Probability Methods for Approximations in Stochastic Control by Harold J. Kushner

By Harold J. Kushner

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If E,N < 00, x E S - dS, then the spectral radius of Q is strictly less than unity, and any of the methods for solving such linear equations can be used. See Varga [Vl] for more details. 4. 8) is via "backward iteration," starting with n = M. ll), we need to calculate the invariant measure p. This can be done by letting uo denote any probability measure on S and using the iteration u"+' = u"P (a-periodic) or via any method that can calculate the eigenvector of P corresponding to eigenvalue unity.

P. 1.. Also, if P { X ( t )= X ( t + ) }= 1, t E (0, T), then we can define the paths to be right continuous, without altering the finite dimensional distributions. In both of these cases, a e . 4 THE SPACE Dm[a,P] 33 we can assume that the process X ( . p. 1, and, indeed, induces the measure P { - } on D"[O, TI. p. 1 in the topology of D"[O, TI. P. ) is continuous. p. p. 1. ,P > u, is defined exactly as D"[O, TI was defined. We simply shift the time origin and set T = P - u. In the special case where a = - P = - T, denote the metric by d'i.

1 . Generally speaking, in our convergence studies, we are concerned mainly with properties of the “limit” X and with convergence of distributions of functionals of the { X , } to those of X . In these cases, we can alter the probability space in any way at all, as long as the distributions of each X , and of X are not changed-it would not matter if we modified the joint distributions of the ({X,}, X ) . Suppose that 6 = [0, 13, 3 is the a-algebra consisting of the Borel sets in [0, I ] , and P is t The subscript n is sometimes put on the random variable, and sometimes on the measure; notation is often abused by interchanging P { X , E A} with Pm(XE A}, but the interchange should cause no confusion even without an identification of the probability space.

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