Backward stochastic partial differential equations related to utility maximization and hedging

We study the utility maximization problem, the problem of minimization of the hedging error and the corresponding dual problems using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an ℝ d -valued continuous semimart...

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Bibliographic Details
Published inJournal of mathematical sciences (New York, N.Y.) Vol. 153; no. 3; pp. 291 - 380
Main Authors Mania, M., Tevzadze, R.
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.09.2008
Subjects
Mathematics
Mathematics and Statistics
Martingale Measure
Stochastic Volatility Model
Hedging
Utility Maximization Problem
Local Martingale
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Summary:We study the utility maximization problem, the problem of minimization of the hedging error and the corresponding dual problems using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an ℝ d -valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic PDEs for the value functions related to these problems, and for the primal problem, we show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward SDE. As examples we consider the mean-variance hedging problem and the cases of power, exponential, logarithmic utilities, and corresponding dual problems.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-008-9129-9