Optimal control for uncertain stochastic dynamic systems with jump and application to an advertising model

Optimal control and zero‐sum game subject to differential equations with Liu processes and random matrices

“…▪ Remark 5. It is worth noting that Theorem 7 provides a sufficient condition for ensuring the saddle-point solution to the game problem (34). The solution of Equations ( 35) and ( 36) exists if the operators sup and inf are interchangeable and J(y 0 , v, w, 0) in ( 34) is concave (convex, respectively) in w (v, respectively).…”

“…Similarly, we adopt J(y, t) to express the optimal value from time t to the terminal time with Y t = y. Employing the Theorem 3, we present equilibrium equations for solving the problem (34).…”

“…random matrices with finite expected values, and C is are independent Liu processes, for i = 1, 2, … , k. The definitions of other signs are identical to (34). Let J(y, t) be the optimal value from time t to the terminal time with Y t = y.…”

“…[28][29][30][31] Qin 32 showed that using chance theory can effectively solve a portfolio selection problem having uncertainty together with randomness. Subsequently, optimal control problems [33][34][35] with both uncertainty and randomness were constructed by exerting chance theory.…”

“…v = {v s ∈ V s , 0 ≤ s ≤ T} and w = {w s ∈ W s , 0 ≤ s ≤ T} are control sequences where V s = [𝛼 1 , 𝛽 1 ] and W s = [𝛼 2 , 𝛽 2 ] are close sets. The other symbols have the same meanings as(34). Let J(y, t) be twice continuously differentiable on R m × [0, T].…”