Unique Continuous Solution Research Articles

On systems of fractional differential equations with the ψ‐Caputo derivative and their applications

Systems of fractional differential equations with a general form of fractional derivative are considered. A unique continuous solution is derived using the Banach fixed point theorem. Additionally, the dependence of the solution on the fractional order and on the initial conditions are studied. Then the stability of autonomous linear fractional differential systems with order 0<α<1 of the ψ‐Caputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leader‐follower consensus for fractional multi‐agent systems is presented. Sufficient conditions are given to ensure that the tracking errors asymptotically converge to zero. The results of the paper are illustrated by some examples.

Zika virus dynamics partial differential equations model with sexual transmission route

Inspired by the system of ordinary differential equations in Agusto et al. (2017) that models Zika virus dynamics by taking into account of both sexual and vector-borne transmissions, we furthermore add diffusive terms in order to capture the movement of human hosts and mosquitoes, considering the unique threat of the sexual transmission route of Zika virus. We conduct complete theoretical analysis. In particular, we show that every initial data that is continuous and non-negative admits a unique continuous and non-negative solution for all positive times. Moreover, we derive the basic reproduction number and when it is beneath one, we prove that the disease-free-equilibrium is globally attractive. Finally, when the basic reproduction number is above one, and any of the exposed males or the exposed females or the exposed mosquitoes is not identically zero, we prove the existence of a positive asymptotic lower bound for every component of the solution which in particular implies the uniform persistence of the disease, as well as the existence of at least one positive steady state.

Existence Result for a Superlinear Fractional Navier Boundary Value Problems

In this paper, we study the following fractional Navier boundary value problem Dβ(Dαu)(x)=u(x)g(u(x)),x∈(0,1),limx⟶0x1-βDαu(x)=-a,u(1)=b,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{lllc} D^{\\beta }(D^{\\alpha }u)(x)=u(x)g(u(x)),\\quad x\\in (0,1), \\\\ \\displaystyle \\lim _{x\\longrightarrow 0}x^{1-\\beta }D^{\\alpha }u(x)=-a,\\quad \\,\\,u(1)=b, \\end{array} \\right. \\end{aligned}$$\\end{document}where alpha ,beta in (0,1] such that alpha +beta >1, D^{beta } and D^{alpha } stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that a+b>0. The function g is a nonnegative continuous function in [0,infty ) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.

Existence of a solution for the problem with a concentrated source in a subdiffusive medium

By using Green's function, the problem is converted into an integral equation. It is shown that there exists a $t_b$ such that, for $0\leq t\lt t_b$, the integral equation has a unique nonnegative continuous solution $u$; if $t_b$ is finite, then $u$ is unbounded in $[0, t_b)$. Then, $u$ is proved to be the solution of the original problem.

Optimal Market Making Under Partial Information With General Intensities

Starting from the Avellaneda–Stoikov framework, we consider a market maker who wants to optimally set bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. We tackle this stochastic control problem under partial information with a model that unifies and generalizes many existing ones under full information, combining several risk metrics and constraints, and using general decreasing intensity functionals. We use stochastic filtering, control and piecewise-deterministic Markov processes theory, to reduce the dimensionality of the problem and characterize the reduced value function as the unique continuous viscosity solution of its dynamic programming equation. We then solve the analogous full information problem and compare the results numerically through a concrete example. We show that the optimal full information spreads are biased when the exact market regime is unknown, and the market maker needs to adjust for additional regime uncertainty in terms of P&L sensitivity and observed order flow volatility. This effect becomes higher, the longer the waiting time in between orders.

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity, as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem. We provide the optimal control as the solution of a reflected diffusion at a suitable boundary surface. Such boundary arises from the analysis of a family of two-dimensional parameter-dependent optimal stopping problems, and it is characterized in terms of the family of unique continuous solutions to parameter-dependent, nonlinear integral equations of Fredholm type.

On Neumann problems for nonlocal Hamilton\u2013Jacobi equations with dominating gradient terms

We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton–Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order strictly less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron’s method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data.

