Sensitivity analysis using Itô-Malliavin calculus and application to stochastic optimal control (2002)
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@MISC{Gobet02sensitivityanalysis,
author = {E. Gobet and R. Munos},
title = {Sensitivity analysis using Itô-Malliavin calculus and application to stochastic optimal control},
year = {2002}
}
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Abstract
We consider a multidimensional diffusion process (X t ) 0tT whose dynamics depends on parameters . Our rst purpose is to give representation formulae of the sensitivity rJ() for the expected cost J() = E(f(X T )) as an expectation: this issue is motivated by stochastic control problems (where the controller is parameterized and the optimization problem is then reduced to a parametric optimization one) or by model misspecifications in nance. Known results concerning the evaluation of rJ() by simulations concern the case of smooth cost functions f or of diusion coecients not depending on (see Kushner and Yang, SIAM J. Control Optim. 29 (5), pp. 12161249, 1991). Here, we handle the general case removing these two restrictions, deriving three new type formulae to evaluate rJ(): we call them Malliavin calculus approach, adjoint approach and martingale approach. For this, our basic tools are Itô calculus, Malliavin calculus and martingale arguments. In the second part of this work, we provide discretization procedures to simulate the relevant random variables and we analyze the associated weak error: the nature of the results are new in that context. We prove that the discretization error is essentially linear w.r.t. the time step. Finally, some numerical experiments deal with some examples in random mechanics and finance: we compare different methods in terms of variance, complexity, computational time and inuence of the time discretization step.
Keyphrases
malliavin calculus stochastic optimal control sensitivity analysis new type formula second part martingale argument adjoint approach relevant random variable parametric optimization expected cost stochastic control problem computational time numerical experiment deal representation formula time discretization step discretization error different method simulation concern general case discretization procedure random mechanic control optim smooth cost function malliavin calculus approach time step optimization problem multidimensional diffusion process basic tool model misspecifications associated weak error diusion coecients sensitivity rj rst purpose martingale approach
