By Bernt Øksendal, Agnès Sulem

ISBN-10: 3540140239

ISBN-13: 9783540140238

The major objective of the ebook is to provide a rigorous, but as a rule nontechnical, creation to crucial and valuable resolution tools of varied kinds of stochastic regulate difficulties for bounce diffusions and its functions. the kinds of keep watch over difficulties lined contain classical stochastic keep an eye on, optimum preventing, impulse keep an eye on and singular keep watch over. either the dynamic programming approach 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 answer formula and numerical equipment. The textual content emphasises purposes, commonly to finance. all of the major effects are illustrated by way of examples and routines appear at the top of every bankruptcy with whole recommendations. it will support the reader comprehend the idea and spot tips on how to practice it. The ebook assumes a few easy wisdom of stochastic research, degree concept and partial differential equations.

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**Extra info for Applied Stochastic Control of Jump Diffusions**

**Sample text**

2 (Optimal consumption and portfolio in a L´ evy type Black-Scholes market [Aa], [FØS1]). Suppose we have a market with two possible investments: (i) a safe investment (bond, bank account) with price dynamics dP1 (t) = rP1 (t)dt ; P1 (0) = p1 > 0 (ii) a risky investment (stock) with price dynamics ∞ − dP2 (t) = P2 (t ) µdt + σdB(t) + z N (dt, dz) , −1 P2 (0) = p2 > 0 42 3 Stochastic Control of Jump Diﬀusions where r > 0, µ > 0 and σ ∈ R are constants. We assume that ∞ |z|dν(z) < ∞ and µ > r . −1 Assume that at any time t the investor can choose a consumption rate c(t) ≥ 0 (adapted, cadlag) and is also free to transfer money from one investment to the other without transaction cost.

R Assume that the matrix γ := γij 1≤i,j≤2 γ −1 = λ = λij ∈ R2 is invertible, with inverse 1≤i,j≤2 and assume that νi (R) > λi1 µ1 + λi2 µ2 for i = 1, 2. 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) 48 3 Stochastic Control of Jump Diﬀusions In the previous chapter we saw how to solve such a problem using dynamic programming and the associated HJB equation. Here we present an alternative approach, based on what is called the maximum principle . In the deterministic case this principle was ﬁrst introduced by Pontryagin and his group [PBGM]. A corresponding maximum principle for Itˆ o diﬀusions was formulated by Kushner [Ku], Bismut [Bi] and subsequently further developed by Bensoussan [Ben], Haussmann [H] and others.

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

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