A white noise approach to optimal insider control of systems with delay

We use a white noise approach to study the problem of optimal inside control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system.These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the optimal insider portfolio in a financial market where the risky asset price is given by a stochastic delay equation. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable. Our results show that with this inside information the market is not viable even if there is delay in the equations.

where Y (t, Z) = X(t − δ, Z), (1.1) s>t H s . We also assume that the Donsker delta functional of Z exists (see below). This assumption implies that the Jacod condition holds, and hence that B(·) and N(·, ·) are semimartingales with respect to H. See e.g. [DØ2] for details. We assume that the value at time t of our insider control process u(t) is allowed to depend on both Z and F t . In other words, u(.) is assumed to be H-adapted, such that u(., z) is F-adapted for each z ∈ R.
Let U denote the set of admissible control values.We assume that the functions b(t, x, y, u, z) = b(t, x, y, u, z, ω) : [0, T ] × R × R × U × R × Ω → R σ(t, x, y, u, z) = σ (t, x, y, u, z, ω) γ(t, x, y, u, z, ζ) = γ(t, x, y, u, z, ζ, ω) are given bounded C 1 functions with respect to x, y and u and adapted processes in (t, ω) for each given x, y, u, z, ζ. Let A be a given family of admissible H−adapted controls u. The performance functional J(u) of a control process u ∈ A is defined by f (t, X(t, Z), u(t, Z), Z))dt + g(X(T, Z), Y (T, Z), Z)], where f (t, x, u, z) : [0, T ] × R × U × R → R g(x, z) : R × R → R (1.5) are given bounded functions, C 1 with respect to x and u. The functions f and g are called the profit rate and terminal payoff, respectively. For completeness of the presentation we allow these functions to depend explicitly on the future value Z also, although this would not be the typical case in applications. But it could be that f and g are influenced by the future value Z directly through the action of an insider, in addition to being influenced indirectly through the control process u and the corresponding state process X.The problem we consider is the following: Problem 1.1 Find u ⋆ ∈ A such that sup u∈A J(u) = J(u ⋆ ).

The Donsker delta functional
To study this problem we adapt the technique of the paper [DØ1] to the SDE with delay situation. For the convenience of the reader we first recall briefly the definition and basic properties of the Donsker delta functional: Definition 2.1 Let Z : Ω → R be a random variable which also belongs to (S) * . Then a continuous functional is called a Donsker delta functional of Z if it has the property that R g(z)δ Z (z)dz = g(Z) a.s.
(2.2) {donsker pr for all (measurable) g : R → R such that the integral converges.
For example, consider the special case when Z is a first order chaos random variable of the form for some deterministic functions β = 0, ψ such that (2.4) and for every ǫ > 0 there exists ρ > 0 such that R\(−ǫ,ǫ) e ρζ ν(dζ) < ∞.
This condition implies that the polynomials are dense in L 2 (µ), where dµ(ζ) = ζ 2 dν(ζ). It also guarantees that the measure ν integrates all polynomials of degree ≥ 2. In this case it is well known (see e.g. [MØP], [DiØ1], Theorem 3.5, and [DØP], [DiØ2]) that the Donsker delta functional exists in (S) * and is given by where exp ⋄ denotes the Wick exponential. Moreover, we have for t < T 0 If D t and D t,ζ denotes the Hida-Malliavin derivative at t and t, ζ with respect to B and N , respectively, we have For more information about the Donsker delta functional, Hida-Malliavin calculus and their properties, see [DØ1].
From now on we assume that Z is a given random variable which also belongs to (S) * , with a Donsker delta functional δ Z (z) ∈ (S) * satisfying (2.11) 3 Transforming the insider control problem to a related parametrized non-insider problem Since X(t) is H-adapted, we get by using the definition of the Donsker delta functional δ Z (z) of Z that for some z-parametrized process X(t, z) which is F-adapted for each z.
Then, again by the definition of the Donsker delta functional we can write, for 0 ≤ t ≤ T Comparing (3.1) and (3.2) we see that (3.1) holds if we for each z choose X(t, z) as the solution of the classical (but parametrized) SPDE As before let A be the given family of admissible H−adapted controls u. Then in terms of X(t, z) the performance functional J(u) of a control process u ∈ A defined in (1.4) gets the form Thus we see that to maximize J(u) it suffices to maximize j(u)(z) for each value of the parameter z ∈ R. Therefore Problem 1.1 is transformed into the problem (3.6) {problem2}

