First Hitting Place Probabilities for a Discrete Version of the Ornstein‐Uhlenbeck Process

Abstract

A Markov chain with state space{0,…,N}and transition probabilities depending on the current state is studied. The chain can be considered as a discrete Ornstein-Uhlenbeck process. The probability that the process hitsNbefore 0 is computed explicitly. Similarly, the probability that the process hitsNbefore−Mis computed in the case when the state space is{−M,…,0,…,N}and the transition probabilitiespi,i+1are not necessarily the same wheniis positive andiis negative.

On a discrete version of the CIR process

“…Previously, the authors had considered such problems for a discrete-time Markov chain that converges to a geometric Brownian motion (see [14]), which is also a fundamental process in financial mathematics. For their part, Lefebvre and Guilbault [13], as well as Guilbault and Lefebvre [9], have treated the case of Markov chains converging to an Ornstein -Uhlenbeck process.…”

Analysis of the Discrete Ornstein-Uhlenbeck Process Caused by the Tick Size Effect

“…Comparing (9) to (8), we see that these two equations differ only in the truncation error term TE = O(h 2 ). This indicates that the effect of using moment matching to carry out the discretisation is essentially equivalent to applying central differencing schemes to the Kolmogorov forward equations.…”

“…For example, Renshaw [10] considered the correlated random walk with friction as a discrete OU process and derived the expressions for its moment generating function and some statistics including variance and kurtosis. Larralde [7] studied the first passage time distribution while Lefebvre and Guilbault [9] studied the first hitting place probabilities. Larralde [8] discussed statistical properties and characteristic functions of time and state discrete OU processes.…”

Numerical determination of hitting time distributions from their Laplace transforms: One dimensional diffusions

“…[15] or [16] to cite two sources explaining classical material. Consider as well Larralde [17,18], Lefebvre and Gilbault [19], Miao [20] for variations on the theme of the discrete versions of the OU process. Here, in order to estimate the Laplace transform φ(α, x) by a Monte Carlo procedure, we will follow a rather simple approach using the explicit version of the solution given by (9).…”