Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the integral and complete the squares on the exponent and compute the Gaussian integral one interval at a time then see what simplification would come out when the number of intervals approaches infinity. Is there a more straightforward and insightful way to accomplish this? Is Fourier transform of any use due to the convolution-like form in the discretized version?

  • $\begingroup$ If you want to check it for yourself here is how it goes (I think): take the representation of BB as $X_t= X_1.t+(1-t).\int_0^t \frac{dW_s}{1-s}$ where W is independent Brownian motion, then use Fubini (stochastic) to derive $\int_0^1 X_s ds$ as a Wiener integral+ deterministic term (so it is a Gaussian r.v.) derive the 1st two moments and conclude. Warning !!! check the integrability conditions for the fubini to to be valid. Best ergards $\endgroup$ – TheBridge Apr 25 '16 at 16:04
  • $\begingroup$ @TheBridge: Excellent. Thank you. Would $X_t=W_t−tW(1)$ where $W_t$ is the Brownian motion be a simpler representation for the Brownian bridge for our purpose? $\endgroup$ – Hans Apr 25 '16 at 19:33
  • $\begingroup$ @TheBridge: Would you please take a look at my other question regarding a generalization of the Brownian bridge math.stackexchange.com/q/1758608/64809? $\endgroup$ – Hans Apr 25 '16 at 22:03

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