Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces

In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls U can be chosen to be non-convex but now the space of controls U can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

A stochastic maximum principle for the risk-sensitive optimal control prob- lem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.

Maximum principle for anticipated recursive stochastic optimal control problem with delay and Lvy processes

In this paper, we study the stochastic maximum principle for optimal control prob- lem of anticipated forward-backward system with delay and Lovy processes as the random dis- turbance. This control system can be described by the anticipated forward-backward stochastic differential equations with delay and L^vy processes （AFBSDEDLs）, we first obtain the existence and uniqueness theorem of adapted solutions for AFBSDEDLs; combining the AFBSDEDLs＇ preliminary result with certain classical convex variational techniques, the corresponding maxi- mum principle is proved.

LOWER INEQUALITIES OF HEAT SEMIGROUPS BY USING PARABOLIC MAXIMUM PRINCIPLE

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly p-local Dirichlet form on the abstract metric measure space. As an application we obtain lower estimates for heat kernels on some Riemannian manifolds.

A MAXIMUM PRINCIPLE FOR DISTRIBUTED PARAMETER SYSTEMS WITH MIXED PHASE-CONTROL CONSTRAINTS ANDENDPOINT CONSTRAINTS

This paper considered the optimal control problem for distributed parameter systems with mixed phase-control constraints and end-point constraints. Pontryagin's maximum principle for optimal control are derived via Duboviskij-Milujin theorem.

OPTIMAL FEED STRATEGY FOR FED-BATCH GLYCEROL FERMENTATION DETERMINED BY MAXIMUM PRINCIPLE

Optimal glucose feed strategy for glycerol fed-batch fermentation was investigated by Pontryagin’s maximum principle to maximize the final glycerol yield. The problem was solved by a nonsingular control approach by selecting the culture volume as the control variable, then the general optimal feed profile was numerically determined.

A MAXIMUM PRINCIPLE APPROACH TO STOCHASTIC H_2/H_∞ CONTROL WITH RANDOM JUMPS

A necessary maximum principle is given for nonzero-sum stochastic Oltterential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2/H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.

MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS

This paper is concerned with a Pontryagin's maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables.By virtue of the duality method and the generalized anticipated backward stochastic differential equations,we establish a necessary maximum principle and a sufficient verification theorem.In particular,we deal with the controlled stochastic system where the distributed delays enter both the state and the control.To explain the theoretical results,we apply them to a dynamic advertising problem.

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.

ANNOUNCEMENT ON“MAXIMUM PRINCIPLE FOR NON-UNIFORMLY PARABOLIC EQUATIONS AND APPLICATIONS”

In this note we announce the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi’s iteration.In particular,the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion coefficients is obtained.

Maximum Principle for Stochastic Control System with Elephant Memory and Jump Diffusion

Motivated by a duopoly game problem,the authors study an optimal control problem where the system is driven by Brownian motion and Poisson point process and has elephant memory for the control variable and the state variable.Firstly,the authors establish the unique solvability of an anticipated backward stochastic differential equation,derive a stochastic maximum principle,and prove a verification theorem for the aforementioned optimal control problem.Furthermore,the authors generalize these results to nonzero-sum stochastic differential game problems.Finally,the authors apply the theoretical results to the duopoly game problem and obtain the corresponding Nash equilibrium solution.

The Accelerated Expansion of the Universe in the Light of the Maximum Ordinality Principle

The main aim of the paper is to present (and at the same time offer) a differ-ent perspective for the analysis of the accelerated expansion of the Universe. A perspective that can surely be considered as being “in parallel” to the tradition-al ones, such as those based, for example, on the hypotheses of “Dark Matter” and “Dark Energy”, or better as a “com-possible” perspective, because it is not understood as being “exclusive”. In fact, it is an approach that, when con-firmed by experimental results, always keeps its validity from an “operative” point of view. This is because, in analogy to the traditional perspectives, on the basis of Popper’s Falsification Principle the corresponding “Generative” Logic on which it is based has not the property of the perfect induction. The basic difference then only consists in the fact that the Evolution of the Universe is now modeled by considering the Universe as a Self-Organizing System, which is thus analyzed in the light of the Maximum Ordinality Principle.

