01:30 PM - 01:55 PM
Optimal Consumption, investment and life-insurance purchase in a diffusive financial market with stochastic coefficients
We find the optimal consumption, investment and life-insurance purchase and selection strategies for economic agents whose lifetime is uncertain. Such agents are allowed to invest their savings in a financial market comprised of one risk-free security and one risky asset. The financial asset prices evolve according to a linear stochastic differential equation modulated by an eventually nonlinear stochastic differential equation representing the evolution of an external process modeling the state of the economy in a given instant of time. Additionally, the Brownian motions driving each of these equations are allowed to be correlated. We assume that the life-insurance market consist of a fixed number of providers offering pairwise distinct contracts. We resort to dynamic programming techniques to characterize the solutions to the problem described above for general family of utility functions, studying the case of discounted constant relative risk aversion utility functions with some more detail.
01:55 PM - 02:20 PM
Dynamic Programming for semi-Markov modulated SDEs
We consider a stochastic optimal control problem with state variable dynamics described by a stochastic differential equation of diffusive type modulated by a semi-Markov process with a finite state space. The time horizon is both deterministic and finite. We obtain a dynamic programming principle and use it to derive the analogue of the classical Hamilton-Jacobi-Bellman equation: a system of coupled second order partial differential equations, one for each state of the underlying semi-Markov process. We illustrate our results with an application to Mathematical Finance: the generalization of Merton's optimal consumption-investment problem to financial markets with semi-Markov switching.
02:20 PM - 02:45 PM
Stochastic Optimal Control of Hybrid Systems with Impulses in a World of Regime Switches and Paradigm Shifts, Impulsiveness, in Finance, Economics and Nature
We contribute to modern OR by hybrid, e.g., mixed continuous-discrete dynamics of stochastic differential equations with jumps and to its optimal control. These hybrid systems allow for the representation of random regime switches or paradigm shifts, and are of growing importance in economics, finance, science, engineering, neuroscience, bio-science, medicine and earth-sciences. We introduce some new approaches to this area of stochastic optimal control and present results. One is analytical and bases on the finding of optimality conditions and, in certain cases, closed-form solutions. We further discuss aspects of differences in information, given by delay or insider information. The presentation ends with a conclusion and an outlook to future studies.
02:45 PM - 03:10 PM
The existence of solution for a stochastic optimal control problem with a random horizon
We provide sufficient conditions guaranteeing the existence of solution to a stochastic optimal control problem with diffusive state variable dynamics and a random horizon. We resort to the weak formulation for the optimal control problem under consideration herein and make use of weak convergence techniques together with Roxin’s condition to obtain the desired existence result.