Operations Research

Download Applied Stochastic Control of Jump Diffusions by Bernt Øksendal, Agnès Sulem PDF

By Bernt Øksendal, Agnès Sulem

The major function of the booklet is to provide a rigorous, but in most cases nontechnical, creation to an important and worthwhile answer equipment of assorted sorts of stochastic keep watch over difficulties for leap diffusions (i.e. ideas of stochastic differential equations pushed via L?vy methods) and its purposes.

The varieties of keep an eye on difficulties coated contain classical stochastic keep watch over, optimum preventing, impulse regulate and singular keep an eye on. either the dynamic programming process 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 tools.

The textual content emphasises purposes, in general to finance. all of the major effects are illustrated via examples and workouts appear at the tip of every bankruptcy with whole strategies. it will aid the reader comprehend the speculation and notice the right way to observe it.
 
The e-book assumes a few simple wisdom of stochastic research, degree thought and partial differential equations.

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We have, by (vi), τm y E [φ(Y (τm ))] = φ(y) + E τm Aφ(Y (t))dt ≤ φ(y) − E y y 0 f (Y (t))dt . 0 Hence by (ii) and the Fatou lemma τm φ(y) ≥ lim inf E y m→∞ f (Y (t))dt + φ(Y (τm )) 0 τ ≥E y f (Y (t))dt + g(Y (τ ))χ{τ <∞} = J τ (y). 0 Hence φ(y) ≥ Φ(y) . 11), so that φ(y) = J τD (y) ≤ Φ(y) . 12) we conclude that φ(y) = Φ(y) and τD is optimal. The following result is sometimes helpful. 3. 2 hold. Suppose g ∈ C 2 (Rk ) and that φ = g satisfies (viii). Define U = {y ∈ S; Ag(y) + f (y) > 0} . s. Then U ⊂ {y ∈ S; Φ(y) > g(y)} = D .

3 (Geometric L´ evy martingales). Let h ∈ L2 (R) be deterministic and define t Y (t) = exp 0 R h(s)z N (ds, dz) − t 0 R eh(s)z − 1 − h(s)z ν(dz)ds . 6 Exercises 23 Show that dY (t) = Y (t− ) R ˜ (dt, dz). 4. Find the generator A of the following jump diffusions: a) (The geometric L´evy process ) dX(t) = X(t− ) µdt + σdB(t) + γ z N (dt, dz) . R b) (The mean-reverting L´evy-Ornstein-Uhlenbeck process ) dX(t) = (m − X(t))dt + σdB(t) + γ ˜ (dt, dz). zN R c) (The graph of the geometric L´evy process ) dt , where X(t) is as in a).

At time t is given by dP (t) = αP (t)dt + βP (t)dB(t) + γP (t− ) (i) z N (dt, dz); P (0) = p > 0 R and the remaining amount of resources Q(t) at time t is (ii) dQ(t) = −λQ(t)dt ; Q(0) = q > 0 where λ > 0 is the (constant) relative extraction rate and α, β, γ ≥ 0 are constants. s. ν. If we decide to stop extraction and close the field at a stopping time τ ≥ 0, the expected discounted total net profit J τ (s, p, q) is given by τ τ J (s, p, q) = E (s,p,q) e−ρ(s+t) (λP (t)Q(t) − K)dt + θe−ρ(s+τ ) P (τ )Q(τ ) , 0 where K > 0 is the (constant) running cost rate, θ > 0 another constant.

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