New PDF release: Applied Stochastic Control of Jump Diffusions (2nd Edition)

By Bernt Øksendal, Agnès Sulem

ISBN-10: 3540698264

ISBN-13: 9783540698265

The most goal of the ebook is to provide a rigorous, but quite often nontechnical, creation to crucial and beneficial resolution equipment of assorted kinds of stochastic keep an eye on difficulties for bounce diffusions and its applications.

The different types of keep watch over difficulties coated comprise classical stochastic keep watch over, optimum preventing, impulse regulate and singular keep an eye on. either the dynamic programming approach and the utmost precept approach 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 purposes, in most cases to finance. all of the major effects are illustrated through examples and routines seem on the finish of every bankruptcy with whole strategies. this can support the reader comprehend the speculation and spot how one can follow it.

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

In the second variation there's a new bankruptcy on optimum keep watch over 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. furthermore, corrections and different advancements were made.

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

Sample text

Y (t− )) ∈ A. Then u∗ is an optimal control and Suppose u∗ (t) := u ∗ φ(y) = Φ(y) = J (u ) (y) for all y ∈ S. 1 Dynamic Programming 47 Proof. (a) Let u ∈ A. For n = 1, 2, . . put τn = min(n, τS ). 24) we have τn E y [φ(Y (τn ))] = φ(y) + E y τn Au φ(Y (t))dt ≤ φ(y) − E 0 f (Y (t), u(t))dt . 0 Hence τn φ(y) ≥ lim inf E y n→∞ f (Y (t), u(t))dt + φ(Y (τn )) 0 τS ≥ Ey 0 f (Y (t), u(t))dt + g(Y (τS )) · X{τS <∞} = J (u) (y). 5) Since u ∈ A was arbitrary we conclude that φ(y) ≥ Φ(y) for all y ∈ S.

3) For other applications of optimal stopping to jump diffusions we refer to [Ma]. 3 Optimal Stopping with Delayed Information This presentation is based on [Ø4]. Let Y (t) be a jump diffusions in Rk . Let δ ≥ 0 be a fixed constant. 1) 0 where we interpret g(Y (α)) as 0 if α = ∞. Here Tδ is the set of δ-delayed stopping times, defined as follows. 8. 2) {ω; α(ω) ≤ t} ∈ Ft−δ for all t ≥ δ or, equivalently, {ω; α(ω) ≤ s + δ} ∈ Fs for all s ≥ 0. 3) The set of all δ-delayed stopping times is denoted by Tδ .

So (iii) holds if T E 0 e−2δt W 2γ (t)dt + R [(1 + θz)γ − 1]ν(dz) < ∞. 20) to hold. 19). 52 3 Stochastic Control of Jump Diffusions x2 ν = 0 (classical Merton line) x2 = ν > 0 (jump Merton line) ∗ θ0 ∗ x1 1−θ0 x2 = θ∗ x 1−θ ∗ 1 x1 0 Fig. 2. The Merton line for ν = 0 and ν > 0 Finally we compare the solution in the jump case (ν = 0) with Merton’s solution in the no jump case (ν = 0). As before let Φ0 , c∗0 , and θ0∗ be the solution when there are no jumps (ν = 0). Then it can be seen that K < K0 and hence Φ(s, w) = e−δs Kwγ < e−δs K0 wγ = Φ0 (s, w) c∗ (s, w) ≥ c∗0 (s, w) θ∗ ≤ θ0∗ .

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


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