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

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By Bernt Øksendal, Agnès Sulem

The major goal of the e-book is to provide a rigorous, but commonly nontechnical, creation to an important and valuable resolution tools of varied forms of stochastic regulate difficulties for bounce diffusions and its functions. the categories of keep watch over difficulties lined comprise classical stochastic regulate, optimum preventing, impulse keep an eye on and singular keep an eye on. either the dynamic programming strategy and the utmost precept strategy 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 tools. The textual content emphasises functions, regularly to finance. the entire major effects are illustrated by means of examples and routines appear at the top of every bankruptcy with whole strategies. this may aid the reader comprehend the speculation and notice easy methods to practice it. The booklet 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

E. that for 0 < x < x∗ . 12) Cxλ1 ≥ x − a Define k(x) = Cxλ1 − x + a. By our chosen values of C and x∗ we have k(x∗ ) = k (x∗ ) = 0. Moreover, k (x) = Cλ1 (λ1 − 1)xλ1 −2 > 0 for x < x∗ . 12) holds and hence (ii) is proved. 34 2 Optimal Stopping of Jump Diffusions (iii): In this case ∂D = {(s, x); x = x∗ } and hence ∞ E ∞ P x [X(t) = x∗ ]dt = 0 . X∂D (Y (t))dt = y 0 0 (iv) and (v) are trivial. 9). s. 5) is given by t X(t) = x exp α− 1 2 2β −γ zν(dz) t + ln(1 + γz)N (dt, dz) + βB(t) . s. s. s. 2 Applications and examples e−ρτ X(τ ) 35 is uniformly integrable.

3) Find an equivalent local martingale measure Q for (S1 (t), S2 (t)) and use this to deduce that there is no arbitrage in this market. 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 .

20) is an optimal control. In feedback form the control can be written (ρt − µt )(φt x + ψt ) . 1. Suppose the wealth X(t) = X (u) (t) of a person with consumption rate u(t) ≥ 0 satisfies the following L´evy type mean reverting Ornstein-Uhlenbeck SDE dX(t) = (µ − ρX(t) − u(t))dt + σdB(t) + θ z N (dt, dz) ; R X(0) = x > 0 Fix T > 0 and define T −s J (u) (s, x) = E s,x e−δ(s+t) 0 uγ (t) dt + λX(T − s) . γ t>0 56 3 Stochastic Control of Jump Diffusions Use dynamic programming to find the value function Φ(s, x) and the optimal consumption rate (control) u∗ (t) such that ∗ Φ(s, x) = sup J (u) (s, x) = J (u ) (s, x) .

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