Neutral stochastic functional differential equations with Levy jumps under the local Lipschitz condition

In this paper, a general neutral stochastic functional differential equations with infinite delay and Lévy jumps (NSFDEwLJs) is studied. We investigate the existence and uniqueness of solutions to NSFDEwLJs at the phase space Cg under the local Carathéodory type conditions. Meanwhile, we also give the exponential estimates and almost surely asymptotic estimates of solutions to NSFDEwLJs

(SI10-083) Approximate Controllability of Infinite-delayed Second-order Stochastic Differential Inclusions Involving Non-instantaneous Impulses

This manuscript investigates a broad class of second-order stochastic differential inclusions consisting of infinite delay and non-instantaneous impulses in a Hilbert space setting. We first formulate a new collection of sufficient conditions that ensure the approximate controllability of the considered system. Next, to investigate our main findings, we utilize stochastic analysis, the fundamental solution, resolvent condition, and Dhage’s fixed point theorem for multi-valued maps. Finally, an application is presented to demonstrate the effectiveness of the obtained results

Fractional Differential Equations, Inclusions and Inequalities with Applications

During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering

Essays on information acquisition

This dissertation studies information acquisition when the choice of information is fully flexible. Throughout the dissertation, I consider a theoretical framework where a decision maker (DM) acquires costly information (signal process) about the payoffs of different alternatives before making a choice. In Chapter 1, I solve a general model where the DM pays a cost that depends on the rate of uncertainty reduction and discounts delayed payoffs. The main finding is that the optimal signal process resembles a Poisson signal --- the signal arrives occasionally according to a Poisson process, and it drives the inferred posterior belief to jump discretely. The optimal signal is chosen to confirm the DM's prior belief of the most promising state. Once seeing the signal, the decision maker is discretely surer about the state and stops learning immediately. When the signal is otherwise absent, the decision maker becomes gradually less sure about the state, and continues learning by seeking more precise but less frequently arriving signals. In Chapter 2, I study the sequential implementation of a target information structure. I characterize the set of decision time distributions induced by all signal processes that satisfy a per-period learning capacity constraint on the rate of uncertainty reduction. I find that all decision time distributions have the same mean, and the maximal and minimal elements by mean-preserving spread order are exponential distribution and deterministic distribution. The result implies that when the time preference is risk loving (e.g. standard or hyperbolic discounting), Poisson signal is optimal since it induces the riskiest exponential decision time distribution. When time preference is risk neutral (e.g. constant delay cost), all signal processes are equally optimal. In Chapter 3, I relax the assumption on information cost by assuming that the measure of signal informativeness is an indirect measure from sequential minimization. I first show that an indirect information measure is supported by sequential minimization iff it satisfies: 1) monotonicity in Blackwell order, 2) sub-additivity in compound experiments and 3) linearity in mixing with no information. Then I study a dynamic information acquisition problem where the cost of information depends on an indirect information measure and the delay cost is fixed (the DM is time-risk neutral). The optimal strategy is to acquire Poisson type signals. The result implies that when the cost of information is measured by an indirect measure, Poisson signals are intrinsically cheaper than other signal processes. Chapter 4 introduces a set of useful technical results on constrained information design that is used to derive the main results in the first three chapters