Analysis of a stochastic 2D–Navier-Stokes model with infinite delay

Some results concerning a stochastic 2D Navier-Stokes system when the external forces contain hereditary characteristics are established. The existence and uniqueness of solutions in the case of unbounded (infinite) delay are first proved by using the classical technique of Galerkin approximations. The local stability analysis of constant solutions (equilibria) is also carried out by exploiting two approaches. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. The asymptotic stability and hence, the uniqueness of equilibrium solution are obtained by constructing Lyapunov functionals. Moreover, some sufficient conditions ensuring the polynomial stability of the equilibrium solution in a particular case of unbounded variable delay will be provided. Exponential stability for other special cases of infinite delay remains as an open problem.Ministerio de Economía y Competitividad (MINECO). EspañaJunta de AndalucíaNational Natural Science Foundation of ChinaScience and Technology Commission of Shanghai Municipalit

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form as well as in the usual sense using acceptable portfolios.Comment: Final version. To appear in Mathematics and Financial Economics. Keywords: Proportional Transaction Costs, Unbounded Random Endowments, Acceptable Portfolios, Super-hedging Theorem, Utility Maximization, Shadow Prices, Convex Dualit

Backward stochastic differential equations with unbounded coefficients and their applications

In this thesis, we focus on problems on the theory of Backward Stochastic Differential Equations (BSDEs). In particular, BSDEs with an unbounded generator are considered, under various conditions (on the generator). Using more general (or weaker) conditions, the classical results on BSDEs are improved and some associated problems on mathematical finance are resolved. Chapter 1 introduces some of the literature, general setting and ideas in this field and emphasises the motivations which has led to the study of these equations. In addition, some mathematical preliminaries we used throughout this thesis are included in Chapter 2. In Chapter 3, we consider nonlinear BSDEs with an unbounded generator. Under a Lipschitz-type condition, we show sufficient conditions for the existence and uniqueness of solutions to nonlinear BSDEs, which are weaker than the existing ones. We also give a comparison theorem as a generalisation of Peng's result. Chapter 4 studies a class of backward stochastic differential equations whose generator satisfies linear growth and continuity conditions, which can also be unbounded. We prove the existence of the solution pair for this class of equations which is more general than the existing ones. In Chapter 5, we consider the problem of solvability for linear backward stochastic differential equations with unbounded coefficients. New and weaker sufficient conditions for the existence of a unique solution pair are given. It is shown that certain exponential processes have stronger integrability in this case. As applications, we solve the problems of completeness in a market with a possibly unbounded coefficients and optimal investment with power utility in a market with unbounded coefficients. Chapter 6 studies the classical Stochastic Differential Equations where the drift and diffusion coefficients satisfy Lipschitz-type and linear growth conditions, which can also be unbounded. We give sufficient conditions for the existence of a unique solution to unbounded SDEs. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Chapter 7 summaries the results in this thesis and outlines possible directions for future works based on current results

Мiшана задача для нелiнiйного параболiчного рiвняння з другою похiдною за часом в необмеженiй областi

We have obtained some sufficient conditions of existence of a generalized solution of the initial boundary-value problem for a nonlinear parabolic equation of the fourth order with the second time derivative in an unbounded domain with respect to spatial variables

Stepping Stone Problem on Graphs

This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1’s to some vertices and assign higher numbers to other vertices by adding up their neighbors’ assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained

Existence of solutions for fourth order three-point boundary value problems on a half-line

WOS: 000365262600001In this paper, we apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory to establish the existence of unbounded solutions for the following fourth order three-point boundary value problem on a half-line x''''(t) + q(t) f(t, x(t), x'(t), x ''(t), x'''(t)) = 0, t is an element of (0, +infinity), x ''(0) = A, x(eta) = B-1, x'(eta) = B-2, x'''(+infinity) = C, where eta is an element of (0, +infinity), but fixed, and f : [0, +infinity) x R-4 -> R satisfies Nagumo's condition. We present easily verifiable sufficient conditions for the existence of at least one solution, and at least three solutions of this problem. We also give two examples to illustrate the importance of our results.Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK)This work was done when the first author was on academic leave, visiting Texas A&M University-Kingsville, Department of Mathematics. He gratefully acknowledges the financial support of The Scientific and Technological Research Council of Turkey (TUBITAK)

Minimum time problem with impulsive and ordinary controls

Given a nonlinear control system depending on two controls u and v, with dynamics affine in the (unbounded) derivative of u and a closed target set depending both on the state and on the control u, we study the minimum time problem with a bound on the total variation of u and u constrained in a closed, convex set U, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function T. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize T as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints

Minimizing Spectral Risk Measures Applied to Markov Decision Processes

We study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in B\"auerle and Ott (2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital