In this paper, we consider a stochastic optimal control problem, in which the cost function is defined through a reflected backward stochastic differential equation in sublinear expectation framework. Besides, we stud...
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In this paper, we consider a stochastic optimal control problem, in which the cost function is defined through a reflected backward stochastic differential equation in sublinear expectation framework. Besides, we study the regularity of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton-Jacobi-Bellman-Isaac equation.
We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, na...
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We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one time-step robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the S&P500$S\&P\nobreakspace 500$. We present two different types of ambiguity sets;one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.
This paper studies the optimal consumption-investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the indiv...
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This paper studies the optimal consumption-investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption-investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption-investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results. (C) 2016 Elsevier B.V. All rights reserved.
We study the stochastic control problem of maximizing expected utility from terminal wealth under a nonbankruptcy constraint. The problem of the agent is to derive the optimal insurance strategy which reduces his expo...
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We study the stochastic control problem of maximizing expected utility from terminal wealth under a nonbankruptcy constraint. The problem of the agent is to derive the optimal insurance strategy which reduces his exposure to the risk. This optimization problem is related to a suitable dual stochastic control problem in which the delicate boundary constraints disappear. We characterize the dual value function as the unique viscosity solution of the corresponding Hamilton Jacobi Bellman Variational Inequality (HJBVI in short). We characterize the optimal insurance strategy by the solution of the variational inequality which we solve numerically by using an algorithm based on policy iterations.
We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the p -Laplacian with 1 < p < 2 . For this version, the asymptotic Holder continuity of solutions can ...
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We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the p -Laplacian with 1 < p < 2 . For this version, the asymptotic Holder continuity of solutions can be directly derived from recent Krylov-Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.
In this paper, we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept o...
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In this paper, we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time consistency as well as any method that is based on Pontryagin's maximum principle. The fundamental obstacle is the dilemma of having to invoke the dynamic programming principle (DPP) in a time-inconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the "dynamic utility" under which the original time-inconsistent problem (under the fixed utility) becomes a time-consistent one. As a benchmark model, we shall consider a stochastic controlled problem with multidimensional backward SDE dynamics, which covers many existing time-inconsistent problems in the literature as special cases;and we argue that the time inconsistency is essentially equivalent to the lack of comparison principle. We shall propose three approaches aiming at reviving the DPP in this setting: the duality approach, the dynamic utility approach and the master equation approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem.
We consider an impulse control problem with switching technology in infinite horizon. We suppose that the firm decides at certain time (impulse time) to switch the technology and the firm value (e.g. a recapitalizatio...
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We consider an impulse control problem with switching technology in infinite horizon. We suppose that the firm decides at certain time (impulse time) to switch the technology and the firm value (e.g. a recapitalization). We show that the value function for such problems satisfies a dynamic programming principle. Our objective is to look for an optimal strategy which maximizes the value function.
In this work we propose a stochastic model for a sequencing-batch reactor (SBR) and for a chemostat. Both models are described by systems of Stochastic Differential Equations (SDEs), which are obtained as limits of su...
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It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In...
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It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the "Optimized Certainty Equivalent" risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.
We prove existence and uniqueness of viscosity solutions for the problem max {-Delta(p1)u(x), -Delta(p2)u(x)} = f(x) in a bounded smooth domain Omega subset of R-N with u = g on partial derivative Omega. Here -Delta(p...
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We prove existence and uniqueness of viscosity solutions for the problem max {-Delta(p1)u(x), -Delta(p2)u(x)} = f(x) in a bounded smooth domain Omega subset of R-N with u = g on partial derivative Omega. Here -Delta(p)u = (N + p)(-1) vertical bar Du vertical bar(2-p) div (vertical bar Du vertical bar(p-2)Du) is the 1-homogeneous p-Laplacian and we assume that 2 <= p(1), p(2) <= infinity. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-of-war game (without noise) or playing at random. Moreover, the operator max {-Delta(p1)u(x), -Delta(p2)u(x)} provides a natural analogue with respect to p-Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.
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