A Finite Volume Method Preserving Maximum Principle for the Conjugate Heat Transfer Problems with General Interface Conditions

In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solution and the DMP-preserving property.Numerical experiments show that the nonlinear iteration numbers of the scheme in[24]increase rapidly when the interfacial coefficients decrease to zero.In contrast,the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero,which reveals that the unified scheme is more robust than the scheme in[24].The accuracy and DMP-preserving property of the scheme are also veri ed in the numerical experiments.

G-stochastic maximum principle for risk-sensitive control problem and its applications

This study advances the G-stochastic maximum principle(G-SMP)from a risk-neutral framework to a risk-sensitive one.A salient feature of this advancement is its applicability to systems governed by stochastic differential equations under G-Brownian motion(G-SDEs),where the control variable may influence all terms.We aim to generalize our findings from a risk-neutral context to a risk-sensitive performance cost.Initially,we introduced an auxiliary process to address risk-sensitive performance costs within the G-expectation framework.Subsequently,we established and validated the correlation between the G-expected exponential utility and the G-quadratic backward stochastic differential equation.Furthermore,we simplified the G-adjoint process from a dual-component structure to a singular component.Moreover,we explained the necessary optimality conditions for this model by considering a convex set of admissible controls.To describe the main findings,we present two examples:the first addresses the linear-quadratic problem and the second examines a Merton-type problem characterized by power utility.

Three-Drug Therapies in Psychiatry in the Light of the Maximum Ordinality Principle and the Explicit Solution to the “Three-Body Problem”—D.D. 23 Luglio 2023, Tempo Ordinario (3.00 e 10.20)

The present paper aims at showing the possible adoption in Psychiatry of a general methodology finalized to prescribe the most appropriate Therapy based on the knowledge of its correlative effects in advance, instead of recognizing them ex post. The specific case here considered is the “bipolar disorder”, in which the adoption of three different drugs is the most common practice, although with a possible differentiation between the prescription in the morning and in the evening, respectively. Thus, the proposed methodology will consider the Ordinal Interactions between the various drugs by evaluating their combined effects, which will result as being not a simple additive “sum”, because they are evaluated on the basis of the Maximum Ordinality Principle (MOP) and, in addition, in Adherence to the Explicit Solution to the “Three-Body Problem”. In this way the Methodology here proposed is able to suggest how to account for the synergistic effects of the various drugs, especially when the latter are characterized by different concentrations and, at the same time, by generally different half-lives respectively.

Solution to the “Three-Body Problem” in the Light of the Maximum Ordinality Principle, as a “Suggestion” for a Ri-Orientation of the Present Scientific Perspective in “Favor” of the “Irreducible Quality”

This paper presents the Solution to the “Three-body Problem” in the Light of the Maximum Ordinality Principle. In the first part, however, it starts with the Solution to the Solar System, made up of “11 Bodies”. This is because, in such a context, the “Three-body Problem” can be analyzed in its all descriptive possibilities. Nonetheless, the paper also presents the Solution to the “Three-body Problem” with reference to Systems totally independent from the Solar System, such as, for example, the “Triple Stars” and the “Triple Galaxies”. In this way, the paper offers a sufficiently complete framework concerning the Solution to the “Three-body Problem”, always in the Light of the Maximum Ordinality Principle, described in detail in Appendix A.

Generativity of Self-Organizing Processes and Their Correlative Description in Terms of a Formal Language of Meta-Ordinal Generative Nature, in the Light of the Maximum Ordinality Principle and the Explicit Solution to the “Three-Body Problem”

The main objective of this paper is to demonstrate that the internal processes of Self-Organizing Systems represent a unique and singular process, characterized by their specific generativity. This process can be modeled using the Maximum Ordinality Principle and its associated formal language, known as the “Incipient” Differential Calculus (IDC).

On Modifications of Continuous and Discrete Maximum Principles for Reaction-Diffusion Problems

In this work,we present and discuss some modifications,in the form of two-sided estimation(and also for arbitrary source functions instead of usual sign-conditions),of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.

IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

Stochastic global maximum principle for optimization with recursive utilities

In this paper,we study the recursive stochastic optimal control problems.The control domain does not need to be convex,and the generator of the backward stochastic differential equation can contain z.We obtain the variational equations for backward stochastic differential equations,and then obtain the maximum principle which solves completely Peng’s open problem.