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arXiv · 1505.04921

Optimal control of predictive mean-field equations and applications to finance

Bernt Øksendal·Agnès Sulem
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Abstract

We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process $X(t)$ and a \emph{predictive mean-field} backward SDE (BSDE) in the unknowns $Y(t), Z(t), K(t,\cdot)$. The driver of the BSDE at time $t$ may depend not just upon the unknown processes $Y(t), Z(t), K(t,\cdot)$, but also on the predicted future value $Y(t+δ)$, defined by the conditional expectation $A(t):= E[Y(t+δ) | \mathcal{F}_t]$. \\ We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems:\\ (i) Optimal portfolio in a financial market with an \emph{insider influenced asset price process.} \\ (ii) Optimal consumption rate from a cash flow modeled as a geometric It\^ o-L\' evy SDE, with respect to \emph{predictive recursive utility}.

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Bernt Øksendal, Agnès Sulem. 2015-05-19. Optimal control of predictive mean-field equations and applications to finance. https://arxiv.org/abs/1505.04921

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