Optional Stopping Theorem - Statement of Theorem

Statement of Theorem

A discrete-time version of the theorem is given below:

Let X = (X_t)_t∈ℕ0 be a discrete-time martingale and τ a stopping time with values in ℕ₀ ∪ {∞}, both with respect to a filtration (F_t)_t∈ℕ0. Assume that one of the following three conditions holds:

(a) The stopping time τ is almost surely bounded, i.e., there exists a constant c ∈ ℕ such that τ ≤ c a.s.

(b) The stopping time τ has finite expectation and the conditional expectations of the absolute value of the martingale increments are almost surely bounded, more precisely, and there exists a constant c such that almost surely on the event {τ > t} for all t ∈ ℕ₀.

(c) There exists a constant c such that |X_t∧τ| ≤ c a.s. for all t ∈ ℕ₀.

Then X_τ is an almost surely well defined random variable and

Similarly, if the stochastic process X is a submartingale or a supermartingale and one of the above conditions holds, then

for a submartingale, and

for a supermartingale.