Controlled Diffusion Processes by Nikolai Vladimirovich Krylov, A.B. Aries

By Nikolai Vladimirovich Krylov, A.B. Aries

This publication offers with the optimum regulate of strategies of totally observable Itô-type stochastic differential equations. The validity of the Bellman differential equation for payoff capabilities is proved and ideas for optimum keep watch over recommendations are developed.

Topics contain optimum preventing; one dimensional managed diffusion; the Lp-estimates of stochastic fundamental distributions; the lifestyles theorem for stochastic equations; the Itô formulation for features; and the Bellman precept, equation, and normalized equation.

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In particular, va(x) and v(x) are Jinite functions. 4. We can consider without loss of generality that r1 = -r,. Let Regardless of the fact that w(x) is the difference between two nondifferentiable functions, we can easily verify that w(x) is twice continuously differentiable and that for each a 2 6, b E [- K,K], x E [rl,r2] In addition, w 2 0 on [rl,r2],w(ri)= 0. By Ito's formula, for each a E x E [r1,r2],t 2 0 a, from which we conclude, using properties of the function w, that w(x) 2 M:(z A t ) and, as t -+ co,w(x) 2 M:z.

However, it is easier, as can be seen later, to find an explicit form for the function w, if we express this function w as a performance function and apply the Bellman equation. Furthermore, the esetimate w(z) I vu(z) is exact (unimprovable) in the class of processes (2) in the case w = v. We shall illustrate this with an example. Consider the process z, = z + Sb o, dw,, where o is a matrix of dimension d x d and w, is a d-dimensional Wiener process. Let E < lzl< R. We estimate from above the probability of the event that this process reaches the closure of a sphere S, = {x:1x1 < E } before it leaves a sphere S,.

Because a, b, c, f are continuous in the argument a We conclude from the above that for each x E [rl,r2] there is an i such that - Next, we denote by i(x) the smallest value of i for which the last given inequality can be satisfied. It is seen that the (measurable) function a(x) a(i(x))yields the function stated in the first assertion. Let a,,,(x) = Ma(n A i(x + w,)) = JzGSm-" a(n - A i(y))e-(Y-x)2p'dt. It is easily seen that a,,,(x) is an infinitely differentiable function. Furthermore, a,,Jx) E A due to the convexity property of the set A.

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