I am trying to show that if for every bounded stopping time $\tau$ it holds that $E(X_{\tau})=E(X_{0})$ then ${X_{n}}$ is a discrete time martingale. I have a few questions on my ideas on solving this. First I consider a set $A \in F_{n}.$ I define $\tau= n-1$ for $\omega \in A$ and $\tau= n$ for […]

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$ $M$ be a local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ $\tau$ be an $\mathcal F$-stopping time Is $M^\tau$ a local $\mathcal F$-martingale? With the given assumptions, this claim can be found in the proof of Lemma 15.1 in Foundations […]

Could you check if my solution is correct? I find the filtrations quite tricky. Here is the problem: Let $\{X_n\}_{n \in \mathbb{N}}$ be a stochastic process and $B$ a borel set in $\mathbb{R}^N$. Prove that $ \tau_k = \inf \{n> \tau_{k-1}: \ X_n \in B\}$ is a stopping time w.r. to the natural filtration generated […]

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq[0,\infty)$ $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $\tau$ be a $\mathbb F$-stopping time $\mathcal F_\tau:=\left\{A\in\mathcal A:A\cap\left\{\tau\le t\right\}\in\mathcal F_t\;\text{for all }t\in I\right\}$ $Y:(\Omega,\mathcal F_\tau)\to\left(\mathbb R,\mathcal B\left(\mathbb R\right)\right)$ be a random variable $s\in I$ and $Z:=1_{\left\{\tau=s\right\}}Y$ Can we show, that $Z$ is $\mathcal F_s$-measurable? Clearly, we’ve […]

Let $(\Omega, \mathcal F,P)$ be a probability space, $I$ a countable subset of $\mathbb R$ and $(\mathcal F_i)_{i\in I}$ a filtration. Let $\tau$ be a random variable with values in $I\cup \{\infty\}$. Prove that $\tau$ is a stopping time if and only if $\forall t\in I, (\tau=t)\in \mathcal F_t$ I’ve found the implication $\tau$ stopping […]

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I’ve been trying to put the set $\{\sigma\le t\}$ equal to a countable union and then showing that this union belongs in $\mathcal F_t^B$. I am struggling to derive a countable union to this as […]

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 heads you flipped and $n$ is the number of times you flipped. I believe that this problem […]

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function $\phi : \mathcal{S} \longrightarrow \mathbb{R}$, there exists a function $\psi: \mathbb{N}\times \mathcal{S} \longrightarrow \mathbb{R}$ such that: $\mathbb{E}[\phi(X_{\tau +1}) | \mathcal{F}_{\tau}] = \psi(\tau, […]

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 measurable (not necessarily a stopping time or an optional time). I first fix $\omega \in \Omega’ \subset \Omega$. […]

Given a probability space $(\Omega,\mathcal{F},P)$ and a process $X_{t}$ defined on it. We consider the natural Filtration generated by the process $\mathcal{F}_{t}=\sigma (X_{s}:s\leq t)$. Let $\tau$ be a stopping time. The corresponding $\sigma$-Algebra of the stopping time $\tau$ is given by \begin{align} F_{\tau}=\left\{A\in \bigcup_{t\geq 0} F_{t}: A\cap\{\tau \leq t\}\in \mathcal{F}_{t} \forall t\geq 0\right\} \end{align} Now […]

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