2

The 1-dimensional equation for a Brownian Bridge, $dY_t=\frac{b-Y_t}{1-t}dt + dB_t$; $0\le t < 1$, $Y_0=a$ has a solution $Y_t=a(1-t)+bt+(1-t)\int_0^t\frac{dBs}{1-s}$; $0 \le t < 1$.

Solutions on this site like Brownian bridge sde and https://math.stackexchange.com/a/410220/311948 use Ito's Isometry $\mathbb{E}\{X_t^2\}$ as the starting point for their proof that $\lim\limits_{t\to 1}Y_t=b$ a.s.

While I understand how Ito's Isometry can be used in general, and I understand how the remainder of the limit argument works afterwards, what is the motivation to start with $\mathbb{E}\{X_t^2\}$ ? It seems to come from nowhere without any explanation. Thank you

user86422
  • 41

1 Answers1

2

First to be clear, in Brownian bridge they are computing the second moment in order to prove $L^{2}$-convergence.

The a.s. convergence cannot be done just with second moment because Why does $L^2$ convergence not imply almost sure convergence.

However, as mentioned in the comment of Brownian bridge sde, one simply observes that

$$(Y_{t})_{t\in [0,1)}\stackrel{law}{=}(B_{t}-tB_{1}))_{t\in [0,1)}$$

as Gaussian processes by comparing their covariances and use that Gaussian processes are uniquely determined by their covariance (Gaussian Process uniquely determined by Covariance and mean?). And from here the a.s. convergence is immediate.

Thomas Kojar
  • 3,596
  • Thank you @thomas-kojar your second half makes sense. For the first part, the question in https://math.stackexchange.com/questions/1369834/brownian-bridge-sde asks for the limit a.s. not in $L^2$ though the comment starts with looking at the second moment and then getting the $L^2$ limit. I am still not understanding why to start with $L^2$ especially from your reference that it doesn't imply a.s. convergence. – user86422 Oct 10 '23 at 14:25
  • 1
    @user86422 In that particular question Shalop doesn't start with L2, he starts with comparing covariances "both are Gaussian processes with the same covariance kernel.". That's very different. Because covariances uniquely determine a Gaussian process. – Thomas Kojar Oct 10 '23 at 15:16