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3
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0answers
102 views

Finite stopping times

I've come across two statements in a proof that I don't really understand. Let $X_{i}$ be iid with values in $\{-1,0,1\}$ all with positive probability. Define $S_{n}=\sum_{i=0}^{n}X_{i}$ and the ...
0
votes
1answer
46 views

What is the optimal stopping point for an experiment when expecting unknown event

Assume we notice that stock prices are rising and we can deduce we are in a bubble. Assume we start at $w(0)=0$ worth at time $t=0$ and the value grows linearly with time $(w(t)=t)$. We know that ...
3
votes
2answers
298 views

Stopping time proof

Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider $$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process ...
0
votes
1answer
168 views

Stopping time for sum of iid random variables.

Suppose we have $m$-sided biased die. Let $X_i$ be the outcome of the $i$'th roll with the die. Furthermore let $\mathbb{P}[X_i=k]=p_k$ with $k \in \{1,...,m\}$. We define $T=\min\{n\text{ ...
3
votes
2answers
161 views

coupon collector problem for different number of copies of each coupon type

I would like to pose a question on a variation on the classical coupon collector's problem: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number of ...
4
votes
0answers
204 views

stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. ...
1
vote
0answers
51 views

Expectation related to renewal measure

Let $X_1,X_2,\ldots$ be i.i.d. random variables, and $S_n=X_1+\cdots+X_n$. Assume that $0 < \mathbf{E}(X_1) < \infty$ (but don't assume that the $X_i$ are $>0$). Let $N$ be the almost surely ...
6
votes
3answers
634 views

Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
1
vote
2answers
206 views

Stopping time for a martingale

Let $X_1,X_2,\ldots$ be iid random variables where $X_i\in\{-1,0,1,2,...\}$, $P(X_i=0)<1$ and $E(X_1)=\mu$. Let $S_n=1+X_1+\cdots+X_n$ and $T=\inf \{n:s_n=0\}$. Show that $E(T)=\infty$ if $\mu=0$ ...
2
votes
1answer
208 views

Doob's stopping time theorem with unbounded stopping time

Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$. ...
4
votes
1answer
279 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
1
vote
1answer
57 views

What is $1_{\{\tau_n>0\}}X^{\tau_n}$ process saying?

As title says, what is $1_{\{\tau_n>0\}}X^{\tau_n}$ process? I do have understanding of what stochastic processes are, but not sure what is this specific process saying.
3
votes
1answer
231 views

Favourable modification of “Double or Nothing”

I am working through 'Great expectations: the theory of optimal stopping' by Y.S. Chow, H. Robbins, D. Siegmund and cannot fill in the gap in reasoning regarding the existence of optimal stopping ...
2
votes
2answers
178 views

Expectation of a stopping time of a Wiener process

How can we calculate $\mathbb{E}(\tau)$ when $\tau=\inf\{t\geq0:B^2_t=1-t\}$? If we can prove that $\tau$ is bounded a.s. (i.e. $\mathbb{E}[\tau]<\infty$), then we can use the fact that ...
3
votes
1answer
178 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
1
vote
2answers
1k 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 ...
1
vote
0answers
920 views

Minimum of two stopping times is a stopping time.

So far I've already shown that the sum and the maximum of two stopping times is a stopping time, but the minimum is giving me some problems which I just can't get around. This is what I've tried: Let ...
3
votes
1answer
243 views

Stopping time on Wiener Process

Let $W_t$ be a Wiener process and for $a\geq0$ $$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$ Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?
7
votes
0answers
164 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...