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 […]

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 < + […]

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$$\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$$

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 […]

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 […]

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 […]

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 […]

From Probability with Martingales:

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 […]

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 […]

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