Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

Abstract

In this paper, we study the optimal singular controls for stochastic recursive systems, in which the control has two components: the regular control, and the singular control. Under certain assumptions, we establish the dynamic programming principle for this kind of optimal singular controls problem, and prove that the value function is a unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman inequality, in a given class of bounded and continuous functions. At last, an example is given for illustration.T ] → R n×m are given deterministic functions, (W s ) s≥0 is an d-dimensional Brownian motion, (x, t) are initial time and state, v (·) : [0, T ] → R k is classical control process, and ξ (·) : [0, T ] → R m , with nondecreasing left-continuous with right limits stands for the singular control.The aim is to minimize the cost functional:where. . . m} are given deterministic functions, where l (·) represents the running cost tare of the problem and K (·) the cost rate of applying the singular control. [40], Karatzas and Shreve [43], and Menaldi and Taksar [52]. The second one is related to probabilistic methods; see Baldursson [6], Boetius [8, 9], Boetius and Kohlmann [10], El Karoui and Karatzas [29, 30], Karatzas [39], and Karatzas and Shreve [41, 42]. Third, the DPP, has been studied in a general context, for example, by Boetius [9], Haussmann and Suo [36], Fleming and Soner [31], and Zhu [66]. At last the maximum principle for optimal singular controls (see, for example, Cadenillas and Haussmann [17], Dufour and Miller [26], Dahl and Øksendal [27] see references therein). The existence for optimal singular control can be found in Haussmann and Suo [36] and Dufour and Miller [25] via different approaches. It is necessary to point out that singular control problems are largely used in diverse areas such as mathematical finance (see Baldursson and Karatzas [12], Chiarolla and Haussmann [18], Kobila [44], and Karatzas and Wang [45], Davis and Norman [24]), manufacturing systems (see, Shreve, Lehoczky, and Gaver [60]), and queuing systems (see Martins and Kushner [53]). Particularly, the application of H-J-B inequality in finance can be seen in Pagès and Possamaï [58], which is employed to investigate the bank monitoring incentives.As is well known for the classical stochastic control problems, DPP is satisfied. Moreover, if the value function has appropriate regularity, it admits a second-order nonlinear partial differential equation (H-J-B equation) (see Fleming and Rishel [32] and Lions [49] reference therein). In the frame work of singular stochastic control, the H-J-B equation becomes a second-order variational inequality (see Fleming and Soner [31], Haussmannand, Suo [35]).Wang [63] firstly introduces and studies a class of singular control problems with recursive utility, where the cost function is determined by a backward stochastic differential equation (BSDE in short). More preciously, the cost functional is defined bywhere Y t,x;v,ξ t is determined by    dX t,x;ξ s = aX t,x;ξ s + b ds + ...

Second‐order necessary optimality conditions for discrete‐time stochastic systems

“…Nevertheless, the reverse does not always hold. For instance, M = [1, 2) ⋃ (3,4] is 𝛾-convex, but it is neither convex set nor starlike relative to each of its point.…”

“…Now, we verify the optimality of the control ū(0) = ū(1) = (0, 0) * , ū( 2 Clearly, the set (f , 𝜎)(0, x(0), U(0)) is 𝛾-convex, the set (f , 𝜎)(1, x(1), U(1)) is convex, and for all x = (x 1 , x 2 , x 3 ) * ∈ R 3 ⧵ {0}, the sets (f , 𝜎)(1, x, U(1)) are starlike relative to the point f (1, x, ū(1)) + 𝜎(1, x, ū( 1))(w(2) − w(1)) = (0, 0, x 2 2 + x 2 2 (w(2) − w(1))) * . Taking into account these calculations, we have p(1) = q(0) = (0, 0, 0) * , further, we see that the control ū(t), t ∈ T is singular at t = 0 relative to the vector (û, v) ∈ U 0 (2) × U(0), where û = (0, û 2 (2)) * , û 2 (2) ∈ {−1, −0.5, 0} and v = (v 1 (0), 0) * , v 1 (0) ∈ [−1, 0] ∪ [2,3], and is also singular at t = 1 relative to the vector (û, ṽ) * ∈ U 0 (2) × U (1), where û = (0, û 2 (2)) * , û 2 (2) ∈ {−1, −0.5, 0} and ṽ = (ṽ 1 (1), ṽ2 (1)) * ∈ U (1). Set û = (0, −1) * , using ( 7), ( 10) and ( 31), (32), we get H xx (0, x(0), ū(0), p(1)) = H xx (1, x(1), ū(1), p(2)) = (0, 0, 0; 0, 0, 0; 0, 0, 0), H xx (2, x(2), ū(2), û(2), p(3)) = P(2) = (0, 0, 0; 0, −2, 0; 0, 0, 0), P(1) = (0, 0, 0; 0, 0, 0; 0, 0, 0), K(0, v, ū(0), û) = 0, K(1, ṽ, ū(1), û) = −2[ṽ 1 (1) + ṽ1 (1)(w(2) − w( 1))] 2 , L(1, ṽ, ū(1), û)[Δ v f (0) + Δ v 𝜎(0)(w(1) − w(0))] = ṽ2 (1)[v 1 (0) + v 1 (0)(w(1) − w(0))].…”

“…Applying the condition (8), we claim that E{2𝜂 0 𝜂 1 ṽ2 (1)[v 1 (0) + v 1 (0)(w(1) − w(0))] − 2𝜂 2 1 [ṽ 1 (1) + ṽ1 (1)(w(2) − w( 1))] 2 } ≤ 0, for any (ṽ 1 (1), ṽ2 (1)) ∈ U(1), v 1 (0) ∈ [−1, 0] ∪ [2,3] and 𝜂 1 ≥ 0, v ∈ U(0). Set (ṽ 1 (1), ṽ2 (1)) * = (0, 1) * ∈ U(1), v = (v 1 (0), v 2 (0)) = (2, 0) * ∈ U(0), 𝜂 1 = 1, then the above inequality can be rewritten as 4 ≤ 0, which indicates that ( 8) is not fulfilled.…”

“…$$ The pair $$$ \left(\mathbf{x},\mathbf{u}\right) $$$ satisfying () is named an admissible pair, and $$$ \left(\overline{\mathbf{x}},\overline{\mathbf{u}}\right) $$$ satisfying () is named an optimal pair. For stochastic optimal control problems, 1–3 one of the main issues is to establish the necessary optimality conditions (maximum principle). In the stochastic setting, maximum principle (MP) for optimal control problems is fundamentally different from the deterministic case.…”