By Wendell H. Fleming, Halil Mete Soner
This ebook is an advent to optimum stochastic regulate for non-stop time Markov strategies and the idea of viscosity options. It covers dynamic programming for deterministic optimum keep watch over difficulties, in addition to to the corresponding conception of viscosity ideas. New chapters during this moment variation introduce the position of stochastic optimum keep an eye on in portfolio optimization and in pricing derivatives in incomplete markets and two-controller, zero-sum differential video games.
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Extra resources for Controlled Markov Processes and Viscosity Solutions (Stochastic Modelling and Applied Probability)
To simplify the problem we assume that all the given data are time inde˜ g˜ and f , are independent of t. With an abuse of notation pendent. Thus L, ˜ v) and g˜(x) respectively. 1′ ) e−βs L(x(s), u(s))ds + e−βτ g(x(τ ))χτ <∞ . J(t, x; u) = t We will take U(t, x) = Ux , where Ux is defined below. 2) ˜). J(t, x; u) = e−βt J(0, x; u Hence, it suffices to consider initial time t = 0. From now on we shall do so, and will write J(x; u) instead of J(0, x; u). Let us now formulate more precisely the class of infinite horizon control problems which we shall consider.
2. 8. Unfortunately, when generalized solutions are considered instead of “classical” solutions of class C 1 (Q), a serious lack of uniqueness is encountered. 3) with given boundary data can have infinitely many generalized solutions. 2 below. This difficulty is circumvented by choosing the unique generalized solution which is also a viscosity solution, according to the definition to be given in Chapter II. Pontryagin’s Principle. During the 1950’s Pontryagin formulated a “maximum principle” which provides a general set of necessary conditions for an extremum in an optimal control problem.
Let (t, x) be a regular point and x∗ (·) the unique minimizer of J for left endpoint (t, x). If (τn , yn ) → (t, x) as n → ∞ and x∗n (·) minimizes J for left endpoint (τn , yn ), then x˙ ∗n (τn ) → x˙ ∗ (t) as n → ∞. Proof. 1) Lx = d ¨∗n Lv = Lvt + Lvx x˙ ∗n + Lvv x ds holds, where Lx , Lv , · · · are evaluated at (s, x∗n (s), x˙ ∗n (s)). Since Lvv > 0 this can be rewritten as x ¨∗n (s) = Φ(s, x∗n (s), x˙ ∗n (s)), τn ≤ s ≤ t1 , I. Deterministic Optimal Control 43 where Φ = L−1 ˙ ∗n (s)| ≤ R1 , where R1 does vv (Lx −Lvt −Lvx v).