scholarly journals Backward stochastic PDEs related to the utility maximization problem

2010 ◽  
Vol 17 (4) ◽  
pp. 705-740
Author(s):  
Michael Mania ◽  
Revaz Tevzadze
Keyword(s):  
Differential Equation ◽  
Utility Maximization ◽  
Market Model ◽  
Maximization Problem ◽  
Programming Approach ◽  
Stochastic Pdes ◽  
Dynamic Programming Approach ◽  
General Utility ◽  
Continuous Semimartingale ◽  
Utility Maximization Problem

Abstract We study utility maximization problem for general utility functions using the dynamic programming approach. An incomplete financial market model is considered, where the dynamics of asset prices is described by an -valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic partial differential equation related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward stochastic differential equation. The cases of power, exponential and logarithmic utilities are considered as examples.

scholarly journals UTILITY MAXIMIZATION WITH RANDOM HORIZON: A BSDE APPROACH

2015 ◽  
Vol 18 (07) ◽  
pp. 1550045 ◽  
Author(s):  
MONIQUE JEANBLANC ◽  
THIBAUT MASTROLIA ◽  
DYLAN POSSAMAÏ ◽  
ANTHONY RÉVEILLAC
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Utility Maximization ◽  
Backward Stochastic Differential Equation ◽  
Maximization Problem ◽  
Default Time ◽  
Random Horizon ◽  
Singular Coefficients ◽  
Uniqueness Of The Solution ◽  
Utility Maximization Problem

In this paper, we study a utility maximization problem with random horizon and reduce it to the analysis of a specific backward stochastic differential equation (BSDE), which we call BSDE with singular coefficients, when the support of the default time is assumed to be bounded. We prove existence and uniqueness of the solution for the equation under interest. Our results are illustrated by numerical simulations.

A Semimartingale Bellman Equation and the Variance-Optimal Martingale Measure

2000 ◽  
Vol 7 (4) ◽  
pp. 765-792 ◽  
Author(s):  
M. Mania ◽  
R. Tevzadze
Keyword(s):  
Bellman Equation ◽  
Generalized Solution ◽  
Market Model ◽  
Martingale Measure ◽  
Programming Approach ◽  
Dynamic Programming Approach ◽  
Volatility Models ◽  
Continuous Semimartingale ◽  
Backward Equation ◽  
Equivalent Change Of Measure

Abstract We consider a financial market model, where the dynamics of asset prices is given by an Rm -valued continuous semimartingale. Using the dynamic programming approach we obtain an explicit description of the variance optimal martingale measure in terms of the value process of a suitable problem of an optimal equivalent change of measure and show that this value process uniquely solves the corresponding semimartingale backward equation. This result is applied to prove the existence of a unique generalized solution of Bellman's equation for stochastic volatility models, which is used to determine the variance-optimal martingale measure.

scholarly journals OPTIMAL INVESTMENT ON FINITE HORIZON WITH RANDOM DISCRETE ORDER FLOW IN ILLIQUID MARKETS

2011 ◽  
Vol 14 (01) ◽  
pp. 17-40 ◽  
Author(s):  
PAUL GASSIAT ◽  
HUYÊN PHAM ◽  
MIHAI SÎRBU
Keyword(s):  
Dynamic Programming ◽  
Optimal Investment ◽  
Market Model ◽  
Programming Approach ◽  
Order Flow ◽  
Short Sale Constraints ◽  
Dynamic Programming Approach ◽  
Short Sale ◽  
Illiquid Markets ◽  
Investor Trading

We study the problem of optimal portfolio selection in an illiquid market with discrete order flow. In this market, bids and offers are not available at any time but trading occurs more frequently near a terminal horizon. The investor can observe and trade the risky asset only at exogenous random times corresponding to the order flow given by an inhomogenous Poisson process. By using a direct dynamic programming approach, we first derive and solve the fixed point dynamic programming equation satisfied by the value function, and then perform a verification argument which provides the existence and characterization of optimal trading strategies. We prove the convergence of the optimal performance, when the deterministic intensity of the order flow approaches infinity at any time, to the optimal expected utility for an investor trading continuously in a perfectly liquid market model with no-short sale constraints.

scholarly journals The Rayleigh quotient and dynamic programming

1978 ◽  
Vol 1 (4) ◽  
pp. 401-405
Author(s):  
Richard Bellman
Keyword(s):  
Differential Equation ◽  
Dynamic Programming ◽  
Nonlinear Partial Differential Equation ◽  
Rayleigh Quotient ◽  
Programming Approach ◽  
Nonlinear Characteristic ◽  
Dynamic Programming Approach ◽  
Computational Aspects ◽  
Characteristic Values ◽  
Matrix Problems

The purpose of this paper is to derive a nonlinear partial differential equation for whichλgiven by (1.3), is one value of the solution. In Section 2, we derive this equation using a straightforward dynamic programming approach. In Section 3, we discuss some computational aspects of derermining the solution of this equation. In Section 4, we show that the same method may be applied to the nonlinear characteristic value problem. In Section 5, we discuss how the method may by applied to find the higher characteristic values. In Section 5, we discuss how the same method may be applied to some matrix problems. Finally, in Section 7, we discuss selective computation.

scholarly journals Dual Formulation of the Utility Maximization Problem Under Transaction Costs

2001 ◽  
Vol 11 (4) ◽  
pp. 1353-1383 ◽  
Author(s):  
Griselda Deelstra ◽  
Huyên Pham ◽  
Nizar Touzi
Keyword(s):  
Transaction Costs ◽  
Utility Maximization ◽  
Maximization Problem ◽  
Dual Formulation ◽  
Utility Maximization Problem
Sign in / Sign up

Export Citation Format

Share Document