Read e-book online Applied Stochastic Control of Jump Diffusions PDF

By Bernt Øksendal, Agnès Sulem

ISBN-10: 3540140239

ISBN-13: 9783540140238

ISBN-10: 3540264418

ISBN-13: 9783540264415

The major goal of the e-book is to offer a rigorous, but normally nontechnical, advent to crucial and priceless resolution equipment of assorted forms of stochastic regulate difficulties for bounce diffusions (i.e. strategies of stochastic differential equations pushed by way of L?vy tactics) and its functions.

The different types of keep an eye on difficulties coated comprise classical stochastic keep an eye on, optimum preventing, impulse keep an eye on and singular keep an eye on. either the dynamic programming technique and the utmost precept procedure 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 resolution formula and numerical equipment.

The textual content emphasises purposes, in general to finance. the entire major effects are illustrated by means of examples and workouts appear at the tip of every bankruptcy with whole recommendations. this may aid the reader comprehend the speculation and notice tips on how to follow it.
 
The e-book assumes a few uncomplicated wisdom of stochastic research, degree thought and partial differential equations.

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T. the probability law P x1 ,x2 of (X1 (t), X2 (t)) when X1 (0− ) = x1 , X2 (0− ) = x2 . Call the control u(t) = (c(t), θ(t)) ∈ [0, ∞) × [0, 1] admissible and write u ∈ A if the corresponding total wealth (u) (u) W (t) = W (u) (t) = X1 (t) + X2 (t) is nonnegative for all t ≥ 0. The problem is to find Φ(s, x1 , x2 ) and u∗ (c∗ , θ∗ ) ∈ A such that ∗ Φ(s, x1 , x2 ) = sup J (u) (s, x1 , x2 ) = J (u ) (s, x1 , x2 ) . u∈A Case 1: ν = 0 . In this case the problem was solved by Merton [M]. 9) γ−1 Moreover, the optimal consumption rate c∗0 (t) is given by .

19). 1) be the bankruptcy time and let T denote the set of all stopping times τ ≤ τS . 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. 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: 28 2 Optimal Stopping of Jump Diffusions Find Φ(y) and τ ∗ ∈ T such that ∗ y ∈ Rk Φ(y) = sup J τ (y) = J τ (y) ; τ ∈T where τ τ J (y) = E f (Y (t))dt + g(Y (τ )) · X{τ <∞} ; y τ ∈T 0 is the performance criterion .

3) converge. From now on we assume that H is differentiable with respect to x. 4) We assume from now on that T σσ T (t, X(t), u(t)) E 0 2 γ (k) (t, X(t), u(t), zk ) νk (dzk ) dt < ∞ + for all u ∈ A . 4 (A sufficient maximum principle [FØS3]). 4) satisfying T n qˆqˆT (t) + E 0 k=1 R |r(k) (t, zk )|2 νk (dzk ) dt < ∞ . 6) Then uˆ is an optimal control. 5. 6) to hold it suffices that the function (x, v) → H(t, x, v, pˆ(t), qˆ(t), rˆ(t, ·)) is concave, for all t ∈ [0, T ] . 6 (Integration by parts). Suppose E[(Y (j) (T )2 ] < ∞ for j = 1, 2, where dY (j) (t) = b(j) (t, ω)dt + σ (j) (t, ω)dB(t) + γ (j) (t, z, ω)N (dt, dz) Rn Y (j) (0) = y (j) ∈R ; n j = 1, 2 where b(j) ∈ Rn , σ (j) ∈ Rn×m and γ (j) ∈ Rn× .

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


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