By Nikolai Vladimirovich Krylov, A.B. Aries
This ebook bargains with the optimum regulate of ideas of totally observable Itô-type stochastic differential equations. The validity of the Bellman differential equation for payoff features is proved and principles for optimum keep watch over recommendations are developed.
Topics contain optimum preventing; one dimensional managed diffusion; the Lp-estimates of stochastic imperative distributions; the life theorem for stochastic equations; the Itô formulation for features; and the Bellman precept, equation, and normalized equation.
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Extra resources for Controlled Diffusion Processes
Definition. A natural strategy a,(~[,,,~) is said to be (stationary) Markov if a , ( ~ ~ , ,= , ~a(x,) ) for some function afx). We denote by %,(x) the set of all Markov strategies admissible at a point x. Note that to each natural strategy a,(~[,,,~) admissible at x, we can set into correspondence a strategy E 2I such that x:" = xf,". In fact, let us take a solution x,(w) = x;,"(w) of Eq. (2), and let us assume that Pt(w)= a , ( ~ ~ , , ~ ~It( wis)seen ) . ". By the uniqueness theorem this equation has no other solutions; therefore, xf," = x:".
Xt) dt + gn(xYod A' Comparing the last equality with (3),we obtain A . which makes a crucial point in our proof. It turns out that if g is replaced with g, = g - (g - En)+, E,(x) will become a payoff function in the optimal stopping problem. 17, f":: = f a + n(g - v",), + n(En- g) + Lug, which En(x)- g(x) = sup M: u s e yields, by Theorem Ji e-"'f "nf (xt)dt. fi 2 fa + Lug2 - (fa + Lag)- ; hence 1 En(x)- g(x)> -sup M e-"'dtsup(/. + L'~)- = --N, n Note that ae'U Jc u,x 1 g - En I - N , n 1 (g- En)+<-N, n (5) which, together with (4), completes the proof of (b).
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. 8. Lemma. Let the function a(x) satisfy a Lipschitz condition.