochastic transport equation. We apply this formula to deduce an explicit Donsker delta function of a diffusion.

Maximum principles for jump diffusion processes with infinite horizon

Abstract We prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information. The results are applied to an optimal consumption and portfolio problem in infinite horizon.

Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R d ∋ x 7−→ φ s,t (x) ∈ R d , s, t ∈ R, for a stochastic differential equation (SDE) of the form dXt = b(t, Xt) dt + dBt, s, t ∈ R, Xs = x ∈ R d . The above SDE is driven by a bounded measurable drift coefficientb : R × R d → R d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow φs,t(·) of the SDE lives in the space L 2 (; W 1,p (R d , w)) for all s, t and all p > 1, where W 1,p (R d , w) denotes a weighted Sobolev space with weight w possessing a p-th moment with respect to Lebesgue measure on R d . This result is counter-intuitive, since the dominant ‘culture’ in stochastic (and deterministic) dynamical systems is that the flow ‘inherits’ its spatial regularity from the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation � dtu(t, x)+ (b(t, x) · Du(t, x))dt + P d=1 ei · Du(t, x) ◦ dB i t = 0, u(0, x) = u0(x), where b is bounded and measurable, u0 is C 1 b and {ei} d=1 a basis for R d . It is well-known that the deterministic counter part of the above equation does not in general have a solution. Using stochastic perturbations and our analysis of the above SDE, we establish a deterministic flow of Sobolev diffeomorphisms for classical one-dimensional (deterministic) ODE’s driven by discontinuous vector fields. Furthermore, and as a corollary of the latter result, we construct a Sobolev stochastic flow of diffeomorphisms for one-dimensional SDE’s driven by discontinuous diffusion coefficients.

Central limit theorem for Banach space valued fuzzy random variables

In this paper we prove a central limit theorem for Borel measurable nonseparably valued random elements in the case of Banach space valued fuzzy random variables.