Random time vs stopping time & expectation

Let $$B_t$$ be a standard Brownian motion. Let us introduce a stopping time $$\tau$$ as $$\tau \ = \ \ \inf\Big\{t:\ |B_t|=1\Big\}.$$ Now, what I am interested in are random variables $$\tau_x$$ defined as $$\tau_x(\omega) \ = \ x\cdot\tau(\omega),$$ for $$x> 0$$. Obviously, $$\tau_x$$ is not a stopping time anymore. Nevertheless, we can still consider a random variable $$B_{\tau_x}(\omega) \ = \ B_{\tau_x(\omega)}(\omega),$$ and it is (if I understand correctly) still well defined, i.e. measurable - as claimed in

My question is, are basic quantities such as $$\mathbb{E}\Big(B_{\tau_\frac{1}{2}}\Big|B_{\tau}=1\Big)$$ computable? How can we calculate it? My crude intuition is, that $$\mathbb{E}\Big(B_{\tau_x}\Big|B_{\tau}=1\Big) \ = \ x,$$ as $$\Delta:=B_{\tau}-B_{0}=1$$ and on average we should make $$x$$ of it in the first $$x$$ part of random $$\tau$$ time.

I would be glad for any help and insight.

• If $x \geq 1$, then $\tau_x$ is a stopping time. Feb 10, 2021 at 1:57
• Though it does seem like you are primarily interested in $x \in (0,1)$. Feb 10, 2021 at 1:57
• Another comment about your intuition. What if $\tau$ were the inf time $t$ such that $B_t = 1$ or $B_t = -\epsilon$ for some small $\epsilon \in (0,1)$? If one is given $B_\tau = 1$, then to me it seems that we should have $E(B_{\tau_{1/2}} | B_\tau = 1) > x$. To me your intuition seems to fit better with $\tau$ being the inf $t$ such that $B_t = 1$. Of course this is all just intuition, but I wanted to ask your perspective. Feb 10, 2021 at 2:03
• Yes, I now see that You are right. Your intuition is better.
– user617199
Feb 10, 2021 at 10:22
• Let $\bar{\tau}$ be the first hitting time of $1$. Let $x\in(0,1)$. Recall that by properties of Brownian bridges, for any $T>0$ the distribution of $B_{xT}$ given $B_T = 1$ is Gaussian with mean $x$ and variance $(1-x)xT$. I think this is the connection to the intuition. But to apply this result directly, one would need the conditional distribution of $B_{x\bar{\tau}}$ given $\bar{\tau} = T$ to correspond to the conditional distribution of $B_{xT}$ given $B_T = 1$. I'm quite skeptical about this. (The same considerations apply for the slightly less simple first passage time $\tau$.)
– user350942
Feb 10, 2021 at 13:34

The following R-code serves to compute the mean of $$B_{\tau/2}$$ conditionally on $$B_\tau = 1$$. Paths of Brownian motions on $$[0,1]$$ are simulated via independent Gaussian increments. For the subsample where $$\tau \leq 1$$ and $$B_{\tau} = 1$$, the values $$B_{\tau/2}$$ are computed, if necessary via linear interpolation. The mean is then approximated by taking the average of these values.

(Intuitively, conditioning on $$\tau \leq 1$$, shouldn't be too problematic; but a rigorous argument escapes me.)

I find that $$\mathbb{E}[B_{\tau/2} \, | \, B_\tau = 1] \approx 0.26$$. Replacing $$\tau$$ by $$\bar{\tau} := \inf\{t > 0 : B_t = 1\}$$, one also finds $$\approx 0.26$$. I find these numerical results to be somewhat consistent with an earlier comment of mine.

set.seed(2)

N <- 5000
n <- 5000

h <- 1/n

X <- apply(rbind(rep(0,N),matrix(rnorm(n*N, mean = 0, sd = sqrt(h)), nrow = n, ncol = N)), 2, cumsum)

tau <- apply(X, 2, function(y){Position(function(x){x>=1 | x<=-1}, y)})

B_tau <- vector()
B_tau_half <- vector()
for (i in 1:N){
if(!is.na(tau[i]) & X[tau[i],i] >= 1){
B_tau <- c(B_tau,X[tau[i],i])
if(tau[i] %% 2 == 0){
B_tau_half <- c(B_tau_half,X[tau[i]/2,i])
}
else{
B_tau_half <- c(B_tau_half,X[(tau[i]-1)/2,i]/2 + X[(tau[i]+1)/2,i]/2)
}
}
}

mean(B_tau_half)

• Thank You very much, this is very informative.
– user617199
Feb 10, 2021 at 16:12
• You're welcome.
– user350942
Feb 10, 2021 at 16:18
• In my own simulation I'm finding the restriction $\tau \leq 1$ plays a significant role. Using simulating up to a max time of $100$ I find $E[B_{\tau/2}] \approx .20$, but simulating up to $1$ I find closer to what you had $\approx .27$. Feb 10, 2021 at 19:20
• Further numerical studies are needed to really understand what is going on, agreed.
– user350942
Feb 10, 2021 at 22:14