VISCOSITY SOLUTION OF LINEAR REGULATOR QUADRATIC FOR DEGENERATE DIFFUSIONS

The paper studied a linear regulator quadratic control problem for degenerate Hamilton-Jacobi-Bellman (HJB) equation. We showed the existence of viscosity properties and established a unique viscosity solution of the degenerate HJB equation associated with this problem by the technique of viscosity solutions. Copyright © 2006 Md. Azizul Baten. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited


Introduction
We are concerned with the linear regulator quadratic control problem to minimize the expected cost with discount factor α > 0: over c ∈ Ꮽ subject to the degenerate stochastic differential equation for nonzero constants A, σ = 0, and a continuous function h on R, where w t is a onedimensinal standard Brownian motion on a complete probability space (Ω,Ᏺ,P) endowed with the natural filtration Ᏺ t generated by σ(w s , s ≤ t), and Ꮽ denotes the class of all Ᏺ t −progressively measurable processes c = (c t ) with J(c) < ∞.This kind of stochastic control problem has been studied by many authors [3,7] for nondegenerate diffusions to (1.1) and (1.2).We also assume that h satisfies the following properties: h(x) ≥ 0 : convex; (1.3) There exists C > 0 such that h(x 2 Viscosity solution for degenerate diffusions for some constant C > 0, n ≥ 0. We refer to [5] for the quadratic case of degenerate diffusions related to Riccati equations in case of h(x) = Cx 2 and n = 2 with infinite horizon.The purpose of this paper is to show the existence of a viscosity solution of u of the associated Hamilton-Jacobi-Bellman (in short, HJB) equation of the form: (1.5) Our method consists in finding a unique viscosity solution for u of (1.5) following Bardi and Capuzzo-Dolcetta [2], Crandall et al. [4], Fleming and Soner [7] through the limit of the solution v = v L , L > 0, to the HJB equation as L → ∞.To show the existence of the viscosity solution v L , we assume that h has the following property: there exists C ρ > 0, for any ρ > 0, such that for a fixed integer n ≥ 2. This condition acts as the uniform continuity of h with order n, and plays an important role for the existence of viscosity solutions [10,11].We notice that (1.7) is fulfilled for In Section 2 we show that the value function v L (x) = inf c∈ᏭL J(c) is a unique viscosity solution of (1.6), where Ꮽ L = {c = (c t ) ∈ Ꮽ : |c t | ≤ L for all t ≥ 0}.Section 3 is devoted to the study of u that u(x) := lim L→∞ v L (x) is a viscosity solution of (1.5).

Viscosity solutions
We here study the properties of the value function v L (x) and show that v L (x) is a viscosity solution of the Bellman equation (1.6) for any fixed L > 0, and then v L converges to a viscosity solution u of the Bellman equation (1.5).
According to Crandall et al. [4] and Fleming and Soner [7] this definition is equivalent to the following: for any x ∈ R, H x,w(x), p, q ≤ 0 for (p, q) ∈ J 2,+ w(x), H x,w(x), p, q ≥ 0 for (p, q) ∈ J 2,− w(x), (2.4) where J 2,+ and J 2,− are the second-order superjets and subjets defined by (2.5) In order to obtain the viscosity property of v L , we assume that there exists β 0 ∈ (0,β) satisfying and we set f k (x) = γ + |x| k for any 2 ≤ k ≤ 2n and a constant γ ≥ 1 chosen later.

Properties of viscosity solutions
Lemma 2.2.Assume (2.6).Then there exist γ ≥ 1 and η > 0, depending on L, k, such that ) where τ is any stopping time and x t is the response to (c t ) ∈ Ꮽ L .
To prove (2.14), we denote by v r (x) the right-hand side of (2.14).By the formal Markov property with c equal to c shifted by τ.Thus (2.20) It is known in [7,12] that this formal argument can be verified, and we deduce v L (x) ≥ v r (x).
To prove the reverse inequality, let ρ > 0 be arbitrary.We set By the same calculation as (2.18), there exists (2.23) (2.24) For any i, we take (2.27) Now, by the definition of v r (x), we can find c ∈ Ꮽ L such that (2.28) Md. Azizul Baten 7 Thus, using the formal Markov property [7], we have where x τ t is the response to c τ t with x τ 0 = x τ .Letting ρ → 0, we deduce v r (x) ≥ v(x), which completes the proof.(2.32) Let (2.33) By Fatou's lemma, we obtain (2.35) Therefore, we obtain (2.30).
To prove (2.31), we have by (2.30) and by the moment inequalities for local martingales [9] and Hölder's inequality for some constant C > 0. Then ( We substract ϕ(z) from both sides and apply Ito's formula to obtain (2.41) Divide by s and let s → 0 and by dominated convergence theorem, 3) is verified.Let ᏻ = ᏻ x be a bounded neighborhood of x and let θ = θ x be the exit time of x t from ᏻ x .Then, by (2.31)

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Theorem 2.6 (Apostol [1] (Dini)).Suppose that X is a compact metric space and {v L (x)} is a sequence of continuous functions that converges pointwise to a continuous function u on X. Suppose also that the sequence is pointwise nonincreasing, that is, for each x ∈ X, there exists v L (x) ≥ v L+1 (x), for all L ∈ N. Then {v L (x)} converges uniformly to u on X.

Uniqueness of HJB.
The most important feature of the theory of viscosity solution is the powerful uniqueness theorem.In the context of optimal control problems, value function is the unique viscosity solutions.
In this section we give a detailed proof of uniqueness result for the quadratic control problem that v is a unique viscosity solution of (1.6).The references for these uniqueness results are Bardi and Capuzzo-Dolcetta [2] and Flacone and Makridakis [6].
Proof.Combining Theorems 2.5 and 2.6, we get the assertion by the stability result of Therefore, we see that u is a viscosity solution of (1.5) and then by Theorem 2.3, it is clear that u fulfills (1.4) and (1.7).
Stochastic control problem: in general we can further study a stochastic control problem for linear degenerate systems to minimize the discounted expected cost: for some constants k 0 ,k 1 > 0 and for a fixed integer m ≥ 2.