0
votes
0answers
37 views

Probability of Stopping Time Taking specific value - Random Walk 1d

We are considering a simple random walk $(X_n)_{n\in\mathbb{N}}$ starting at $X_0=0$ with $X_n=\sum_{i=1}^nY_i$ where $Y_i$ are iid and $\mathbb{P}(Y_i=1)=\mathbb{P}(Y_i=-1)=\frac{1}{2}$. We want to ...
0
votes
1answer
22 views

Expressing units of time

How would you express 8/3 seconds as time after 3pm ? 8/3 = 2.66666 0.66*60 =40 miliseconds = 0.04 seconds so 2.04 seconds after 3 3:00:02:04 pm ? Is this correct?
0
votes
2answers
54 views

Show $L$ is not a stopping time

Let $L = \sup\{ n : n \le 10; A_n \in B \}$, $B \in \mathcal B$, $\sup\{\emptyset \}=0$. $(A_n)_{n \ge1}$ is a process adapted by a natural filtration $\{\mathcal F_n\}.$ Show that $L$ is NOT a ...
0
votes
1answer
44 views

Expectation of stopping times

Let B = (Bt)t¸0 be a standard Brownian motion started at zero, let $X_t$ be a non negative stochastic process solving: $dX_t=1/X_tdt+dB_t$ Compute $E[\sigma]$ when $\sigma=\inf \{ t\ge 0 : X_t= 1 ...
2
votes
1answer
82 views

Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...
1
vote
2answers
179 views

$(S_n^2-n)_{n\ge 0}$ martingale and bounded stopping times

Consider the random walk $$S_n=\sum_{k}^{n}X_{k}$$ Where $X_k$'s are iid, $$\mathbb P(X_1=1)=\mathbb P(X_1=-1)=\frac{1}{2}$$ and $\mathcal{F}_{n}=\sigma(X_i,0\leq i\leq n)$. How do I prove that ...
0
votes
2answers
146 views

Martingale Stopping Time

Let $(S_n)_{n \ge 0}$ be a $(\mathcal F_n)$-martingale and $\tau$ a stopping time with finite expectation. Assume that there is a $c > 0$ such that, $\forall n, \mathbb E (|S_{n+1} - S_n | | ...
1
vote
2answers
901 views

Sum of two stopping times is a stopping time?

Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. Question: Is $\sigma + \tau$ a stopping time? Attempt at a solution: I ...