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

By Bernt Øksendal, Agnès Sulem

Here's a rigorous advent to an important and helpful answer equipment of assorted varieties of stochastic keep watch over difficulties for bounce diffusions and its functions. dialogue comprises the dynamic programming strategy and the utmost precept approach, and their dating. The textual content emphasises real-world purposes, essentially in finance. effects are illustrated through examples, with end-of-chapter routines together with whole ideas. The 2d variation provides a bankruptcy on optimum keep an eye on of stochastic partial differential equations pushed through L?vy tactics, and a brand new part on optimum preventing with not on time details. uncomplicated wisdom of stochastic research, degree thought and partial differential equations is believed.

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2). 6. Show that t t γ(s, z)N (ds, dz) E exp {eγ(s,z) − 1 − γ(s, z)}ν(dz)ds , = exp 0 R 0 R provided that the right hand side is finite. 7. Let dXi (t) = γi (t, z)N (dt, dz) ; i = 1, 2 R be two 1-dimensional Itˆ o-L´evy processes. 16) to prove the following integration by parts formula : t t − X1 (t)X2 (t) = X1 (0)X2 (0) + X2 (s− )dX1 (s) X1 (s )dX2 (s) + 0 0 t + γ1 (s, z)γ2 (s, z)N (ds, dz) . 6 Exercises 25 Remark. 2) is called the quadratic covariation of X1 and X2 . 27. 8. Consider the following market (Bond price) dS0 (t) = 0; S0 (0) = 0 (Stock price 1) dS1 (t) = S1 (t− )[µ1 dt + γ11 dη1 (t) + γ12 dη2 (t)]; S1 (0) = x1 > 0 (Stock price 2) dS2 (t) = S2 (t− )[µ2 dt + γ21 dη1 (t) + γ22 dη2 (t)]; S2 (0) = x2 > 0 where µi and µij are constants and η1 (t), η2 (t) are independent L´evy martingales of the form dηi (t) = z Ni (dt, dz); i = 1, 2.

2 The maximum principle x2 47 ν>0 (jump Merton line) ν = 0 (classical Merton line) x2 = θ∗ x 1−θ ∗ 1 x2 = θ0 x 1−θ0 1 0 x1 Fig. 2. The Merton line for ν = 0 and ν > 0 As before N (dt, dz) = (N1 (dt, dz1 ), . . 16). The process u(t) = u(t, ω) ∈ U ⊂ Rk is our control . 1) has a unique strong solution X (u) (t); t ∈ [0, T ]. Such controls are called admissible. The set of admissible controls is denoted by A. Suppose the performance criterion has the form T J(u) = E f (t, X(t), u(t))dt + g(X(T )) ; u∈A 0 where f : [0, T ] × Rn × U → R is continuous, g : Rn → R is C 1 , T < ∞ is a fixed deterministic time and T f − (t, X(t), u(t))dt + g − (X(T )) < ∞ E for all u ∈ A .

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].

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