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Register a new userOptimal Consumption in the Stochastic Ramsey Problem without Boundedness Constraints
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This paper investigates optimal consumption in the stochastic Ramsey problem with the Cobb-Douglas production function. Contrary to prior studies, we allow for general consumption processes, without any a priori boundedness constraint. A non-standard stochastic differential equation, with neither Lipschitz continuity nor linear growth, specifies the dynamics of the controlled state process. A mixture of probabilistic arguments are used to construct the state process, and establish its non-explosiveness and strict positivity. This leads to the optimality of a feedback consumption process, defined in terms of the value function and the state process. Based on additional viscosity solutions techniques, we characterize the value function as the unique classical solution to a nonlinear elliptic equation, among an appropriate class of functions. This characterization involves a condition on the limiting behavior of the value function at the origin, which is the key to dealing with unbounded consumptions. Finally, relaxing the boundedness constraint is shown to increase, strictly, the expected utility at all wealth levels.
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We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a consumption constraint. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion $0<alphale1$ of her consumption habit, which is the exponentially-weighted average of past consumption rates. $alpha=1$ prohibits the habit process to decrease and corresponds to the completely addictive model. $alpha=0$ makes the habit-formation constraint moot and corresponds to the non-addictive model. $0<alpha<1$ leads to partially addictive models, with the level of addictiveness increasing with $alpha$. In contrast to the existing habit-formation literature, our constraint cannot be incorporated in the objective function through infinite marginal utility. Assuming that the individual invests in a risk-free market, we formulate and solve an infinite-horizon, deterministic control problem to maximize the discounted CRRA utility of the consumption-to-habit process subject to the habit-formation constraint. Optimal consumption policies are derived explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals consume at the minimum rate if their wealth-to-habit ratio is below a threshold, and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either: (1) larger than a high threshold; or (2) below a low threshold and the agent is less risk averse. Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.
A price-maker company extracts an exhaustible commodity from a reservoir, and sells it instantaneously in the spot market. In absence of any actions of the company, the commoditys spot price evolves either as a drifted Brownian motion or as an Ornstein-Uhlenbeck process. While extracting, the company affects the market price of the commodity, and its actions have an impact on the dynamics of the commoditys spot price. The company aims at maximizing the total expected profits from selling the commodity, net of the total expected proportional costs of extraction. We model this problem as a two-dimensional degenerate singular stochastic control problem with finite fuel. To determine its solution, we construct an explicit solution to the associated Hamilton-Jacobi-Bellman equation, and then verify its actual optimality through a verification theorem. On the one hand, when the (uncontrolled) price is a drifted Brownian motion, it is optimal to extract whenever the current price level is larger or equal than an endogenously determined constant threshold. On the other hand, when the (uncontrolled) price evolves as an Ornstein-Uhlenbeck process, we show that the optimal extraction rule is triggered by a curve depending on the current level of the reservoir. Such a curve is a strictly decreasing $C^{infty}$-function for which we are able to provide an explicit expression. Finally, our study is complemented by a theoretical and numerical analysis of the dependency of the optimal extraction strategy and value function on the models parameters.
In this work we analytically solve an optimal retirement problem, in which the agent optimally allocates the risky investment, consumption and leisure rate to maximise a gain function characterised by a power utility function of consumption and leisure, through the duality method. We impose different liquidity constraints over different time spans and conduct a sensitivity analysis to discover the effect of this kind of constraint.
While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.
Optimized certainty equivalents (OCEs) is a family of risk measures widely used by both practitioners and academics. This is mostly due to its tractability and the fact that it encompasses important examples, including entropic risk measures and average value at risk. In this work we consider stochastic optimal control problems where the objective criterion is given by an OCE risk measure, or put in other words, a risk minimization problem for controlled diffusions. A major difficulty arises since OCEs are often time inconsistent. Nevertheless, via an enlargement of state space we achieve a substitute of sorts for time consistency in fair generality. This allows us to derive a dynamic programming principle and thus recover central results of (risk-neutral) stochastic control theory. In particular, we show that the value of our risk minimization problem can be characterized via the viscosity solution of a Hamilton--Jacobi--Bellman--Issacs equation. We further establish the uniqueness of the latter under suitable technical conditions.
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