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Brownian motion is a continuous-time stochastic process M(t) with the following properties:

• M(0) = 0
• M(t) is normally distributed with mean 0 and variance t
• M(t) - M(s) is normally distributed with mean 0 and variance t-s for s < t
• M(t) is a martingale
• M(t) is a submartingale
• M(t) is a supermartingale

If M(t) is a martingale, then the expected value of M(t) given the information up to time s is equal to M(s) for all s < t.

If M(t) is a submartingale, then the expected value of M(t) given the information up to time s is greater than or equal to M(s) for all s < t.

If M(t) is a supermartingale, then the expected value of M(t) given the information up to time s is less than or equal to M(s) for all s < t.

If M(t) is a martingale and f is a convex function, then the integral of f(M(t)) with respect to t is a martingale.

If M(t) is a submartingale and f is a convex function, then the integral of f(M(t)) with respect to t is increasing.

If M(t) is a supermartingale and f is a convex function, then the integral of f(M(t)) with respect to t is decreasing.

If M(t) is a martingale and f is a convex function, then the integral of f(M(t)) with respect to t is a submartingale.

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