Medium · Stochastic Calculus · Free response · ~10 min · #27

Show W² − t Is a Martingale

Real interview · J.P. Morgan

Let $W_t$ be a standard Brownian motion with natural filtration $\mathcal{F}_t$ . Prove that $M_t = W_t^2 - t$ is a martingale, i.e. for $s < t$ ,

\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s .

(You may take integrability for granted.)

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Concepts

Martingale propertyIndependent incrementsQuadratic variation

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