Question: Write down a formula for (ar) where X-N(4.0) and, by differentiating. or otherwise, derive an expression for E(Xeax). (ii) Show that: [2] X =(B,

Write down a formula for E(eat) where X-N(4.0) and, by differentiating. or otherwise, derive an expression Let B, (120) be a standard Brownian motion process starting with B=0. Show that, when s <1, E(B,B,)=s. Hence A student has said: (i) What is meant by saying that the process (Y) is a martingale with respect to another process {X,)? [2]

Write down a formula for (ar) where X-N(4.0) and, by differentiating. or otherwise, derive an expression for E(Xeax). (ii) Show that: [2] X =(B, -at)e-, is a martingale, where B, is a standard Brownian motion, and a is an arbitrary constant. You may assume that E[X] < [5] [Total 7] Let B, (120) be a standard Brownian motion process starting with B=0. Show that, when s

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