Articles of stopping times

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 optional sampling theorem implies that $$ \mathbb E \left\{ \exp(\lambda \ B_{T_a\ \wedge\ n} -\frac{\lambda^2}{2}({T_a\ \wedge\ n})) \ |\ \mathcal […]

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 $B=(B_t)_{t\geq 0}$ is a standard brownian motion in $\mathbb R$ starting from zero. To show that $T_x < + […]

Exist $\alpha < \infty$, $\beta > 0$ such that $\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$

Let $B_t$ be a standard one-dimensional Brownian motion. Suppose $\lambda > 0$ and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist $\alpha < \infty$ and $\beta > 0$ (which may depend on $\lambda$) such that for all $t$, we have$$\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$$

If $\tau$ is an $(\mathcal F_t)_{t\in I}$-stopping time and $t\in I$, then $\tau\wedge t$ is $\mathcal F_t$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq\mathbb R$ $(\mathcal F_t)_{t\in I}$ be a filtration of $\mathcal A$ $\tau$ be an $\mathcal F$-stopping time, i.e. $\tau:\Omega\to I\cup\sup I$ is $\mathcal A$-$\mathcal B(I\cup\sup I)$-measurable ($\mathcal B(E)$ denotes the Borel $\sigma$-algebra on $E\subseteq[-\infty,\infty])$ and $$\left\{\tau\le t\right\}\in\mathcal F_t\;\;\;\text{for all }t\in I\tag1$$ $t\in I$ How can we show that […]

Reference request: proof that the first hitting time of a Borel set is a stopping time

Where exactly (book and page number) can I find the proof that the first hitting time of a Borel set a “Stopping time” (continuous time). My notes say it is a deep theorem, particularly hard to prove but didn’t give any reference. Moreover my prof mentioned that is is so hard to prove that the […]

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 seems that every theorem in intro stoch. processes is about when something is a martingale (super/sub) under various […]

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 $$T=\inf\{t\geq0,B_t\geq e^{-\lambda t}\}$$ with $\lambda>0$, what is $$E[T]$$ If it is known is it also known when, instead of a Brownian Motion, one has a simple […]

Prove previsibility and $E \le E$

From Probability with Martingales:

Asymmetric Random Walk / Prove $E = \frac{b}{p-q}$ / How do I use hint?

Given random variables $Y_1, Y_2, \ldots \stackrel{\mathrm{iid}}{\sim} P(Y_i = 1) = p = 1 – q = 1 – P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$, define $X = (X_n)_{n \ge 0}$ where $X_0 […]

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 $F_{S \vee T}$ and show it is in $\sigma(F_S,F_T)$. I also notice that if $A$ is in $F_{S […]