On Generalized Proportional Allocation Policies for Traffic Signal Control

The fast-increasing demand and relatively slow growth of infrastructure capacity are providing a strong motivation for research in real-time urban traffic controls that make the best use of novel sensing in order to increase efficiency and resilience of the transportation system. In our contribution, we focus on a class of dynamic feedback traffic signal control policies that are based on a generalized proportional allocation rule. The proposed traffic signal controls are decentralized (they make use of local information only), scalable (they are independent of the network size and topology), and universal (they do not rely on any information about external inflows or turning ratios). In spite of their fully distributed nature, we prove that such control policies achieve a global objective, maximum throughput, in that they stabilize the urban traffic network whenever possible under the given capacity constraints.The traffic model we consider consists in a network of interconnected vertical queues with deterministic dynamics driven by physical laws (conservation of mass and preservation of non-negativity of the traffic volumes) as well as scheduling constraints (described as a set of phases, each phase consisting in a subset of lanes that can be be given green light simultaneously). This results in a differential inclusion for which we prove existence and, in the special case of orthogonal phases, uniqueness of continuous solutions via a generalization of the reflection principle. Stability is then proved by interpreting the generalized proportional allocation controllers as minimizers of a certain entropy-like function that is then used as a Lyapunov function for the closed-loop system.

On the Solutions of Systems of Nonlinear Functional Equations Continuous in t ∈

For a broad class of systems of nonlinear functional equations, we establish conditions for the existence and uniqueness of continuous solutions. A method for the construction of solutions of this kind and investigation of their properties as e → 0 is proposed.

An age-structured SIR epidemic model with fixed incubation period of infection

This paper is devoted to the study of an age-structured SIR epidemic system on the basis of the model of polycyclic population dynamics of susceptible, infected and recovered individuals. This model was considered as a nonlinear competitive system of three initial–boundary value problems for the nonlinear transport equations with non-local integral boundary conditions and discrete time delay (fixed incubation period of infection). It was obtained the explicit recurrent formulae for computing the traveling wave solution of such system provided the model parameters (coefficients of equations and initial values) satisfy the restrictions that guarantee the existence and uniqueness of continuous solution. Using some insignificant simplifications the age-structured SIR model was reduced to the nonlinear autonomous system of delay ODE. It was studied the dimensionless indicators and conditions of existence of trivial disease-free equilibrium, two non-trivial endemic equilibriums for the incurable and curable infection-induced diseases, respectively. By carrying out an analysis of the local asymptotical stability of the solution of such system we obtain the restrictions for the time delay that guarantee the stability of solutions in the neighborhood of equilibriums. The numerous simulations of dynamics of SIR population illustrate the results of theoretical analysis.

SIMULTANEOUS TRADING IN ‘LIT’ AND DARK POOLS

We consider an optimal trading problem over a finite period of time during which an investor has access to both a standard exchange and a dark pool. We take the exchange to be an order-driven market and propose a continuous-time setup for the best bid price and the market spread, both modeled by Lévy processes. Effects on the best bid price arising from the arrival of limit buy orders at more favorable prices, the incoming market sell orders potentially walking the book, and deriving from the cancellations of limit sell orders at the best ask price are incorporated in the proposed price dynamics. A permanent impact that occurs when ‘lit’ pool trades cannot be avoided is built in, and an instantaneous impact that models the slippage, to which all lit exchange trades are subject, is also considered. We assume that the trading price in the dark pool is the mid-price and that no fees are due for posting orders. We allow for partial trade executions in the dark pool, and we find the optimal trading strategy in both venues. Since the mid-price is taken from the exchange, the dynamics of the limit order book also affects the optimal allocation of shares in the dark pool. We propose a general objective function and we show that, subject to suitable technical conditions, the value function can be characterized by the unique continuous viscosity solution to the associated partial integro-differential equation. We present two explicit examples of the price and the spread models, derive the associated optimal trading strategy numerically. We discuss the various degrees of the agent's risk aversion and further show that roundtrips are not necessarily beneficial.

Systems of Linear and Nonlinear Equations with Partial Integrals

We establish the existence and uniqueness of continuous solutions to systems of linear and nonlinear integral equations with partial integrals, with partial integrals and potential type kernels, and fractional partial integrals. We obtain conditions guaranteeing that the solutions have continuous partial derivatives and are continuously differentiable.

