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Center for Nonlinear Studies |
We show that stochastic control problems with a particular cost structure involving a relative entropy term admit a purely probabilistic solution, without the necessity of applying the dynamic programming principle. The argument is as follows. Minimization of the expectation of a random variable with respect to the underlying probability measure, penalized by relative entropy, may be solved exactly. In the case where the randomness is generated by a standard Brownian motion, this exact solution can be written as a Girsanov density. The stochastic process appearing in the Girsanov exponent has the role of control process, and the relative entropy of the change of probability measure is equal to the integral of the square of this process. An explicit expression for the control process may be obtained in terms of the Malliavin derivative of the density process. The theory is applied to the problem of minimizing the maximum of a Brownian motion (penalized by the relative entropy), leading to an explicit expression of the optimal control law in this case. The link to linearization of the Hamilton-Jacobi-Bellman equation is made for the case of diffusion processes.
Host: Misha Chertkov