A sufficient-type maximum principle
In this section we will establish a sufficient maximum principle for Problem 3.1.
Problem 3.1 is a stochastic control problem with a standard (albeit parametrized) stochastic partial differential equation (3.3) for the state process X(t, z), but with a non-standard performance functional given by (3.5). We can solve this problem by a modified maximum principle approach, as follows: R denotes the set of all functions r(·) : R → R such that the last integral above converges. The quantities p, q, r(·) are called the adjoint variables. The adjoint processes p(t, z), q(t, z), r(t, z, ζ) are defined as the solution of the z-parametrized advanced backward stochastic differential equation (ABSDE) We can now state the first maximum principle for our problem (3.6): Theorem 4.1 [Sufficient-type maximum principle] Letû ∈ A, and denote the associated solution of (3.3) and (4.2) byX(t, z) and (p(t, z),q(t, z),r(t, z, ζ)), respectively. Assume that the following hold: Then u(·, z) is an optimal insider control for Problem 3.1.

Proof.
By considering an increasing sequence of stopping times τ n converging to T , we may assume that all local integrals appearing in the computations below are martingales and hence have expectation 0. See [ØS2]. We omit the details. Choose arbitrary u(., z) ∈ A, and let the corresponding solution of (3.3) and (4.2) be X(t, z), p(t, x, z), q(t, x, z), r(t, x, z, ζ). For simplicity of notation we write x, X(t, z), u(t, z)) and similarly with b, b, σ, σ and so on. Moreover putĤ and where (4.5) {eq4.7} By the definition of H and the concavity of H, we have Since g is concave with respect to x we have and hence Combining (4.6) and (4.8) we obtain using the fact X(t) The last inequality holds because of the maximum condition of H. Hence j(u) ≤ j(û). Since u ∈ A was arbitrary, this shows thatû is optimal.

A necessary-type maximum principle
In some cases the concavity conditions of Theorem 4.1 do not hold. In such situations a corresponding necessary-type maximum principle can be useful. For this, instead of the concavity conditions we need the following assumptions about the set of admissible control values: • A 1 . For all t 0 ∈ [0, T ] and all bounded H t 0 -measurable random variables α(z, ω), the control θ(t, z, ω) := 1 [t 0 ,T ] (t)α(z, ω) belongs to A.
• A3. For all β as in (5.2) the derivative process exists, and belongs to L 2 (λ × P) and

(5.3) {d chi}
Theorem 5.1 [Necessary maximum principle] Letû ∈ A. Then the following are equivalent: Proof. For simplicity of notation we write u instead ofû in the following. By considering an increasing sequence of stopping times τ n converging to T , we may assume that all local integrals appearing in the computations below are martingales and have expectation 0. See [ØS2]. We omit the details. We can write d da J((u + aβ)(., z))| a=0 = I 1 + I 2 where By our assumptions on f and g and by (4.1) we have By the Itô formula Summing (5.4) and (5.6) we get In particular, applying this to Differentiating with respect to s, we get Since this holds for all s ≥ t 0 and for all α, we conclude that 6 Optimal insider portfolio in a financial market with delay As an application of the results above, consider a financial market with the following two investment possibilities: (i) A risk free asset, with unit price S 0 (t) = 1 for all times t ≥ 0.
(ii) A risky asset, in which the investments have a delayed effect, in the following sense: If we at time t invest in this asset the fraction π(t, Z) of the current wealth X(t, Z), then we assume that the dynamics of the wealth X(t, Z) = X π (t, Z) is described by a stochastic delay equation of the form Here α 0 (t) and β 0 (t) are given bounded adapted processes and ξ is a given bounded deterministic function. The performance functional is defined by Let A H be the set of H-adapted controls π(t) = π(t, Z) such that there is a unique solution X(t) = X(t, Z) of (6.1) with X(T, Z) > 0 a.s. Note that equation (6.1) can be solved inductively step by step in each interval [kδ, (k + 1)δ] for k = 0, 1, 2, .... We study the following problem: Problem 6.1 Find π * ∈ A H (called an optimal control) such that: This is a problem of the type investigated in the previous sections, in the special case with no jumps and with controls π(t, z), and we can apply the results in Theorem 5.1 to study it.
For studies of financial markets modelled by stochastic delay equations we refer to [AHMP1], [AHMP2] and [KMT].
Remark 6.3 (The no-delay case) Note that the above theorem also applies to the special case when there is no delay, i.e. δ = 0. In this case we see that , and (6.13)reduces to (6.14) This result has been proved in [ØR] and [DØ1] by different methods.