Malliavin Calculus and Optimal Control of Stochastic Volterra Equations

Abstract

Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore, classical methods, such as dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that using Malliavin calculus, it is possible to formulate modified functional types of maximum principle suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a Mangasarian sufficient condition and a Pontryagin-type maximum principle of this type, and then, we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory.

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

“…The literature on Gaussian Hilbert space, white noise analysis, and its relevance to Malliavin calculus is vast; we limit ourselves here to citing [17,[36][37][38][39][40][41], and the papers cited there.…”

“…Fix a Hilbert space L over R with inner product ·, · L . Then (see [17,42,43]) there is a probability space (Ω, F , P), depending on L , and a real linear mapping Φ : L −→ L 2 (Ω, F , P), i.e., a Gaussian field as specified in Definition 2.10, satisfying…”

“…We refer the reader to the papers [4][5][6][7][8], and also see [9,10]. Of more recent papers dealing with results which have motivated our present paper are [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Malliavin method for optimal investment in financial markets with memory

“…We refer to [21] for Malliavin calculus applied to optimal control of stochastic partial differential equations with jumps, and to [22] for Malliavin calculus applied to optimal control of stochastic Volterra equations but without the mean-field framework.…”

“…In the present paper following the idea of [22] we study the optimal investment problem in a finance market modeled by mean-field stochastic Volterra equation, in which both the financial market with an infinite number of rational agents in competition and non-Markov type solutions are all taken into account. The method of Malliavin calculus to solve this optimal investment problem in this paper is still adopted.…”