Finding the expectation of a characteristic function conditioned on a gaussian random variable.

Question

For $X_t$ is a Brownian bridge, I have to find $E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}]$.

I can find the distribution of $X_{4/5}-\frac{1}{2} X_{3/5}$ with no issues, and I can see that $E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}]$ appears to be the characteristic function of $X_{4/5}-\frac{1}{2} X_{3/5}$. However, I'm not sure how to proceed when the function is conditional on a variable. Would appreciate some assistance.

Answer 1

$E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}]=E[e^{iu([X_{4/5}-X_{3/5}]+\frac{1}{2} X_{3/5})}|X_{3/5}]=e^{iu/2X_{3/5}} E[e^{iu(X_{4/5}-X_{3/5})}]$. Now you can write down $E[e^{iu(X_{4/5}-X_{3/5})}]$ using the fact that $X_{4/5}-X_{3/5} \sim N(0,1/5)$