Uniform Integrability of a Single Jump Local Martingale with State-Dependent Characteristics

2017 ◽  
Author(s):  
Michael Schatz ◽  
Didier Sornette
Keyword(s):  
Local Martingale ◽  
Uniform Integrability ◽  
State Dependent

scholarly journals Optimal Portfolios in Lévy Markets under State-Dependent Bounded Utility Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-27
Author(s):  
José E. Figueroa-López ◽  
Jin Ma
Keyword(s):  
Random Measure ◽  
Local Martingale ◽  
Risk Minimization ◽  
Lévy Measure ◽  
State Dependent ◽  
Deterministic Function ◽  
Bounded State ◽  
Market Driven ◽  
Portfolio Optimization Problem ◽  
Merton’S Portfolio

Motivated by the so-called shortfall risk minimization problem, we consider Merton's portfolio optimization problem in a non-Markovian market driven by a Lévy process, with a bounded state-dependent utility function. Following the usual dual variational approach, we show that the domain of the dual problem enjoys an explicit “parametrization,” built on a multiplicative optional decomposition for nonnegative supermartingales due to Föllmer and Kramkov (1997). As a key step we prove a closure property for integrals with respect to a fixed Poisson random measure, extending a result by Mémin (1980). In the case where either the Lévy measure ν of Z has finite number of atoms or ΔSt/St−=ζtϑ(ΔZt) for a process ζ and a deterministic function ϑ, we characterize explicitly the admissible trading strategies and show that the dual solution is a risk-neutral local martingale.

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