New answers tagged

0 votes

Show that this stopped process converges ucp to the original process

I supplement some details to Dawkins' answer. (1) Show that $\tau_n\to \infty$ almost surely. Throughout, fix some $\omega\in \Omega$. When $M$ is bounded, that is, $\exists K>0, \forall t>0, |M(...
Mingzhou Liu's user avatar
3 votes
Accepted

Show that this stopped process converges ucp to the original process

You have $\sup_{s\in[0,t]}|M^{\tau_n}_s-M_s|=0$ on the event $\{\tau_n\ge t\}$, and so $$ P\left(\sup_{s\in[0,t]}|M^{\tau_n}_s-M_s|>\epsilon\right)\le P(\tau_n<t)\to 0,\quad n\to\infty, $$ ...
John Dawkins's user avatar
  • 26.1k
0 votes

Optional Stopping Theorem for martingales bounded except at the stopping time

My friend pointed out a simple counterexample: $X_0 = 0$, $$X_{n+1} = \begin{cases} X_n & \text{with probability $1/2$} \\ X_n - 2^n & \text{with probability $1/4$} \\ X_n + 2^n & \text{...
lily's user avatar
  • 3,727
0 votes

The distribution of the first hitting time for the Constant Elasticity of Variance process.

We are going to prove that the Laplace transform $(3)$ approaches the correct limit when $\beta \rightarrow 1_+$. We denote ${\mathfrak N}:= 1/(2(-1+\beta)$ and $\theta := 2 \mu/\sigma^2$ and we have: ...
Przemo's user avatar
  • 11.4k
1 vote
Accepted

Optional Stopping Theorem and Stopped $\sigma$-fields

For real $x$ you have $$ \{X_T\le x\}\cap\{S\le n\} = \cup_{k=0}^n\{X_k\le x, T=k, S\le n\}. $$ which is clearly $\mathcal F_n$-measurable, for each $n$. It follows that $X_T$ is $\mathcal F_S$-...
John Dawkins's user avatar
  • 26.1k
1 vote
Accepted

Is $\inf\left\{t\in\left[0,1\right]\vert t+B^2_t=1\right\}$ a stopping time?

(Edited) Useful theorems Assuming nothing about the filtration: If $X=(X_t)_{t\geq 0}$ is a continuous $\mathbb{R}^d$-valued $(\mathcal{F}_t)_{t\geq 0}$-adapted process, then $T_A:=\inf\left\{t\geq ...
Wilfred Montoya's user avatar
1 vote
Accepted

Expected value of the square of a stopping time

$\def\={\mathrel{\phantom=}}$Your calculation writes\begin{gather*} E( (a^2 - S_a) I_{\{ B_{S_a} = a \}} ) + E( ((-a)^2 - S_a) I_{\{ B_{S_a} = -a \}} )\\ = a^2 - E( S_a I_{\{ B_{S_a} = a \}} ) + (-a)^...
Ѕᴀᴀᴅ's user avatar
  • 34.3k
1 vote
Accepted

Expected value of the exponential of a stopping time

All good. Perhaps, you can add more details on the "by symmetry" Independence of $T$ and $B_T$ i.e. we have $B_t\stackrel{d}{=}-B_{t}$ and $S_{a}$ is only a function of $|B_{t}|$, which is ...
Thomas Kojar's user avatar
  • 3,621
0 votes
Accepted

Stirling approximation of the probability that the stopping time is finite

Consider $x=0$. Using Stirling's approximation you find $$P(S_{2m}=0)=\binom{2m}{m}2^{-2m}\sim \frac{1}{\sqrt{\pi m}},$$ which yields $$\sum_{m=0}^\infty P(S_{m}=0)= \infty.$$ Define $$\tau_{0}^{(m)}:=...
user408858's user avatar
  • 2,487
0 votes
Accepted

First hitting time of biased random walk in 2 dimensions

Let $B_t = (B_t^x, B_t^y)$ be a 2-dimensional Brownian motion. In particular $B_t^x$ and $B_t^y$ are independent $1$-dimensional Brownian motions. Define the first hitting time of the line $L_{x_0} = \...
Jose Avilez's user avatar
  • 12.9k

Top 50 recent answers are included