One kind of corporate optimal portfolio and consumption choice problem is studied for a investor who can invest his wealth in the bond (bank account) and in a real project which has the production. The bank pays at an...
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(纸本)9787810778022
One kind of corporate optimal portfolio and consumption choice problem is studied for a investor who can invest his wealth in the bond (bank account) and in a real project which has the production. The bank pays at an interest rate for any deposit and takes at a large rate for any loan. The optimal strategies are obtained by Hamilton-Jacobi-Bellman equation which is derived from dynamic programming principle. We also give the economic analysis to the optimal choice using the investment theory. For the specific Hyperbolic Absolute Risk Aversion case, we get the explicit optimal investment and consumption solution. At last, we give some simulation results to illustrate the optimal result and the influence of the volatility parameter on the optimal choice.
In this paper we introduce a new approach to discrete-time semi-Markov decision processes based on the sojourn time process. Different characterizations of discrete-time semi-Markov processes are exploited and decisio...
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In this paper we introduce a new approach to discrete-time semi-Markov decision processes based on the sojourn time process. Different characterizations of discrete-time semi-Markov processes are exploited and decision processes are constructed by their means. With this new approach, the agent is allowed to consider different actions depending also on the sojourn time of the process in the current state. A numerical method based on Q-learning algorithms for finite horizon reinforcement learning and stochastic recursive relations is investigated. Finally, we consider two toy examples: one in which the reward depends on the sojourn-time, according to the gambler's fallacy;the other in which the environment is semi-Markov even if the reward function does not depend on the sojourn time. These are used to carry on some numerical evaluations on the previously presented Q-learning algorithm and on a different naive method based on deep reinforcement learning.
We provide a representation formula for viscosity solutions to an elliptic Dirichlet problem involving Pucci's extremal operators. This is done through a dynamic programming principle derived from Denis et al. (20...
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We provide a representation formula for viscosity solutions to an elliptic Dirichlet problem involving Pucci's extremal operators. This is done through a dynamic programming principle derived from Denis et al. (2010). The formula can be seen as a nonlinear extension of the Feynman-Kac formula. (C) 2020 Elsevier Ltd. All rights reserved.
We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Diri...
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We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.
The study of epidemics using mathematical modelling is critical in understanding its dynamics and proposing potential control measures. We propose a generalised epidemiological model corresponding to a pandemic wherei...
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The study of epidemics using mathematical modelling is critical in understanding its dynamics and proposing potential control measures. We propose a generalised epidemiological model corresponding to a pandemic wherein its dynamics is represented as a novel hybrid system obtained by coupling a deterministic model with a stochastic model. The hybrid system dynamics is established in individualistic (macroscopic) and intraindividualistic (microscopic) scales. The established hybrid system is then considered the basis for an optimal control problem, with the rate of vaccination and velocity of spatial dynamics taken as the control parameters affecting the system's trajectory. We define the cost functional constituted by the continuous cost corresponding to the deterministic model and discrete costs corresponding to the transitions in the microscopic scale. The objective of the control problem is to find an optimal control pair of vaccination rate and spatial velocity, which minimises the cost functional. We use the dynamic programming principle (DPP) as the optimisation technique, followed by verification of the value function obtained by DPP as a viscosity solution of the appropriate Hamilton-Jacobi-Bellman equation to analyse the existence of an optimal control pair to the hybrid system. We prove the existence of optimal controls to the multi -scale dynamics for pandemic modelling, along with an abstract method to synthesise it.
We investigate a model of hybrid control system in which both discrete and continuous controls are involved. In this general model, discrete controls act on the system at a given set interface. The state of the system...
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We investigate a model of hybrid control system in which both discrete and continuous controls are involved. In this general model, discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where the controller can choose to jump or not. At each jump, the trajectory can move to a different Euclidean space. We prove the continuity of the associated value function V with respect to the initial point. Using the dynamic programming principle satisfied by V, we derive a quasi-variational inequality satisfied by V in the viscosity sense. We characterize the value function V as the unique viscosity solution of the quasi-variational inequality by the comparison principle method.
We obtain an asymptotic Holder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where t...
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We obtain an asymptotic Holder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.
This paper is concerned with the two-player zero-sum stochastic differential game in a regime switching model with an infinite horizon. The state of the system is characterized by a number of diffusions coupled by a c...
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This paper is concerned with the two-player zero-sum stochastic differential game in a regime switching model with an infinite horizon. The state of the system is characterized by a number of diffusions coupled by a continuous-time finite-state Markov chain. Based on the dynamic programming principle (DPP), the lower and upper value functions are shown to be the unique viscosity solutions of the associated lower and upper Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Moreover, the lower and upper value functions coincide under the Isaacs' condition, which implies that the game admits a value. All the proofs in this paper are markedly different from those for the case when there is no regime switching. (C) 2020 Elsevier Ltd. All rights reserved.
In this paper, we revisit the optimal consumption and portfolio selection problem for an investor who has access to a risk-free asset (e.g. bank account) with constant return and a risky asset (e.g. stocks) with const...
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In this paper, we revisit the optimal consumption and portfolio selection problem for an investor who has access to a risk-free asset (e.g. bank account) with constant return and a risky asset (e.g. stocks) with constant expected return and stochastic volatility. The main contribution of this study is twofold. Our first objective is to provide an explicit solution for dynamic portfolio choice problems, when the volatility of the risky asset returns is driven by the Ornstein-Uhlenbeck process, for an investor with a constant relative risk aversion (CRRA). The second objective is to carry out some numerical experiments using the derived solution in order to analyze the sensitivity of the optimal weight and consumption with respect to some parameters of the model, including the expected return on risky asset, the aversion risk of the investor, the mean-reverting speed, the long-term mean of the process and the diffusion coefficient of the stochastic factor of the standard Brownian motion.
The problem of ergodic control of a reflecting diffusion in a compact domain is analysed under the condition of partial degeneracy, i.e. when its transition kernel after some time is absolutely continuous with respect...
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The problem of ergodic control of a reflecting diffusion in a compact domain is analysed under the condition of partial degeneracy, i.e. when its transition kernel after some time is absolutely continuous with respect to the Lebesgue measure on a part of the state space. Existence of a value function and a “martingale dynamic programming principle” are established by mapping the problem to a discrete time control problem. Implications for existence of optimal controls are derived.
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