By Bernt Øksendal, Agnès Sulem

The major objective of the booklet is to offer a rigorous, but more often than not nontechnical, advent to crucial and worthwhile answer tools of assorted kinds of stochastic keep an eye on difficulties for bounce diffusions and its functions. the categories of regulate difficulties coated contain classical stochastic regulate, optimum preventing, impulse regulate and singular keep an eye on. either the dynamic programming procedure and the utmost precept process 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 equipment. The textual content emphasises purposes, in general to finance. all of the major effects are illustrated through examples and routines appear at the top of every bankruptcy with whole strategies. this may aid the reader comprehend the speculation and notice the right way to practice it. The booklet assumes a few simple wisdom of stochastic research, degree thought and partial differential equations.

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

**Sample text**

1 by using the stochastic maximum principle. 3. Deﬁne ⎤ u(t, ω) z N (dt, dz) dX1 (t) ⎥ ⎢ R 2 =⎣ dX (u) (t) = dX(t) = ⎦∈R dX2 (t) z 2 N (dt, dz) ⎡ R and, for ﬁxed T > 0 (deterministic) J(u) = E − (X1 (T ) − X2 (T ))2 . Use the stochastic maximum principle to ﬁnd u∗ such that J(u∗ ) = sup J(u) . u z 2 N (T, dz). We may regard F as a given Interpretation: Put F (ω) = R T -claim in the normalized market with the two investment possibilities bond and stock, whose prices are (bond) dS0 (t) = 0 ; (stock) dS1 (t) = S0 (0) = 1 z N (dt, dz), a L´evy martingale.

19). Finally we compare the solution in the jump case (ν = 0) with Merton’s solution in the no jump case (ν = 0): As before let Φ0 , c∗0 and θ0∗ be the solution when there are no jumps (ν = 0). 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.

1 In particular, if we try ψ(w) = Kwγ we get (u) A0 ψ(w) + f (w, u) = −ρKwγ + [r(1 − θ) + µθ]w − c Kγwγ−1 ∞ + K · 12 σ 2 θ2 w2 γ(γ − 1)wγ−2 + Kwγ {(1 + θz)γ − 1 − γθz}ν(dz) + −1 cγ . γ Let h(c, θ) be the expression on the right hand side. e. 1 Dynamic programming ∂h = (µ−r)Kγwγ +Kσ 2 θγ(γ−1)wγ +Kwγ ∂θ 45 ∞ {γ(1+θz)γ−1z−γz}ν(dz) = 0 . 15) we get that θ = θˆ should solve the equation ∞ 2 Λ(θ) := µ − r − σ θ(1 − γ) − 1 − (1 + θz)γ−1 zν(dz) = 0 . 18) −1 then there exists an optimal θ = θˆ ∈ (0, 1].