Bernt Øksendal, Agnès Sulem's Applied Stochastic Control of Jump Diffusions (2nd Edition) PDF

By Bernt Øksendal, Agnès Sulem

ISBN-10: 3540698264

ISBN-13: 9783540698265

The most goal of the publication is to offer a rigorous, but usually nontechnical, advent to an important and precious resolution tools of varied kinds of stochastic keep an eye on difficulties for leap diffusions and its applications.

The forms of regulate difficulties lined contain classical stochastic regulate, optimum preventing, impulse keep an eye on and singular keep watch over. either the dynamic programming procedure and the utmost precept process are mentioned, in addition to the relation among them. Corresponding verification theorems concerning 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 purposes, in most cases to finance. all of the major effects are illustrated by way of examples and workouts seem on the finish of every bankruptcy with entire suggestions. this can aid the reader comprehend the idea and notice the best way to observe it.

The e-book assumes a few simple wisdom of stochastic research, degree idea and partial differential equations.

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

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

Sample text

6) (b) Now apply the above argument to u(t) = u ˆ(Y (t)), where u ˆ is as in (v). 5) and hence φ(y) = J (ˆu) (y) ≤ Φ(y) for all y ∈ S. 4). 2 (Optimal Consumption and Portfolio in a L´ evy Type Black–Scholes Market [Aa, FØS1]). Suppose we have a market with two possible investments: (i) A safe investment (bond, bank account) with price dynamics dP1 (t) = rP1 (t)dt, P1 (0) = p1 > 0. (ii) A risky investment (stock) with price dynamics dP2 (t) = P2 (t− ) μ dt + σ dB(t) + ∞ z N (dt, dz) , −1 P2 (0) = p2 > 0, where r > 0, μ > 0, and σ ∈ R are constants.

Then E[Y (1) (T ) · Y (2) (T )] T Y (1) (t− )dY (2) (t) = y1 · y2 +E 0 T T Y (2) (t− )dY (1) (t) + + 0 T tr[σ (1) σ (2) ](t)dt 0 n T (1) + 0 j=1 i=1 R (2) γij (t, zj )γij (t, x) νj (dzj ) dt . Proof. 16). 4. Let u ∈ A be an admissible control with corresponding state process X(t) = X (u) (t). Then T ˆ ˆ ))−g(X(T )) .

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, . . we have, by (vi), τm E y [φ(Y (τm ))] = φ(y) + E y 0 τm Aφ(Y (t))dt ≤ φ(y) − E y f (Y (t))dt . 0 30 2 Optimal Stopping of Jump Diffusions Hence by (ii) and the Fatou lemma τm φ(y) ≥ lim inf E y f (Y (t))dt + φ(Y (τm )) m→∞ τ ≥ Ey 0 0 f (Y (t))dt + g(Y (τ ))χ{τ <∞} = J τ (y).

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Applied Stochastic Control of Jump Diffusions (2nd Edition) (Universitext) by Bernt Øksendal, Agnès Sulem


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