1,721,121 research outputs found
A pension fund in the accumulation phase: a stochastic control approach
No full textIn this paper we propose and study a continuous time stochastic model of optimal allocation for a defined contribution pension fund in the accumulation phase. The level of wealth is constrained to stay above a "solvency level". The fund manager can invest in a riskless asset and in a risky asset, but borrowing and short selling are prohibited. The model is naturally formulated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation. In the special case when the boundary is absorbing we show that it is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation
Erratum to: “Elasticity and permeability of porous fibre-reinforced materials under large deformations [Mech. Mater., 44, 58–71, 2012]”
Full text linkEfficient evaluation of the material response of tissues reinforced by statistically oriented fibres
Full text linkFor several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation
A Singular Stochastic Control Problem with Interconnected Dynamics
No full textFederico S, Ferrari G, Schuhmann P. A Singular Stochastic Control Problem with Interconnected Dynamics . SIAM Journal on Control and Optimization. 2020;58(5):2821-2853.In this paper we study a Markovian two-dimensional bounded-variation stochastic control problem whose state process consists of a diffusive mean-reverting component and of a purely controlled one. The main problem's characteristic lies in the interaction of the two components of the state process: the mean-reversion level of the diffusive component is an affine function of the current value of the purely controlled one. By relying on a combination of techniques from viscosity theory and free-boundary analysis, we provide the structure of the value function and we show that it satisfies a second-order smooth-fit principle. Such a regularity is then exploited in order to determine a system of functional equations solved by the two monotone continuous curves (free boundaries) that split the control problem's state space into three connected regions. Further properties of the free boundaries are also obtained
Irreversible reinsurance: minimization of capital injections in presence of a fixed cost
Full text linkWe propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cram & eacute;r-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model's parameters
Optimal Control of Stochastic Delay Differential Equations and Applications to Path-Dependent Financial and Economic Models
Full text linkIn this manuscript we consider a class of optimal control problems of stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we characterize the value function of the problem as the unique viscosity solution of the associated infinite-dimensional Hamilton-Jacobi-Bellman equation. Finally, we prove a C 1 Alpha partial regularity of the value function. We apply these results to path dependent financial and economic problems (Merton-like portfolio problem and optimal advertising)
On mean field games in infinite dimension
No full textWe study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman (HJB) equation in the paper, coupled with a nonlinear Fokker-Planck (FP) equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions
Linear Elastic Composites with Statistically Oriented Spheroidal Inclusions
Full text linkThe purpose of this chapter is to critically review some results that our
groups obtained in previous works, which were devoted to the investigation of the
elastic properties of composite materials with a statistical distribution of spheroidal
inclusions. These studies were motivated by our interest in the description of
mechanical properties of fibre-reinforced biological tissues (such as articular cartilage), starting from the internal structure of these tissues. After an introduction to
tensor algebra, which defines the notation and clarifies the mathematical framework
adopted in the chapter, we present, in a covariant setting inspired by Differential
Geometry, Walpole’s representation of isotropic and transversely isotropic secondand fourth-order tensors, along with its properties. Hence, starting from Eshelby’s
seminal work on the problem of an inclusion in an infinite matrix, we briefly review
the theories developed by Hill, Walpole and Weng for the determination of the
overall elasticity tensor of materials with one or more inclusion phases. Then, we
discuss in detail the cases of composite materials with aligned spheroidal inclusions
and with statistically oriented spheroidal inclusions. Emphasis is put on extending
Walpole’s formula to the case of inclusions aligned according to some probability
density of orientation, both in the transversely isotropic and the isotropic case
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