Wald's martingale

In probability theory Wald's martingale, named after Abraham Wald and more commonly known as the geometric Brownian motion, is a stochastic process of the form

\left\{\exp \left(\lambda W_{t}-{\frac {1}{2}}\lambda ^{2}t\right),t\geq 0\right\}

for any real value λ where W_t is a Wiener process.^[1]^: 32^[2]^: 261^[3] The process is a martingale.^[1]

Notes

^ ^a ^b Hunt, P. J.; Kennedy, J. E. (2005). "Martingales". Financial Derivatives in Theory and Practice. Wiley Series in Probability and Statistics. p. 31. doi:10.1002/0470863617.ch3. ISBN 9780470863619.
^ Chang, F. R. (2004). "Boundaries and Absorbing Barriers". Stochastic Optimization in Continuous Time. pp. 225–287. doi:10.1017/CBO9780511616747.008. ISBN 9780511616747.
^ Asmussen, S. R.; Kella, O. (2000). "A multi-dimensional martingale for Markov additive processes and its applications". Adv. Appl. Probab. 32 (2): 376–393. CiteSeerX 10.1.1.49.9292. doi:10.1239/aap/1013540169. JSTOR 1428194.

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