We develop a continuous-time Markov chain (CTMC) approximation of one-dimensional diffusions with sticky boundary or interior points. Approximate solutions to the action of the Feynman–Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second-order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short-rate model for a low-interest environment and option pricing under a geometric Brownian motion price model with a sticky interior point.

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching processes. Later, this method was extended to prove the L log L theorem for multiple- and general multiple-type branching processes in Biggins and Kyprianou (2004), Kurtz et al. (1997), and Lyons (1997). In this paper we extend this method to a class of superdiffusions and establish a Kesten–Stigum L log L type theorem for superdiffusions. One of our main tools is a spine decomposition of superdiffusions, which is a modification of the one in Englander and Kyprianou (2004).

We define sets with finitely ramified cell structure, which are generalizations of post-critically finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

We describe and implement a novel methodology for Monte Carlo simulation of one-dimensional killed diffusions. The proposed estimators represent an unbiased and efficient alternative to current Monte Carlo estimators based on discretization methods for the cases when the finite-dimensional distributions of the process are unknown. For barrier option pricing in finance, we design a suitable Monte Carlo algorithm both for the single barrier case and the double barrier case. Results from numerical investigations are in excellent agreement with the theoretical predictions.

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

Dynamic asset allocation strategies that are continuously rebalanced so as to always keep a fixed constant proportion of wealth invested in the various assets at each point in time play a fundamental role in the theory of optimal portfolio strategies. In this paper we study the rate of return on investment, defined here as the net gain in wealth divided by the cumulative investment, for such investment strategies in continuous time. Among other results, we prove that the limiting distribution of this measure of return is a gamma distribution. This limit theorem allows for comparisons of different strategies. For example, the mean return on investment is maximized by the same strategy that maximizes logarithmic utility, which is also known to maximize the exponential rate at which wealth grows. The return from this policy turns out to have other stochastic dominance properties as well. We also study the return on the risky investment alone, defined here as the present value of the gain from investment divided by the present value of the cumulative investment in the risky asset needed to achieve the gain. We show that for the log-optimal, or optimal growth policy, this return tends to an exponential distribution. We compare the return from the optimal growth policy with the return from a policy that invests a constant amount in the risky stock. We show that for the case of a single risky investment, the constant investor's expected return is twice that of the optimal growth policy. This difference can be considered the cost for insuring that the proportional investor does not go bankrupt.

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.

This paper is concerned with the problem of estimation for the diffusion coefficient of a diffusion process on R, in a non-parametric situation. The drift function can be unknown and considered as a nuisance parameter. We propose an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval. We describe the asymptotic behaviour of this estimator when the step of discretization tends to zero. We prove consistency and asymptotic normality, the rate of convergence to the normal law being a random variable linked to the local time of the diffusion or to its suitable discrete approximation. This can also be interpreted as a convergence to a mixture of normal law.

In this paper we extend the results of Meyn and Tweedie (1992b) from discrete-time parameter to continuous-parameter Markovian processes Φ evolving on a topological space.

We consider a number of stability concepts for such processes in terms of the topology of the space, and prove connections between these and standard probabilistic recurrence concepts. We show that these structural results hold for a major class of processes (processes with continuous components) in a manner analogous to discrete-time results, and that complex operations research models such as storage models with state-dependent release rules, or diffusion models such as those with hypoelliptic generators, have this property. Also analogous to discrete time, ‘petite sets', which are known to provide test sets for stability, are here also shown to provide conditions for continuous components to exist.

New ergodic theorems for processes with irreducible and countably reducible skeleton chains are derived, and we show that when these conditions do not hold, then the process may be decomposed into an uncountable orbit of skeleton chains.

The theory of robust non-linear filtering in Clark (1978) and Davis (1980), (1982) is used to evaluate the limiting conditional distribution of a diffusion, given an observation of a ‘rare-event' sample-path of the diffusion, as the signal-to-noise ratio and the diffusion noise-intensity converge to infinity and zero respectively. Under mild conditions it is shown that the limiting conditional distribution is a Dirac measure concentrated at a trajectory which solves a variational problem parametrised by the sample-path of the observed signal.

In this article we examine R&D projects where the project status changes according to a general dynamic stochastic equation. This allows for both continuous and jump behavior of the project status. The time parameter is continuous. The decision variable includes a non-stationary resource expenditure strategy and a stopping policy which determines when the project should be terminated. Characterization of stationary policies becomes straightforward in the present setting. A non-linear equation is determined for the expected discounted return from the project. This equation, which is of a very general nature, has been considered in certain special cases, where it becomes manageable. The examples include situations where the project status changes according to a compound Poisson process, a geometric Brownian motion, and a Brownian motion with drift. In those cases we demonstrate how the exact solution can be obtained and the optimal policy found.

The coupling method is well fitted to be used in the study of the asymptotics of one-dimensional diffusion processes. We give an elementary proof of Orey's theorem in the recurrent case, and establish rate results for tendency towards equilibrium under moment conditions on the speed measure and the initial distributions.

This paper uses martingale calculus in order to study the Markov properties of diffusion local time first discovered by Ray and Knight. The approach enables us to calculate the laws of the processes involved and is easily modified to deal with conditioning with respect to the excursion σ-fields.

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.