TY - JOUR

T1 - Explicit solution of relative entropy weighted control

AU - Bierkens, Joris

AU - Kappen, Hilbert J.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider the minimization over probability measures of the expected value of a random variable, regularized by relative entropy with respect to a given probability distribution. In the general setting we provide a complete characterization of the situations in which a finite optimal value exists and the situations in which a minimizing probability distribution exists. Specializing to the case where the underlying probability distribution is Wiener measure, we characterize finite relative entropy changes of measure in terms of square integrability of the corresponding change of drift. For the optimal change of measure for the relative entropy weighted optimization, an expression involving the Malliavin derivative of the cost random variable is derived. The theory is illustrated by its application to several examples, including the case where the cost variable is the maximum of a standard Brownian motion over a finite time horizon. For this example we obtain an exact optimal drift, as well as an approximation of the optimal drift through a Monte-Carlo algorithm.

AB - We consider the minimization over probability measures of the expected value of a random variable, regularized by relative entropy with respect to a given probability distribution. In the general setting we provide a complete characterization of the situations in which a finite optimal value exists and the situations in which a minimizing probability distribution exists. Specializing to the case where the underlying probability distribution is Wiener measure, we characterize finite relative entropy changes of measure in terms of square integrability of the corresponding change of drift. For the optimal change of measure for the relative entropy weighted optimization, an expression involving the Malliavin derivative of the cost random variable is derived. The theory is illustrated by its application to several examples, including the case where the cost variable is the maximum of a standard Brownian motion over a finite time horizon. For this example we obtain an exact optimal drift, as well as an approximation of the optimal drift through a Monte-Carlo algorithm.

KW - Brownian motion

KW - Itô calculus

KW - Malliavin calculus

KW - Monte-Carlo sampling

KW - Path integral control

KW - Relative entropy

KW - Stochastic optimal control

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U2 - 10.1016/j.sysconle.2014.08.001

DO - 10.1016/j.sysconle.2014.08.001

M3 - Article

AN - SCOPUS:84908550389

VL - 72

SP - 36

EP - 43

JO - Systems and Control Letters

JF - Systems and Control Letters

SN - 0167-6911

ER -