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 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 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 Derivatives pricing via p-optimal martingale measures: some extreme cases

2006 ◽  
Vol 43 (03) ◽  
pp. 634-651
Author(s):  
Marina Santacroce
Keyword(s):  
Asset Price ◽  
Martingale Measure ◽  
Power Utility ◽  
Derivatives Pricing ◽  
Trade Off ◽  
Continuous Semimartingale ◽  
Backward Equation ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Mean Variance

In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for the p-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.

scholarly journals Derivatives pricing via p-optimal martingale measures: some extreme cases

2006 ◽  
Vol 43 (3) ◽  
pp. 634-651 ◽  
Author(s):  
Marina Santacroce
Keyword(s):  
Asset Price ◽  
Martingale Measure ◽  
Power Utility ◽  
Derivatives Pricing ◽  
Trade Off ◽  
Continuous Semimartingale ◽  
Backward Equation ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Mean Variance

In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for the p-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.

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