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Register a new userConstrained Optimal Transport
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The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice $cal{X}$ with a order unit. The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of $cal{X}$ and the dual problem is defined on the bi-dual of $cal{X}$. These results are then applied to several extensions of the classical optimal transport.
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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.
Let $Y$ be a sublattice of a vector lattice $X$. We consider the problem of identifying the smallest order closed sublattice of $X$ containing $Y$. It is known that the analogy with topological closure fails. Let $overline{Y}^o$ be the order closure of $Y$ consisting of all order limits of nets of elements from $Y$. Then $overline{Y}^o$ need not be order closed. We show that in many cases the smallest order closed sublattice containing $Y$ is in fact the second order closure $overline{overline{Y}^o}^o$. Moreover, if $X$ is a $sigma$-order complete Banach lattice, then the condition that $overline{Y}^o$ is order closed for every sublattice $Y$ characterizes order continuity of the norm of $X$. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.
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.
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.
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.
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