# Tagged Questions

22 views

### Distribution of the first exit time of a one-dimensional diffusiom/ Brownian motion

I have a one-dimensional diffusion on $[0,1]$ and I need to calculate the distribution of the first exit time of the interval $(-\epsilon,\epsilon)$ for an $\epsilon > 0$. A good first step would ...
30 views

### Exercise on stopping times

Let $(Y_n)_{n \geq 1}$ be a sequence of independent r.v.'s s.t. $$P(Y_n=y) = {n \choose k } \left(\frac1n\right)^y \left(1-\frac1n\right)^{n-y}\quad {\rm if }\;y \in \{0,1,\dots,n\}$$ How to show ...
55 views

### Uniform integrability and stopping times

I want to know whether there is any example where $X_n$ is uniformly integrable, $N$ is a stopping time and $E[X_N] =\infty$? Or uniform integrability of $X_n$ implies that $E[X_N]< \infty$?
34 views

### Stopped process not uniformly integrable

I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT ...
66 views

### Counterexample for uniform integrability of a stopped process

I want to find an example where $X_n$ is uniformly integrable, $N$ is a stopping time, but $X_n^N = X_{\min\{n,N\}}$ in not uniformly integrable. There is a theorem saying that if $M_n$ is a uniform ...
33 views

28 views

### Martingale representation theorem , optimal stopping time and the principal agent problem

I am self-learning some Econ papers. Any suggestion will be appreciated. Even though the questions are from an Econ paper, they are math-related. I provide the economic interpretation as background ...
32 views

### Simple question regarding stopping times.

I have this exercise regarding stopping times, but I am not able to solve it. You have a probability space $(\Omega, \mathcal{F},P)$, with a filtration $\{\mathcal{F}_t\}$}. You have two stopping ...
58 views

### Showing a stopping time is finite

Let $T = \inf\{ n : S_n = a \text{ or } S_n = -b\}$ be a stopping time, where $S_n = X_1 + \dots +X_n$ and each $X_n$ is a martingale. I am looking at a proof which shows that $T < \infty$ almost ...
44 views

35 views

### Finiteness of the hitting time of random walk

Let $X_1,X_2,\ldots$ be an iid sequence such that $P\{X_1 = 1\} = u$, $P\{X_1 = -1\} = d$ and $P\{X_1 = 0\} = 1-(u+d)$. We have that $E[X_1] \neq 0$. Define $S_n = \sum_{i=1}^nX_i$ and $S_0 = 0$ and ...
36 views

### defining the stopping time sigma algebra

For a stopping time T, define $\mathcal{F}_T$ by $\mathcal{F}_T={A \in \mathcal{F}:A \cap \{T \le t\} \in \mathcal{F}_t, \text{for each t.}}$ Verify that $\mathcal{F}_T$ is a $\sigma$-algebra. ...
53 views

### Hitting times for Brownian Motion (2)

In this post there is shown that for a standard Brownian motion $\mathbb{E}[\tau^p]<\infty$ for all $p \geq 1$, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\...
58 views

### Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
23 views

### Martingales with bounded increments

It is known that if $T$ is a stopping time such that $E[T] < \infty$ and $(M_n)$ is a martingale with bounded increments, i.e. $\lvert M_n - M_{n-1}\rvert \leq K < \infty$ for every $n$, almost ...
121 views

### Showing that the first hitting time of a closed set is a stopping time.

I found this exercise online: I am stuggling with the last part of the second exercise, that is I am not able to show that $\tau = \sup_i \tau_i$. Obviously we have that $\tau \ge \sup_i \tau_i$, ...
51 views

### Conditional Expectation.

Let $X$ be a random variable in $L^2(\Omega, \Sigma, P)$ and $\mathcal G$ a sub-$\sigma$-algebra of $\Sigma$. Prove that $E[(X-E[X\mid\mathcal G])^2] \le E[(X-E[X])^2]$. As conditional expectation ...
65 views

### Measurability of the zero-crossing time of Brownian motion

I have the following random time $\tau = \inf\{t > 0: W_t = 0\}$ where $(W_t)_{t\geq 0}$ is Brownian motion with almost surely continuous paths and $W_0 = 0$ a.s. I need to prove that $\tau$ is ...
98 views

36 views

### If a stopping time, $T$, satisfies $P(T>k\alpha)\leq (1-\epsilon)^k$ then $E(T)<\infty$

Suppose that $T$ is a stopping time such that for some positive integer $\alpha$ and some $\epsilon>0$ we have for every $n$: $$P(T\leq n+\alpha|\mathcal{F}_n)>\epsilon \text{ a.s.}$$ I ...
86 views

26 views

### For the continuous time case, is there any example such that $\tau_1, …, \tau_n$ are stopping times, but $\inf_n \tau_n$ is not a stopping time?

For the continuous time case, is there any example such that $\tau_1, ..., \tau_n$ are stopping times, but $\inf_n \tau_n$ is not a stopping time? We know if the filtration is right continuous, then ...
44 views