It is a pleasure to announce that on March, 14 in Room 2BC/30 of the Department of Mathematics at the University of Padova (via Trieste, 63) there will be two seminars and the discussion of a PhD thesis.
The plan of the morning is as follows:
10am Prof. Carlo Sgarra (Politecnico di Milano):
"American options valuation in stochastic volatility models with transaction costs"
(joint with A. Cosso and D. Marazzina)
11am Prof Fausto Gozzi (LUISS Roma):
"HJB equations for stochastic control problems with delay in the control: regularity and feedback controls"
(joint with Federica Masiero)
12am Dr Matteo Basei will defense his PhD thesis titled
"Topics in stochastic control and differential game theory, with application to mathematical finance"
Here below the abstracts of the presentations.
ABSTRACT SGARRA
In the present paper we analyze the American option valuation problem in a stochastic volatility model when transaction costs are taken into account. We shall show that it can be formulated as a singular stochastic optimal control problem, proving the existence and uniqueness of the viscosity solution for the associated Hamilton-Jacobi-Bellman partial differential equation. Moreover, after performing a dimensionality reduction through a suitable choice of the Utility Function, we shall provide a numerical example illustrating how American options prices can be computed in the present modeling framework.
ABSTRACT GOZZI
Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than the the ones when the delay appears only in the state. This is particularly true when we look at the associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even in the simplified setting (introduced first by Vinter and Kwong [44] for the deterministic case) the HJB equation is an infinite dimensional second order semilinear Partial Differential Equation (PDE) that does not satisfy the so-called "structure condition" which substantially means that "the noise enters the system with the control". The absence of such condition, together with the lack of smoothing properties which is a common feature of problems with delay, prevents the use of the known techniques (based on Backward Stochastic Differential Equations (BSDEs) or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation and so no results on this direction have been proved till now.
In this paper we provide a result on existence of regular solutions of such kind of HJB equations and we use it to solve completely the corresponding control problem finding optimal feedback controls also in the more difficult case of pointwise delay. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results hold for a specific class of equations and data which arises naturally in many applied problems.
ABSTRACT BASEI:
We consider three problems in stochastic control and differential game theory, arising from practical situations in mathematical finance and energy markets. First, we address the problem of optimally exercising swing contracts in energy markets. Our main result consists in characterizing the value function as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. The case of contracts with penalties is straightforward. Conversely, the case of contracts with strict constraints gives rise to stochastic control problems where a non-standard integral constraint is present: we get the anticipated characterization by considering a suitable sequence of unconstrained problems. The approximation result is proved for a general class of problems with an integral constraint on the controls. Then, we consider a retailer who has to decide when and how to intervene and adjust the price of the energy he sells, in order to maximize his earnings. The intervention costs can be either fixed or depending on the market share. In the first case, we get a standard impulsive control problem and we characterize the value function and the optimal price policy. In the second case, classical theory cannot be applied, due to the singularities of the penalty function; we then outline an approximation argument and we finally consider stronger conditions on the controls to characterize the optimal policy. Finally, we focus on a general class of non-zero-sum stochastic differential games with impulse controls. After defining a rigorous framework for such problems, we prove a verification theorem: if a couple of functions is regular enough and satisfies a suitable system of quasi-variational inequalities, it coincides with the value functions of the problem and a characterization of the Nash equilibria is possible. We conclude by a detailed example: we investigate the existence of equilibria in the case where two countries, with different goals, can affect the exchange rate between the corresponding currencies.
Tiziano
-------------------------------------------------------------------------- Tiziano Vargiolu Dipartimento di Matematica Phone: +39 049 8271383 Universita' di Padova Fax: +39 049 8271428 Via Trieste, 63 E-mail: vargiolu@math.unipd.it I-35121 Padova (Italy) WWW: http://www.math.unipd.it/~vargiolu --------------------------------------------------------------------------