- Vivi Padova
- Il Bo
Monday 14 March 2016 h 10:00, Room 2BC30
"Half day in stochastic processes and mathematical finance" and the discussion of a PhD thesis
10:00 - Prof. Carlo Sgarra (Politecnico di Milano)
"American options valuation in stochastic volatility models with transaction costs"
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.
(joint with A. Cosso and D. Marazzina)
11:00 - Prof. Fausto Gozzi (LUISS Roma)
"HJB equations for stochastic control problems with delay in the control: regularity and feedback controls"
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  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.
(joint with Federica Masiero)
12:00 - Dr. Matteo Basei will defense his PhD thesis titled
"Topics in stochastic control and differential game theory, with application to mathematical finance"