By Bernt Øksendal, Agnès Sulem

The most objective of the ebook is to offer a rigorous, but as a rule nontechnical, creation to an important and valuable resolution equipment of varied kinds of stochastic regulate difficulties for leap diffusions and its applications.

The varieties of keep watch over difficulties lined comprise classical stochastic keep an eye on, optimum preventing, impulse regulate and singular keep watch over. either the dynamic programming procedure and the utmost precept procedure are mentioned, in addition to the relation among them. Corresponding verification theorems related to the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There also are chapters at the viscosity answer formula and numerical methods.

The textual content emphasises functions, usually to finance. the entire major effects are illustrated via examples and routines look on the finish of every bankruptcy with entire ideas. this may support the reader comprehend the speculation and spot how one can practice it.

The ebook assumes a few uncomplicated wisdom of stochastic research, degree concept and partial differential equations.

In the second version there's a new bankruptcy on optimum regulate of stochastic partial differential equations pushed by means of Lévy approaches. there's additionally a brand new part on optimum preventing with not on time info. in addition, corrections and different advancements were made.

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Additional info for Applied Stochastic Control of Jump Diffusions (2nd Edition) (Universitext)

Example text

The results below remain valid, with the natural modifications, if we allow S to be any Borel set such that S ⊂ S 0 where S 0 denotes the interior of S, S 0 its closure. Let f : Rk → R and g : Rk → R be continuous functions satisfying the conditions τS Ey f − (Y (t))dt < ∞ for all y ∈ Rk . 2) 0 The family {g − (Y (τ )) · X{τ <∞} , τ ∈ T } is uniformly integrable, for all y ∈ Rk. ) The general optimal stopping problem is the following: Find Φ(y) and τ ∗ ∈ T such that ∗ Φ(y) = sup J τ (y) = J τ (y), τ ∈T y ∈ Rk , 28 2 Optimal Stopping of Jump Diffusions where τ J τ (y) = E y 0 f (Y (t))dt + g(Y (τ )) · X{τ <∞} , τ ∈T is the performance criterion.

Vi) Aφ + f ≤ 0 on S \ ∂D. s. on {τS < ∞} and limt→τ − φ(Y (t)) = g(Y (τS )) · S χ{τS <∞} . τS (viii) E y |φ(Y (τ ))| + |Aφ(Y (t))|dt < ∞ for all τ ∈ T . 0 ¯ Then φ(y) ≥ Φ(y) for all y ∈ S. (b) Moreover, assume (ix) Aφ + f = 0 on D. s. for all y. (xi) {φ(Y (τ )); τ ∈ T , τ ≤ τD } is uniformly integrable, for all y. Then φ(y) = Φ(y) and τ ∗ = τD is an optimal stopping time. 2 (Sketch). (a) Let τ ≤ τS be a stopping time. 1 we can assume that φ ∈ C 2 (S). 24) applied to τm := min(τ, m), m = 1, 2, .

Moreover, φ = g outside D. , that Cxλ1 ≥ x − a for 0 < x < x∗ . 12) Define k(x) = Cxλ1 − x + a. By our chosen values of C and x∗ we have k(x∗ ) = k (x∗ ) = 0. Moreover, k (x) = Cλ1 (λ1 − 1)xλ1 −2 > 0 for x < x∗ . 12) holds and hence (ii) is proved. (iii): In this case ∂D = {(s, x); x = x∗ } and hence ∞ Ey ∞ X∂D (Y (t))dt = 0 (iv) and (v) are trivial. 0 P x [X(t) = x∗ ]dt = 0. 34 2 Optimal Stopping of Jump Diffusions (vi): Outside D we have φ(s, x) = e−ρs (x − a) and therefore Aφ + f (s, x) = e−ρs (−ρ(x − a) + αx) φ(s, x + γxz) − φ(s, x) − + R −ρs (α − ρ)x + ρa =e + ≤e ∂φ (s, x)γxz ν(dz) ∂x x+γxz

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