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Existence of linear-quadratic regulator for degenerate diffusions

This paper studies a linear regulatory quadratic control problem for degenerate Hamilton-Jacobi-Bellman (HJB) equation. We establish the existence of a unique viscosity and a classical solution of the degenerate HJB equation associated with this problem by the technique of viscosity solutions, and, hence, derive an optimal control from the optimality conditions in the HJB equation. | Turk J Math 30 (2006) , 309 – 328. ¨ ITAK ˙ c TUB Existence of Linear-Quadratic Regulator for Degenerate Diffusions Azizul Baten Abstract This paper studies a linear regulatory quadratic control problem for degenerate Hamilton-Jacobi-Bellman (HJB) equation. We establish the existence of a unique viscosity and a classical solution of the degenerate HJB equation associated with this problem by the technique of viscosity solutions, and, hence, derive an optimal control from the optimality conditions in the HJB equation. Key words and phrases: Stochastic differential equation, Hamilton-Jacobi-Bellman equation, Linear-Quadratic problem, Viscosity solutions, Applications to control theory. 1. Introduction We are concerned with the quadratic control problem to minimize the expected cost with discount factor β > 0: Z J(c) = E ∞ e−βt {h(xt ) + |ct |2 }dt (1) 0 over c ∈ A and subject to the degenerate stochastic differential equation dxt = [Axt + ct ]dt + σxt dwt , x0 = x ∈ R, t ≥ 0. (2) 2000 AMS Mathematics Subject Classification: 60H10, 49N10, 49J15, 49L25, 58E25. 309 BATEN Here, A consists of non-zero constants, σ 6= 0, and a continuous function h on R. xt is the state variable of the system at time t, ct is the control variable of the system at time t, wt is a one-dimensinal standard Brownian motion on a complete probability space (Ω, F , P ) endowed with the natural filtration Ft generated by σ(ws , s ≤ t), x0 = x is the initial value of the state variable, and A denotes the class of all Ft −progressively measurable processes c = (ct ) with J(c) 0 such that h(x) ≤ C(1 + |x|n), x R, (4) for some constant C > 0, n ≥ 2. We refer to [11] for the quadratic case of degenerate diffusions related to Ricatti equations in case of h(x) = Cx2 and n = 2 with infinite horizon. The purpose of this paper is to show the existence of a smooth solution u of the associated Hamilton-Jacobi-Bellman (in short, HJB) equation of the form: 1 −βu + σ 2 x2 u00 + Axu0 + min(r