This is a stochastic calculus question being asked by a physicist who has not previously delved deeply into stochastic calculus. I apologise therefore for (presumably) mangling the terminology.
I am trying to find the variance of a stochastic random variable $X(t)$ in two different (but related) contexts. In both cases, the variable $X(t)$ represents the displacement from the origin, at time $t$, of a particle that moves in only one dimension. The particle always starts at the origin $X(0)=0$ and is assumed to have its position vary as the result of integrating its velocity over time:$$X(t) = \int_{s=0}^t V(t) dt$$ where the velocity $V(t)$ is itself assumed to be proportional to a standard Brownian Bridge $B(t)$ having $B(0)=B(T)=0$ and Var$(B(t))=t(T-t)/T$. Specifically, we can take the constant of proportionality to be a constant $\lambda$ so that $$V(t) = \lambda B(t).$$
The two cases for which the variance of $X(t)$ are to be compared are:
- In the presence of no other constraints, and
- In the presence of the additional requirement that everything is conditional on $X(T)=0$.
Crudely put, we want to look at how the variance of $X(t)$ is affected by the presence (or absence) of conditioning on the requirement that the particle ends back at the origin. In this latter case, not only is the velocity a "bridge", but the position is a "bridge" too!
I am reasonably sure that I can solve by myself the case where there is no conditional constraint on $X(T)$ - but I do not propose to put my solution here lest it distract from the main issue -- which is how to get an answer with the additional $X(T)=0$ constraint. That I have no idea how to tackle.
Looking through maths stack exchange, I get the impression that my question may just be a special case of the more general question asked here: "Conditional distribution" of Brownian sample paths , and indeed it is possible that the answer given there can be used somehow to answer mine. However, the more general nature of the question posed there has perhaps contributed to its answer being posed in terminology and notation that is way beyond my level of understanding. I am hoping, therefore, that by making my question concrete, it may be possible to give answers that are more at my level.
After posting the above question, a friend pointed me toward the paper by Alain Mazzolo published in J. Stat. Mech. (2017) 023203 at https://doi.org/10.1088/1742-5468/aa4f15 .
In particular, the quantity called $B^c_t$ in equation (49) of the above paper is identical to the the $B(t)$ in the doubly constrained version of my question, provided that the values $A$ and $x_f$ it contains are both set to zero.
We may therefore calculate Var$(X(t))$ by integrating that expression in (49):$$X(t)=\lambda \int_{s=0}^t B^c_s ds$$ and then crunching through the calculation of Var$(X(t))$.
At the end of that calculation I find that Var$(X(t))$ for the doubly constrained case is:$$\lambda^2 \frac {\frac 1 3 t^3 (T-t)^3}{T^3}.$$
Denoting the variance of $X(t)$ in the singly and doubly constrained cases as $S(t)$ and $D(t)$ respectively, we therefore seem to have:$$ D(t) = \frac{S(t) S(T-t)}{S(T)}$$ which is just the sort of nice relationship I was hoping to see.