A General Verification Result for Stochastic Impulse Control Problems

This paper establishes existence of optimal controls for a general stochastic impulse control problem. For this, the value function is characterized as the pointwise minimum of a set of superharmonic functions, as the unique continuous viscosity solution of the quasi-variational inequalities (QVIs), and as the limit of a sequence of iterated optimal stopping problems. A combination of these characterizations is used to construct optimal controls without relying on any regularity of the value function beyond continuity. Our approach is based on the stochastic Perron method and the assumption that the associated QVIs satisfy a comparison principle.

Continuous dependence of the solution of a stochastic differential equation with nonlocal conditions

In this paper we are concerned with a nonlocal problem of a stochastic differential equation that contains a Brownian motion. The solution contains both of mean square Riemann and mean square Riemann-Steltjes integrals, so we study an existence theorem for unique mean square continuous solution and its continuous dependence of the random data X 0 and the (non-random data) coefficients of the

Utility Maximization in a Factor Model with Constant and Proportional Costs

We study the problem of maximizing expected utility of terminal wealth under constant and proportional transactions costs in a multi-asset market in which prices are driven by a multidimensional factor process. We characterize the value function as the pointwise minimum of a set of superharmonic functions and as the unique continuous viscosity solution of the quasi-variational inequalities associated with the optimization problem. A combination of these characterizations is then used as a basis for the construction of optimal strategies.

Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact

In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control andmarket regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.

Regular finite fuel stochastic control problems with exit time

We consider a class of exit time stochastic control problems for diffusion processes with discounted criterion, where the controller can utilize a given amount of resource, called “fuel”. In contrast to the vast majority of existing literature, concerning the “finite fuel” problems, it is assumed that the intensity of fuel consumption is bounded. We characterize the value function of the optimization problem as the unique continuous viscosity solution of the Dirichlet boundary value problem for the correspondent Hamilton–Jacobi–Bellman (HJB) equation. Our assumptions concern the HJB equations, related to the problems with infinite fuel and without fuel. Also, we present computer experiments for the problems of optimal correction and optimal tracking of a simple stochastic system with the stable or unstable equilibrium point.

A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients

We study the level set equation in a bounded domain when the velocity of the interface is given by the mean curvature plus a discontinuous velocity. We prove a comparison principle for the initial-boundary value problem whose consequence is uniqueness of continuous solutions and well- posedness of the level set method.

Continuous dependence of the solution of Ito stochastic differential equation with nonlocal conditions

In this paper we are concerned with a nonlocal problem of a stochastic differential equation that contains a Brownian motion. The solution contains both of mean square Riemann and mean square Riemann-Steltjes integrals, so we study an existence theorem for unique mean square continuous solution and its continuous dependence of the random data X 0 and the (non-random data) coefficients of the nonlocal condition ak. Also, a stochastic differential equation with the integral condition will be considered.

Integral equations, transformations, and a Krasnoselskii-Schaefer type fixed point theorem

In this paper we extend the work begun in 1998 by the author and Kirk for integral equations in which we combined Krasnoselskii’s fixed point theorem on the sum of two operators with Schaefer’s fixed point theorem. Schaefer’s theorem eliminates a difficult hypothesis in Krasnoselskii’s theorem, but requires an a priori bound on solutions. Here, we simplify the work by means of a transformation which often reduces the a priori bound to a triviality. Our work is focused on an integral equation in which the goal is to prove that there is a unique continuous positive solution on [0, ∞). In addition to the transformation, there are two techniques which we would emphasize. A technique is introduced yielding a lower bound on the solutions which enables us to deal with problems threatening non-uniqueness. The technique offers a solution to a classical problem and it seems entirely new. We show that when the equation defines the sum of a contraction and a Lipschitz operator, then we first get existence on arbitrary intervals [0, E] and then introduce a technique which we call a progressive contraction which allows us to prove uniqueness and then parlay the solution to [0, ∞). The technique is well suited to integral equations.