Stochastic differential equations with polar-decomposed Lévy measures and applications to stochastic optimization

Abstract

This paper is concerned with the optimal control of jump-type stochastic differential equations associated with polar-decomposed Lévy measures with the feature of explicit construction on the jump term. The concrete construction is then utilized for analysis of two portfolio optimization problems for financial market models driven by stable-like processes.

Anticipating linear stochastic differential equations driven by a Lévy process

“…Here X 0 is a random variable, a, b and v(y), for any y ∈ R, y = 0, are random processes no necessarily adapted to the underlying filtration, W is the canonical Wiener process, N is the canonical Poisson random measure with parameter ν (see Section 2.2 for details), d Ñ (t, y) := dN (t, y) − dt ν(dy), and the integral with respect to W (respectively the integrals with respect to N and Ñ ) is in the Skorohod sense (respectively are pathwise defined). In the adapted case (i.e., deterministic initial condition and adapted coefficients to the filtration generated by W and N ), the stochastic differential equation (1.1) with no necessarily linear coefficients has been analyzed by several authors (see, for instance, [1,2,3,9,10,14,15,22,23]). For example, Ikeda and Watanabe [10] have considered this equation with no necessarily linear coefficients and have used the Picard iteration procedure and Gronwall´s lemma to show existence and uniqueness of the solution, respectively.…”

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

“…In Ref. [5], we examined the sufficient maximum principle and considered the portfolio optimization problems in the financial markets modeled by stablelike Lévy processes with polar-decomposed Lévy measures.…”