By Wendell H. Fleming, Halil Mete Soner
This ebook is meant as an creation to optimum stochastic keep an eye on for non-stop time Markov approaches and to the idea of viscosity suggestions. Stochastic regulate difficulties are handled utilizing the dynamic programming technique. The authors method stochastic keep watch over difficulties via the strategy of dynamic programming.
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Extra info for Controlled Markov Processes and Viscosity Solutions
If L is of class C ∞ (Q0 × IRn ) then the method of characteristics shows that V ∈ C ∞ (N ). 1. In dimension n = 1, consider the problem of minimizing J= 1 2 t1 t 2 |x(s)| ˙ ds + ψ(x(t1 )). I. 1 it is seen that any minimizing x∗ (·) is a straight line segment. 6), 1 V (t, x) = min1 [ (t1 − t)v 2 + ψ(x + (t1 − t)v)] v∈IR 2 where v is the slope of the line y = x + (s − t)v. In this example, L(v) = 12 v 2 and H(p) = 12 p2 . 5) becomes X(t1 , α) = α, P (t1 , α) = ψ ′ (α).
For each (t, x) ∈ Q0 , there exists u∗R (·) ∈ UR (t) such that J(t, x; u∗R ) ≤ J(t, x; u) for all u(·) ∈ UR (t). 1, which is proved below. The next lemma is a special case of Pontryagin’s principle. 3. 4) t t1 (b) PR (s) = s u∗R (r)dr, Lx (r, x∗R (r), u∗R (r))dr, t ≤ s ≤ t1 . 5). 1. 5) L(s, x∗R (s), u∗R (s)) + u∗R (s) · PR (s) = min [L(s, x∗R (s), v) + v · PR (s)]. 6) VR (t, x) = J(t, x; u∗R ) = min J(t, x; u). UR (t) We are going to find bounds for VR and u∗R (s) which do not depend on R. 2c).
For a definition of approximately continuous function, see [EG] or [McS, p. ) Given v ∈ U and 0 < δ < t1 − s, let ⎧ ∗ ⎨ u (r) if r ∈ [s, s + δ] uδ (r) = ⎩ v if r ∈ [s, s + δ], 24 I. 2) with u(r) = uδ (r) and with xδ (t) = x. 12) 0≤ 1 δ + 1 δ s+δ [L(r, xδ (r), v) − L(r, x∗ (r), u∗ (r)]dr s t1 s+δ [L(r, xδ (r), u∗ (r)) − L(r, x∗ (r), u∗ (r))]dr 1 + [ψ(xδ (t1 )) − ψ(x∗ (t1 ))]. δ The first term on the right side tends to L(s, x∗ (s), v) − L(s, x∗ (s), u∗ (s)) as δ → 0. By the mean value theorem, the second term equals 1 t1 0 s+δ Lx (r, x∗ (r) + δλζδ (r), u∗ (r)) · ζδ (r)dλdr where Lx = (∂L/∂x1 , .