First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion

Abstract

We consider a discrete-time Markov chain with state space {1, 1 Δx, . . . , 1 kΔx N}. We compute explicitly the probability p j that the chain, starting from 1 jΔx, will hit N before 1, as well as the expected number d j of transitions needed to end the game. In the limit when Δx and the time Δt between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that p j and d j Δt tend to the corresponding quantities for the geometric Brownian motion.

First Passage Time of a Markov Chain That Converges to Bessel Process

“…}, starting from 0 = 1 + Δ , takes to reach either 1 or 1 + Δ = . Unlike the results of [6], we find the explicit formula of the average number of transitions needed to end the game. …”

“…In Section 2, we briefly describe the transition probability derived from the Bessel stochastic differential equation. Our main contribution is in the third section (Section 3.2): We find the explicit formula of the average number of transitions needed to end the game that was impossible to obtain in [6]. We also show the sequence of the probability of the first passage times and the average number of transitions to end the game converges (with Euclidean metric) to the corresponding values in the continuous case.…”

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.…”