Optimal Impulse Control for Systems Deriven by Stochastic Delayed Di ﬀ erential Equations

. In this paper we study the problem of optimal impulse control for stochastic systems with delay in the case when the value function of the impulse problem depends only on the initial data of the given process through its initial value (value at zero) and some weighted averages. A veriﬁcation theorem for such impulse control problem is given. As an example the optimal stream of dividends with transaction costs is solved.


Introduction
A stochastic impulse control policy can be characterized by the following factors: The first factor is known to be the random dates at which the considered policies are exercised while the second one is the size of the applied policies.Such characterization indicates that times and size are factors that can be studied separately by depending on the nature of the considered applications such as forest economic and cash flow management.In such applications both the timing and size of an admissible impulse policy have to be simultaneously determined.The mathematical analysis of stochastic impulse control in most cases is based on a combination of dynamic programming techniques and quasi-variational inequalities.The approach is general, its typically results into functional inequalities which, depending on the nature of the considered problem.
In what follows, we refer to the law of the solution X ξ (t) of (2.1) by ρ ξ (t) and the corresponding expectation by E ξ .
If the impulse control v is applied to system (2.1)-(2.2), the process X (ξ,v) (t) is defined by where Suppose that the profit rate is a function u : R 3 → R which is continuous, increasing and concave.Let g : ∂S → R be a given bequest function, where ∂S denotes for the boundary of S.Moreover, suppose that the profit performing an intervention K where K : R 3 × H → R; K(t, x, y, η) is a given function.
Let ∆ be the set of admissible controls includes the set of impulse controls v = (τ 1 , τ 2 , . . .; η 1 , η 2 , . . . ) such that X (ξ,v) (t) ∈ S for all t ≤ T, T * = ∞ and lim τ n = Ta.s.p s,ξ,v for all s, ξ, v where p s,ξ,v is the law of the time space harvested process Assume the following conditions to be hold: Then the total expected profit J (s,ξ) when v ∈ ∆ is applied to the system (2.1)-(2.2) is defined by: (2.6) Now, our optimal impulse control problem for systems (2.1)-(2.2) is to find the value function Φ(s, ξ) and the optimal impulse control v * ∈ ∆ such that:

.7)
A problem of this type for systems without delay had been studied in [2,4].Problem (2.7) in general is infinite dimensional.The purpose of this paper is to reduce problem (2.7) for system (2.1)-(2.2) to finite dimensional one when we restrict its value function Φ to depends only on the initial path ξ through the three linear functionals namely: (2.10) In this case Φ can be written as: where Ψ : R 4 → R.

A Quasi-variational Inequality Formulation
In this section we prove a verification theorem for problem (2.7) in view of (2.8)-(2.10).To start with, first let X t (S) = X(t + s) for t ≥ 0, −δ ≤ S ≤ 0 to be the segment of the path of X from t − δ to t.
where the differential operator L acting on f as Proof.See [1].

The intervention operator.
If we denote by G for the space of all measurable functions : S → R, then the intervention operator denoted by M where M : G → G is denoted to be: where ∈ G and (x, y) ∈ S.
(iii) Second order derivatives of ϕ with respect to x are locally bounded near ∂D.Then there exists a sequence of functions ϕ j , j = 1, 2, . . .such that (a) ϕ j → ϕ uniformly on compact subsets S as j → ∞.
Suppose the following hold: Suppose that the continuation region D has the form: (iv) D := {(s, x, y) ∈ S : ω(x, y) ⊆ ω * } for some function ω : R 2 → R and some constant ω * and ∂D is Lipschitz surface.
Proof.As in [4] for systems without delay we give the following details of the proof for systems with constant delay as follows: First when (i)-(iv) is satisfied and by Lemma 3.2, we can find such sequence of functions ϕ j , j = 1, 2 . . . in C 1,2,1 (S 0 ) ∩ C(S), such that (a) ϕ j → ϕ uniformly on compact subsets of S, j → ∞.
and v * is optional.

Application
This application is an extension to the no-delay case of a problem of optional stream of dividends with transaction costs.Suppose that if we make no interventions the amount X(t) = X ξ (t) available (cash flow) is given by where Suppose that at any time t we are free to take out dividend η from X(t) by applying the transaction cost K(η) = c + γη, where c > 0 and γ > 0 are constants.The constant c is called the fixed part and the quantity γη is called the proportional part, respectively, of the transaction cost.The resulting cash flow X (ξ,v) is given by (4.1)-( 4.2) and for ρ is a constant (discounted exponent).The functional J s,ξ represents the total expected discounted dividend up to time T, k ∈ (0, 1) is a constant.The problem is to find the optional impulse v * ∈ ν and the value function Φ such that we try to find a function ϕ(s, ξ(0), y(ξ)) of the form ϕ(s, x, y) = exp(−ρs)ψ(x, y) satisfying the conditions of Theorem 3.1.Assume that the continuation region D has the form: where ψ(x, y), ω * are to be determined.
which has a general solution for some arbitrary constants C, D, where where p := (θ + βe λδ ) are the solutions of the equation: If we assume that ρ ≥ p := (θ + βe λδ ).which implies that K > 0 since 0 < k < 1. Choose h(ω) of the form: where r is given by (4.14).If this is the case, then our solution ψ(s, x, y) gets the form ψ(s, x, y) = e −ρs φ(x, y); ω ≤ ω 1 , where Mψ is: The supremum of where D 1 is the derivative of φ with respect to the first variable and ω 0 , x 0 are as follows: In particular, where ω 1 = ω(x 1 , y) and
In other words, if ω > ω 1 , then i.e., we harvest exactly enough to bring x-level down to the value x 1 .We summarize what we have proved in the following theorem: )ξ(s)ds.
R 3 be a given Borel set (solvency set) with the property that S = (S 0 ) where S 0 and S − denotes for the interior and closure of S respectively.Define T = inf{t ∈ (0, T * (ω)); (s + t, X(t), Y(t)) not belongs to S} where Y(t) := 0 −δ exp(λs)X(t + s)ds.

Lemma 3 . 3 .
(Approximation): Let D ⊆ S be open, ∂S is Lipschitz and ϕ : S → R satisfy the following