Introduction to Stochastic Calculus With Applications: Second Edition by Fima C Klebaner

Introduction to Stochastic Calculus With Applications: Second Edition by Fima C Klebaner

Author:Fima C Klebaner
Language: eng
Format: epub
Publisher: Imperial College Press


Thus we have shown that when p ≥ 1/2 any positive state will be reached from 0 in a finite time, but when p = 1/2 the average time for it to happen is infinite.

The results obtained above are known as transience (p ≠ 1/2) and recurrence (p = 1/2) of the Random Walk, and are usually obtained by Markov Chains Theory.

Example 7.9: (Optional stopping of discrete time martingales)

Let M(t) be a discrete time martingale and be a stopping time such that E|M()| < ∞.

1. If E < ∞ and |M(t + 1) − M(t)| ≤ K, then EM() = EM(0).

2. If E < ∞ and E(|M(t + 1) − M(t)||Ft) ≤ K, then EM() = EM(0).

PROOF: We prove the first statement.

M(t) = M(0)|+ (M(i+1) − M(i)). This together with the bound on increments gives

M(t) ≤ |M(0)| +|M(i + 1) − M(i)| ≤ |M(0)| + Kt.

Take for simplicity non-random M(0) Then

EM(t)I( > t) ≤ |M(0)|P( > t) + KtP( > t).

The last term converges to zero, tP( > t) ≤ E( I( > t)) → 0, by dominated convergence due to E( ) < ∞. Thus condition (7.8) holds, and the result follows. The proof of the second statement is similar and is left as an exercise.



Download



Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.