We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

In contrast to the standard approach in energy economics, this paper assumes that physical capital accumulation is relatively more energy-intensive than consumption. By means of Gaussian hypergeometric functions, we provide a closed-form representation of the optimal solution paths of our variables in levels, whatever the initial conditions (i.e., global dynamics). We find that, in general, the optimal trajectories are nonmonotonic. P.S. Dasgupta and G.M. Heal [Review of Economic Studies (special number), 3–28 (1974)] pointed out this result (local dynamics) for the optimal consumption in a model with identical technology for both physical capital accumulation and consumption. However, our paper introduces five novelties with respect to their study, namely, global dynamics, the importance of the proportion of nonrenewable energy resources to endowment of physical capital [S(0)/K(0)], the role of technical progress, U-shaped behavior of consumption, and nonmonotonicity of resource extraction.

Let Mn denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of Mn/n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

Convergence results are given for transient characteristics of an M/M/∞ system such as the period of time the occupation process remains above a given state, the area swept by this process above this state and the number of customers arriving during this period. These results are precise in contrast to approximations derived in the framework of the Poisson clumping heuristic introduced by Aldous.