Integral Operator Riccati Equations Arising in Stochastic Volterra Control Problems

Linear--Quadratic control for a class of stochastic Volterra equations: solvability and approximation

“…Proof of Theorem 3.4. First, the existence and uniqueness of a solution Γ ∈ C([0, T ], L 1 (µ⊗ µ)) to the Riccati equation (3.7) satisfying the estimate (3.11) and such that Γ t ∈ S d + (µ⊗µ), for all t ≤ T , follow from [5,Theorem 2.3].…”

“…The existence and uniqueness of a solution to the Riccati system follows from [5], and is stated in the next theorem. Its proof is given in Section 6.…”

“…Next, in the LQ case, i.e., when b, σ are of linear form, and the cost function is linear-quadratic, we prove by means of a refined martingale verification argument combined with a squares completion technique, that the value function is of quadratic form while the optimal control is in linear feedback form with respect to these lifted state variables. The coefficients of the quadratic and linear form of the value function and optimal control are expressed in terms of a non-standard system of integral operator Riccati equations whose solvability (existence and uniqueness) is proved in [5]. A related infinite-dimensional Riccati equation appeared in [7] for the minimization problem of an energy functional defined in terms of a non-singular (i.e.…”

Linear-quadratic stochastic Volterra controls II: Optimal strategies and Riccati--Volterra equations

“…We say that a pair (X t0,x , Θ t0,x ) satisfying the above system a causal feedback solution of the controlled SVIE (1.1) at (t 0 , x) corresponding to the causal feedback strategy (Ξ, Γ, v). This framework is different from that of [1,2,4,6] and more reasonable in view of the (generalized) flow property and the time-consistency of the state dynamics. For more detailed theory on the causal feedback strategies and the associated causal feedback solutions, see our previous paper [11].…”

“…Specifically, Abi Jaber, Miller and Pham [1] studied LQ stochastic Volterra control problems with completely monotone and convolution-type kernels. Based on an infinite-dimensional approach, they obtained a kind of a linear feedback representation of the optimal control; see also [2] for the study on the associated integral operator Riccati equation. We emphasize that the approaches of [1,2,4,6] heavily rely on the special structure of the completely monotone and convolution-type kernels, and they cannot be applied to the non-convolution-type SVIE (1.1).…”