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

By Bernt Øksendal, Agnès Sulem

Here's a rigorous creation to an important and worthwhile answer equipment of varied varieties of stochastic regulate difficulties for bounce diffusions and its functions. dialogue comprises the dynamic programming strategy and the utmost precept strategy, and their courting. The textual content emphasises real-world purposes, basically in finance. effects are illustrated by way of examples, with end-of-chapter routines together with whole recommendations. The second variation provides a bankruptcy on optimum keep watch over of stochastic partial differential equations pushed by way of L?vy strategies, and a brand new part on optimum preventing with behind schedule info. simple wisdom of stochastic research, degree thought and partial differential equations is believed.

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10) V (s, x) = sup J (u) (s, x) . 7 (FØS3). 1). Define pi (t) = ∂V (t, X ∗ (t)) ; ∂xi n σik (t, X ∗ (t), u∗ (t)) qjk (t) = i=1 rik (t, z) = 1≤i≤n ∂2V (t, X ∗ (t)) ; ∂xi ∂xj 1 ≤ j ≤ n, 1 ≤ k ≤ m ∂V ∂V (t, X ∗ (t) + γ (k) (t, X ∗ (t), u∗ (t), zk )) − (t, X ∗ (t)) ; ∂xi ∂xi 1 ≤ i ≤ n, 1 ≤ k ≤ . 4). For a proof see [FØS3]. 8. A general discussion of impulse control for jump diffusions can be found in [F]. A study with vanishing impulse costs is given in [ØUZ]. 3 Application to finance The following example is from [FØS3].

A portfolio in this market is a two-dimensional cadlag, adapted process θ(t) = (θ0 (t), θ1 (t)) giving the number of units of bonds and stocks, respectively, held at time t by an agent. The corresponding wealth process X(t) = X (θ)(t) is defined by X(t) = θ0 (t)S0 (t) + θ1 (t)S1 (t) ; t ∈ [0, T ] . 5) The portfolio θ is called self-financing if t X(t) = X(0) + t θ0 (s)dS0 (s) + 0 0 or, in short hand notation, dX(t) = θ0 (t)dS0 (t) + θ1 (t)dS1 (t) . 6) Alternatively, the portfolio can also be expressed in terms of the amounts w0 (t), w1 (t) invested in the bond and stock, respectively.

Iv) {φ− (Y (τ ))}τ ≤τS is uniformly integrable for all u ∈ A and y ∈ S. Then φ(y) ≥ Φ(y) for all y ∈ S . 1 Dynamic programming 41 and {φ(Y (ˆu) (τ ))}τ ≤τS is uniformly integrable. (vi) Suppose u∗ (t) := u ˆ(Y (t− )) ∈ A. Then u∗ is an optimal control and ∗ φ(y) = Φ(y) = J (u ) (y) for all y ∈ S . 4) Proof. a) Let u ∈ A. For n = 1, 2, . . put τn = min(n, τS ). 24) we have τn y E [φ(Y (τn ))] = φ(y) + E τn A φ(Y (t))dt ≤ φ(y) − E y u f (Y (t), u(t))dt . 0 0 Hence τn φ(y) ≥ lim inf E n→∞ y f (Y (t), u(t))dt + φ(Y (τn )) 0 τS ≥E f (Y (t), u(t))dt + g(Y (τS )) · X{τS <∞} = J (u) (y) .

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