Linear Feedback of Mean-Field Stochastic Linear Quadratic Optimal Control Problems on Time Scales

This paper addresses a version of the linear quadratic control problem for mean-field stochastic differential equations with deterministic coefficients on time scales, which includes the discrete time and continuous time as special cases. Two coupled Riccati equations on time scales are given and the optimal control can be expressed as a linear state feedback. Furthermore, we give a numerical example.

Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities [Formula: see text] and [Formula: see text] and show that they satisfy the coupled Riccati equations, from which we find that [Formula: see text] satisfies the Painlevé V equation. Furthermore, we prove that [Formula: see text], a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo–Miwa–Okamoto [Formula: see text]-form of the Painlevé V. In the end, by using Dyson’s Coulomb fluid approach, we consider the large [Formula: see text] asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.