G. B. Di Masi

Mean-variance hedging of options on stocks with Markov volatilities

We consider the problem of hedging an European call option for a diffusion model where drift and volatility are functions of a Markov jump process. The market is thus incomplete implying that perfect hedging is not possible. To derive a hedging strategy, we follow the approach based on the idea of hedging under a mean-variance criterion as suggested by Follmer, Sondermann, and Schweizer. This also leads to a generalization of the Black–Scholes formula for the corresponding option price which, for the simplest case when the jump process has only two states, is given by an explicit expression involving the distribution of the integrated telegraph signal (known also as the Kac process). In the Appendix we derive this distribution by simple considerations based on properties of the order statistics.

Infinite horizon risk sensitive control of discrete time Markov processes with small risk

A control problem with risk sensitive ergodic performance criterion is considered for a discrete time Feller process. Using assumptions of uniform ergodicity and small risk factor, the existence and uniqueness of the solution to the Bellman equation is proved. Uniform approximations to such solution in terms of discounted cost and discounted game problems are also shown.

Hedging of Options under Discrete Observation on Assets with Stochastic Volatility

The paper considers the hedging of contingent claims on assets with stochastic volatilities when the asset price is only observable at discrete time instants. Explicit formulae are given for risk-minimizing hedging strategies.

Ergodicity of hidden Markov models

In this paper we study ergodic properties of hidden Markov models with a generalized observation structure. In particular sufficient conditions for the existence of a unique invariant measure for the pair filter-observation are given. Furthermore, necessary and sufficient conditions for the existence of a unique invariant measure of the triple state-observation-filter are provided in terms of asymptotic stability in probability of incorrectly initialized filters. We also study the asymptotic properties of the filter and of the state estimator based on the observations as well as on the knowledge of the initial state. Their connection with minimal and maximal invariant measures is also studied.

Risk sensitive control of discrete time partially observed markov processes with infinite horizon

In this paper existence of solutions to the Bellman equation corresponding to risk sensitive control of partially observed discrete time Markov processes is shown; this in turn leads to the existence of optimal strategies. The method used in the paper is based on discounted risk sensitive approximation

Almost sure optimality and optimality in probabilityfor stochastic control problems over aninfinite time horizon

A pathwise optimality criterion is proposed for stochastic control problems in order toreduce the risk connected with the fluctuations of the cost around its expected value. Thisapproach may be of relevance also in economic applications, where risky situations appearparticularly dangerous. Some examples of applications are examined, in particular for thelinear quadratic Gaussian model.

Design of an L.Q.G. controller for single point moored large tankers

Abstract Single point mooring is a commonly used method to keep large tankers moored at offshore loading terminals. When environmental forces increase however, loading operations become increasingly difficult. Dangerous oscillations are observed, with periods of several hundreds of seconds, which may cause mooring line breakage, wear and tear of the loading and mooring system etc. To counteract them the use of an active control system has been suggested. The system commands the main propeller and a bank of lateral thrusters placed in bow and astern position. The objectives and structure of such a control system are quite different from dynamic positioning systems currently used for drilling and support vessels.

Backward representation for nonstationary Markov processes with finite state space

Abstract We consider a problem of time reversal for a nonstationary continuous-time Markov processes with k states. The proposed approach is strongly based on the backward representation of the Gaussian analog of the original Markov process.

Generalized Finite-Dimensional Filters in Discrete Time

Nonlinear filtering problems in discrete-time are studied using the Bayesian approach. The concept of a generalized finite-dimensional filter is introduced and it is shown that under suitable assumptions such a filter exists and can be implemented in a recursive way. The resulting technique is successfully applied to various situations.

An Approximation to Optimal Nonlinear Filtering with Discontinuous Observations

The paper deals with a possible approach to the problem of finite- dimensional recursive filtering in the nonlinear case, when the signal is a diffusion process and the observations have both continuous and discontinuous components. The approach used consists in the approximation of the optimal filter by means of a converging sequence of finite-dimensional filters. These are given in terms of funtionals of continuous-time Markov chains and can be recursively computed via a finite-dimensional Zakai equation.