South Africa is a net importer of fertilizer products, importing all of its potassium, as well as 60–70% of its nitrogen requirements. Thus, domestic prices are impacted significantly by international prices, shipping costs, and exchange rates. Producing these fertilizers locally would be far more economical. Phlogopite, a rich source of potassium, is discarded in large quantities during mining operations; the objective of the present study, therefore, was to determine the acid-leaching characteristics and behavior of phlogopite as a means of releasing potassium. Phlogopite samples were leached with nitric acid (source of nitrogen for fertilizers) at various concentrations, temperatures, and reaction times. The feed phlogopite and leached residue samples corresponding to conversions of 14% (LP1), 44% (LP2), and 100% (LP3) were collected and analyzed using X-ray fluorescence spectroscopy (XRF), X-ray diffractometry (XRD), attenuated total reflectance-Fourier transform infrared spectroscopy (ATR-FTIR), Brunauer–Emmett–Teller surface area and porosity analysis (BET), thermogravimetric analysis (TGA), and field emission gun-scanning electron microscopy (FEG-SEM). The feed phlogopite was highly crystalline. The absence of defects in the lattice meant that the motion of H+ atoms penetrating into the lattice was restricted, suggesting internal diffusion-controlled leaching. Furthermore, results obtained from the various analytical techniques corroborated each other in terms of the release of cations during leaching. All leaching experiments were conducted batchwise, in a closed system. The gravimetric data from the experiments were used to identify a suitable model which predicts accurately the leaching behavior. The reaction was found to be internal diffusion-controlled, and the D1 model, which represents one-dimensional diffusion through a flat plate, predicts the leaching behavior most accurately. The observed activation energies (Ea) and pre-exponential constants (k0) varied with initial nitric acid concentration ([H+]0).

We study the static maximization of long-term averaged profit, when optimal preset thresholds are determined to describe a pairs trading strategy in a general one-dimensional ergodic diffusion model of a stochastic spread process. An explicit formula for the expected value of a certain first passage time is given, which is used to derive a simple equation for determining the optimal thresholds. Asymptotic arbitrage in the long run of the threshold strategy is observed.

In this paper we establish a new connection between a class of two-player nonzero-sum games of optimal stopping and certain two-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover, a differential link between the players' value functions holds across the two games.

In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

In this note we propose an exact simulation algorithm for the solution of (1)

\begin{equation} \label{eds-intro} {\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t,\quad X_0=x, \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}\mathrm{\right)}\mathrm{d}\mathit{t,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x,}$

\hbox{$\bar{b}$}b̅

\hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}b̅(0 + ) ≠b̅(0 −)

(2)

\begin{equation} \label{edsbeta} {\rm d}X^\beta_t={\rm d}W_t+\bar{b}(X^\beta_t){\rm d}t + \beta {\rm d}L^0_t(X^\beta),\quad X_0=x \end{equation}$\mathrm{d}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{=}\mathrm{d}{\mathit{W}}_{\mathit{t}}\mathrm{+}\mathit{b̅}\mathrm{\left(}{\mathit{X}}_{\mathit{t}}^{\mathit{\beta }}\mathrm{\right)}\mathrm{d}\mathit{t}\mathrm{+}\mathit{\beta }\mathrm{d}{\mathit{L}}_{\mathit{t}}^{\mathrm{0}}\mathrm{\left(}{\mathit{X}}^{\mathit{\beta }}\mathrm{\right)}\mathit{,}\hspace{1em}{\mathit{X}}_{\mathrm{0}}\mathrm{=}\mathit{x}$

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d Xt=σ(Xt,Xt)dWt, where Xt= m ∧ inf0 ≤s≤tXs. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula for X. We also show that (Xt,Xt) admits a joint density, which can be characterized in terms of two independent tied-down Brownian meanders (or, equivalently, two independent Bessel-3 bridges). Finally, we show that the joint density satisfies a generalized forward Kolmogorov equation in a weak sense, and we derive a new forward equation for down-and-out call options.

We study quasistationary distributions on a drifted Brownian motion killed at 0, when +∞ is an entrance boundary and 0 is an exit boundary. We prove the existence of a unique quasistationary distribution and of the Yaglom limit.

We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as the expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order 1 / √N, where N is the sample size, is achieved, which is in sharp contrast to the slower nonparametric rates achieved by kernel smoothing of cumulative distribution functions.

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.