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Expected exponential of Brownian motion

MediumStochastic Processes~4m

For standard Brownian motion, what is E ⁣[eW4]E\!\left[e^{W_4}\right] (the expected value of ee raised to the position at time 44, to four decimals)?

Show hints (2)+
  1. Jensen: E[eW4]≠eE[W4]E[e^{W_4}]\ne e^{E[W_4]}; use the Gaussian MGF.
  2. E[eX]=em+s2/2E[e^{X}]=e^{m+s^2/2} with m=0m=0, s2=4s^2=4, so e2e^{2}.

Answer

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7.3891 (± 0.001)

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