Optimal growth trajectories with finite carrying capacity

In this work, a growing network model that can generate a random network with finite degree in infinite time is studied. The dynamics are governed by a rule where the degree increases under a scheme similar to the Malthus-Verhulst model in the context of population growth. The degree distribution is analysed in both stationary and time-dependent regimes through some exact results and simulations, and a scaling behaviour is found in asymptotically large time. For finite times,...

Modern ecology has re-emphasized the need for a quantitative understanding of the original 'survival of the fittest theme' based on analyzis of the intricate trade-offs between competing evolutionary strategies that characterize the evolution of life. This is key to the understanding of species coexistence and ecosystem diversity under the omnipresent constraint of limited resources. In this work we propose an agent based model replicating a community of interacting individua...

We consider the problem of utility maximization for investors with power utility functions. Building on the earlier work Larsen et al. (2016), we prove that the value of the problem is a Frechet-differentiable function of the drift of the price process, provided that this drift lies in a suitable Banach space. We then study optimal investment problems with non-Markovian driving processes. In such models there is no hope to get a formula for the achievable maximal utility. A...

In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to study a manager's decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the...

This paper studies the investment exercise boundary erasing in a stochastic, continuous time capacity expansion problem with irreversible investment on the finite time interval $[0, T]$ and a state dependent scrap value associated with the production facility at the finite horizon $T$. The capacity process is a time-inhomogeneous diffusion in which a monotone nondecreasing, possibly singular, process representing the cumulative investment enters additively. The levels of capa...

We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the ...

Reflected diffusions naturally arise in many problems from applications ranging from economics and mathematical biology to queueing theory. In this paper we consider a class of infinite time-horizon singular stochastic control problems for a general one-dimensional diffusion that is reflected at zero. We assume that exerting control leads to a state-dependent instantaneous reward, whereas reflecting the diffusion at zero gives rise to a proportional cost with constant margina...

The original Kelly criterion provides a strategy to maximize the long-term growth of winnings in a sequence of simple Bernoulli bets with an edge, that is, when the expected return on each bet is positive. The objective of this work is to consider more general models of returns and the continuous time, or high frequency, limits of those models.

Von Neumann-Gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of such systems is played by a special class of paths (trajectories) called rapid: they grow over each time period t-1,t in a sense faster than others. The paper establishes existence and characterization theorems for such paths showing, in particular, that any trajectory maximizi...

This paper is concerned with solutions to a one dimensional linear diffusion equation and their relation to some problems in stochastic control theory. A stochastic variational formula is obtained for the logarithm of the solution to the diffusion equation, with terminal data which is the characteristic function of a set. In this case the terminal data for the control problem is singular, and hence standard theory does not apply. The variational formula is used to prove conve...