On the Uniqueness of Martingales with Certain Prescribed Marginals

This note contains two main results. (1) (Discrete time) Suppose S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose the initial option prices agree with the Cox–Ross–Rubinstein binomial tree model.) Then S is a geometric simple random walk. (2) (Continuous time) Suppose S = S0eσX−σ 〈X〉/2 is a continuous martingale whose marginal laws agree with a geometric Brownian motion. (In financial terms, let S be a risk-neutral asset price and suppose the initial option prices agree with the Black–Scholes model with volatility σ > 0.) Then there exists a Brownian motion W such that Xt = Wt + o(t1/4+ ) as t ↑ ∞ for any > 0.