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.

scholarly journals The identification problem for BSDEs driven by possibly non-quasi-left-continuous random measures

2020 ◽  
Vol 20 (06) ◽  
pp. 2040011
Author(s):  
Elena Bandini ◽  
Francesco Russo
Keyword(s):  
Random Measure ◽  
Jump Process ◽  
Identification Problem ◽  
Local Martingale ◽  
Random Measures ◽  
Underlying Process ◽  
Process Solution ◽  
Deterministic Function ◽  
Singular Coefficients

In this paper, we focus on the so-called identification problem for a BSDE driven by a continuous local martingale and a possibly non-quasi-left-continuous random measure. Supposing that a solution [Formula: see text] of a BSDE is such that [Formula: see text] where [Formula: see text] is an underlying process and [Formula: see text] is a deterministic function, solving the identification problem consists in determining [Formula: see text] and [Formula: see text] in terms of [Formula: see text]. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when [Formula: see text] is a non-semimartingale jump process solution of an SDE with singular coefficients.

scholarly journals COMPLETENESS OF BOND MARKET DRIVEN BY LÉVY PROCESS

2010 ◽  
Vol 13 (05) ◽  
pp. 635-656 ◽  
Author(s):  
MICHAŁ BARSKI ◽  
JERZY ZABCZYK
Keyword(s):  
Dense Subset ◽  
Random Measure ◽  
Bond Market ◽  
Contingent Claims ◽  
Market Model ◽  
Time Interval ◽  
Poisson Random Measure ◽  
Finite Time Interval ◽  
Lévy Measure ◽  
Market Driven

The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a countable, dense subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the Lévy measure consists of a finite number of points. Explicit constructions of contingent claims which cannot be replicated are provided.

scholarly journals On intensities of modulated Cox measures

1998 ◽  
Vol 11 (3) ◽  
pp. 411-423 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Stochastic Process ◽  
Stochastic Models ◽  
Queueing Systems ◽  
Random Measure ◽  
Random Measures ◽  
Explicit Formulas ◽  
State Dependent

In this paper we introduce and study functionals of the intensities of random measures modulated by a stochastic process ξ, which occur in applications to stochastic models and telecommunications. Modulation of a random measure by ξ is specified for marked Cox measures. Particular cases of modulation by ξ as semi-Markov and semiregenerative processes enabled us to obtain explicit formulas for the named intensities. Examples in queueing (systems with state dependent parameters, Little's and Campbell's formulas) demonstrate the use of the results.

scholarly journals Unifying the Dynkin and Lebesgue–Stieltjes formulae

2017 ◽  
Vol 54 (1) ◽  
pp. 252-266 ◽  
Author(s):  
Offer Kella ◽  
Marc Yor
Keyword(s):  
Bounded Variation ◽  
Sufficient Conditions ◽  
Jump Diffusion ◽  
Continuous Functions ◽  
Local Martingale ◽  
Functions Of Bounded Variation ◽  
Deterministic Function ◽  
Stieltjes Integration ◽  
Change Of Variable ◽  
Martingale Convergence

AbstractWe establish a local martingaleMassociate withf(X,Y) under some restrictions onf, whereYis a process of bounded variation (on compact intervals) and eitherXis a jump diffusion (a special case being a Lévy process) orXis some general (càdlàg metric-space valued) Markov process. In the latter case,fis restricted to the formf(x,y)=∑k=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes anL2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both inL2and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

scholarly journals The Markov Chain Market

2003 ◽  
Vol 33 (02) ◽  
pp. 265-287 ◽  
Author(s):  
Ragnar Norberg
Keyword(s):  
Markov Chain ◽  
Financial Market ◽  
Homogeneous Markov Chain ◽  
Risk Minimization ◽  
Interest Rate Derivatives ◽  
First Order ◽  
Chain Conditions ◽  
The Matrix ◽  
Market Driven ◽  
Absence Of Arbitrage

We consider a financial market driven by a continuous time homogeneous Markov chain. Conditions for absence of arbitrage and for completeness are spelled out, non-arbitrage pricing of derivatives is discussed, and details are worked out for some cases. Closed form expressions are obtained for interest rate derivatives. Computations typically amount to solving a set of first order partial differential equations. An excursion into risk minimization in the incomplete case illustrates the matrix techniques that are instrumental in the model.

Sign in / Sign up

Export Citation Format

Share Document