Question: Let the process {X(t), t 0} be defined by X(t) = B2(t) t, where {B(t), t 0} is a standard Brownian motion.

Let the process {X(t), t ≥ 0} be defined by X(t) = B2(t) − t, where {B(t), t ≥ 0} is a standard Brownian motion.

a. What is E[X(t)]?

b. Show that {X(t), t ≥ 0} is a martingale.

Hint: Start by computing E[X(t)|B(v), 0 ≤ v ≤ s].

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