By Bernt Øksendal, Agnès Sulem

ISBN-10: 3540140239

ISBN-13: 9783540140238

ISBN-10: 3540264418

ISBN-13: 9783540264415

The major function of the booklet is to provide a rigorous, but usually nontechnical, advent to crucial and helpful resolution equipment of varied kinds of stochastic regulate difficulties for leap diffusions (i.e. ideas of stochastic differential equations pushed by means of L?vy approaches) and its purposes.

The sorts of keep an eye on difficulties coated contain classical stochastic keep an eye on, optimum preventing, impulse keep an eye on and singular regulate. either the dynamic programming procedure and the utmost precept technique 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 equipment.

The textual content emphasises functions, quite often to finance. the entire major effects are illustrated by means of examples and routines appear at the top of every bankruptcy with whole options. it will aid the reader comprehend the speculation and spot how you can observe it.
 
The ebook assumes a few simple wisdom of stochastic research, degree conception and partial differential equations.

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Extra resources for Applied Stochastic Control of Jump Diffusions

Example text

Define D = {y ∈ S; φ(y) > g(y)} (the continuation region). s. on {τS < ∞} and lim φ(Y (t)) = g(Y (τS )) · χ{τS <∞} t→τS− and τS |Aφ(Y (t))| + |σ T (Y (t))∇φ(Y (t))|2 (viii) E |φ(Y (τ ))| + y 0 |φ(Y (t) + γ (j) (Y (t), z)) − φ(Y (t))|2 νj (dzj ) dt < ∞ + j=1 R for all τ ∈ T . ¯ Then φ(y) ≥ Φ(y) for all y ∈ S. s. for all y (xi) {φ(Y (τ )); τ ∈ T } 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).

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 .

Then it can be seen that K < K0 Φ(s, w) = e−δs Kwγ < e−δs K0 wγ = Φ0 (s, w) and hence c∗ (s, w) ≥ c∗0 (s, w) θ∗ ≤ θ0∗ . So with jumps it is optimal to place a smaller wealth fraction in the risky investment, consume more relative to the current wealth and the resulting value is smaller than in the no-jump case. For more details we refer to [FØS1]. 3. For more information and other applications of stochastic control of jump diffusions see [GS], [BKR], [Ma] and the references therein. 2 The maximum principle Suppose the state X(t) = X (u) (t) of a controlled jump diffusion in Rn is given by dX(t) =b(t, X(t), u(t))dt + σ(t, X(t), u(t))dB(t) γ(t, X(t− ), u(t− ), z)N (dt, dz).

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


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