Carlo Marinelli

LOCAL WELL-POSEDNESS OF MUSIELA’S SPDE WITH LÉVY NOISE

We determine sufficient conditions on the volatility coefficient of Musiela’s stochastic partial differential equation driven by an infinite dimensional Levy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight.

The stochastic goodwill problem

Stochastic control problems related to optimal advertising under uncertainty are considered. In particular, we determine the optimal strategies for the problem of maximizing the utility of goodwill at launch time and minimizing the disutility of a stream of advertising costs that extends until the launch time. We also consider some generalizations, such as problems with constrained budget, optimization under partial information, and discretionary launching.

Subordinated exchange rate models: evidence for heavy tailed distributions and long-range dependence

We investigate the main properties of high-frequency exchange rate data in the setting of stochastic subordination and stable modeling, focusing on heavy-tailedness and long memory, together with their dependence on the sampling period. We show that the intrinsic time process exhibits strong long-range dependence and has increments well described by a Weibull law, while the return series in intrinsic time has weak long memory and is well approximated by a stable Levy motion. We also show that the stable domain of attraction offers a good fit to the returns in physical time, which leads us to consider as a realistic model for exchange rate data a process Z(t) subordinated to an @a-stable Levy motion S(t) (possibly fractional stable) by a long-memory intrinsic time process T(t) with Weibull-distributed increments.

Existence of Weak Solutions for a Class of Semilinear Stochastic Wave Equations

We prove existence of weak solutions (in the probabilistic sense) for a general class of stochastic semilinear wave equations on bounded domains of

Subordinated Stock Price Models: Heavy Tails and Long-Range Dependence in the High-frequency Deutsche Bank Price Record

R^d

On Maximal Inequalities for Purely Discontinuous Martingales in Infinite Dimensions

driven by a possibly discontinuous square integrable martingale.

A variational approach to dissipative SPDEs with singular drift

Following a previous study on subordinated exchange rate models, we investigate the main properties of the high—frequency Deutsche Bank price record in the setting of stochastic subordination and stable modeling, focusing on heavytailedness and long memory, together with their dependence on the sampling period. We find that the market time process has increments described well by Gamma distributions, and that the log price process in intrinsic time can be approximated, at different time scales, by α—stable Levy processes. Most importantly, long—range dependence with strong intensity is present in the market time process, with an estimated Hurst index H ≈ 0.9. Finally, the stable domain of attraction offers a good fit of the returns in physical time, which display weak long memory. As a consequence, we propose as a realistic model for the stock prices a process Z(t) subordinated to an α-stable Levy motion S(t) by a long—memory intrinsic time process T(t) with Gamma—distributed increments.

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.

STRONG SOLUTIONS FOR STOCHASTIC POROUS MEDIA EQUATIONS WITH JUMPS

We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the evaluation operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations. AMS Subject Classification: 60H15; 47H06; 46N30.

Strong solutions to SPDEs with monotone drift in divergence form

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in