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3
votes
1answer
193 views

A martingale with bounded increments either converges or diverges to both infinities a.s.

I am reading page 236 "Probability : theory and examples" by R. Durrett. Theorem 31. Let $X_1, X_2,\ldots$ be a martingale with $|X_{n+1}-X_n|\leq M<\infty$. Let $C=\{\lim X_n \;\;\; \text{exists ...
2
votes
2answers
566 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
7
votes
3answers
1k 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 ...
5
votes
1answer
168 views

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
1
vote
1answer
103 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
1
vote
0answers
106 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
3
votes
1answer
234 views

Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a ...
2
votes
1answer
588 views

proof that a stopped martingale is a martingale?

Defenition. $\mathcal{F}_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\}, F\cap(\tau\leq n)\in \mathcal{F}_{n}$} is a sigma-algebra. Defenition. $\forall \omega \in \Omega: ...
2
votes
2answers
383 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 ...
1
vote
3answers
177 views

Optimal stopping in coin tossing with finite horizon

There's a classic coin toss problem that asks about optimal stopping. The setup is you keep flipping a coin until you decide to stop, and when you stop you get paid $H/n%$ where $H$ is the number of ...
0
votes
1answer
25 views

Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
0
votes
1answer
271 views

Snowplow Problem

A snowplow can remove snow at a constant rate (in cubic feet per minute). One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, ...
0
votes
1answer
101 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
0
votes
1answer
64 views

equality of value implies equality of stopping time

Question: Let X be a stochastic process and T a stopping time of ${\mathcal{F}^{X}_{t}}$. Suppose that for some pair $\omega$, $\omega$' $\in$ $\Omega$, we have $X_{t}(\omega)=X_{t}(\omega')$ for all ...
0
votes
1answer
86 views

How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?

i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...