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1answer
15 views

Illegal lottery problem (Merging dependent bernoulli trials)

Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability $p$. Each time I buy a ticket, the police may catch me and confiscate ...
4
votes
2answers
454 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 ...
-1
votes
2answers
45 views

Expectation of first passage time of Brownian Motion [closed]

Let $B(t)$ be Brownian Motion beginning at zero. Define $T_{\alpha} = inf\{t>0 ; B(t) = \alpha\}$ to be the first passage time. I need to calculate the expectation of the first passage time for a ...
0
votes
0answers
38 views

Secretary Problem - Optimal algorithm for expected value of candidate.

I recently encountered the secretary problem and there are essentiall two problems: Maximizing the probability of choosing the best candidate. Gnedin proved in his paper ...
-1
votes
1answer
31 views

Martingale: Show $p\{T<+\infty \}=1$.

Let $(X_i)$ i.i.d. such that $p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2}$ and let $(S_n)$ the martingale define by $S_0=0$ and $S_n= X_1+...+X_n$. Moreover, let $$T=\begin{cases}\inf\{n\geq 0\mid ...
2
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0answers
17 views

charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) ...
-1
votes
0answers
30 views

Hitting time vs supremum of a càdlàg process

A question on stopping times that came up while I was trying to prove Doob's inequality. Let $X$ be a càdlàg, nonnegative submartingale. Define $X^*_t = \sup_{0\le s\le t} X_s$ and, for $K \ge 0$, ...
1
vote
1answer
21 views

How to show that consecutive hitting times are a sequence of stopping times?

Let $\{X_n:n=0,1,\ldots\}$ be a martingale with respect to a filtration $\{\mathcal F_n\}$. Let $A,B$ be nonempty, disjoint Borel sets and define $T_0=0$, \begin{align} S_n &= \inf\{m\geqslant ...
2
votes
2answers
50 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
2
votes
1answer
54 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
0
votes
0answers
14 views

Expected difference between a process and a stopped process

Let $\tau$ be a stopping-time such that for some $t\geq 0$ the probability $P(\tau<t)=\epsilon$ is small. I am interested in quantifying the following difference $$E[f(X_t)] - E[f(X_{t\wedge\tau})] ...
2
votes
0answers
49 views

Brownian Motion Hitting Times

I am reading through Walsh's Knowing the Odds book and came across this problem. Let $B_t$ be Brownian motion. Find the probability that $B_t$ hits plus one and then minus one before time one. I am ...
2
votes
0answers
24 views

Limit of decreasing sequences of markov time (stopping time) is markov time?

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geqslant 0}, \mathbb{P})$ be a filtered probability space and let $\tau_n \geqslant \tau_{n+1}$ be a markov time (stopping time) with respect to ...
0
votes
0answers
30 views

Proving Galmarino's Test

Galmarino's Test gives a condition equivalent to being a stopping time. It says: Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous ...
2
votes
0answers
85 views

Brownian motion stopped at the hitting time of an independent Brownian Motion

While I was working on the exit time of planar BM out of a square I came across the following observation, which I cannot grasp. I define this exit time as $$\tau = \inf\{t \geq 0: \lvert B(t)\rvert ...
2
votes
0answers
32 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. For every $s\geq0$, let $$\tau_s = \inf \left\{t \geq 0 \mid X_1(t) - x_1 > s \right\}\qquad Y_s = ...
1
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0answers
18 views

jump-diffusion hitting time

Suppose I have a stochastic process $dS_t= rS_t dt + \sigma S_t dW_t + dJ_t$ where $W_t$ is a brownian motion and $J_t$ a compound poisson process of parameter $\lambda$ with lognormal jump size, ...
0
votes
1answer
39 views

Escape probabilities in a random walk.

So, a lot of theory in symmetric random walks seems to concentrate on 'hitting/stopping times' and things like that. So I started wondering... How would I go about calculating the probability of ...
1
vote
1answer
22 views

Properties of Stochastic Interval

I'm reading "Limit Thoerem for Stochastic Processes" and finding it hard to calculate the Stochastic interval.For example : In proposition 2.10,$T$ is a stopping time: If $A\in\mathcal F_0$,I need ...
1
vote
1answer
33 views

Does Brownian Motion return to the origin infinitely soon? [closed]

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
2
votes
0answers
35 views

Why do we use an exponential Martingale for the stopping time of a BIASED random walk?

The following is a passage from the lecture notes: Let a simple random move to the right with probability $p$ and to the left with probability $q = 1 − p$. We want the probability that it hits ...
10
votes
3answers
842 views

What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the ...
0
votes
1answer
26 views

Brownian Motion Hitting Time Distribution

Define $\tau_a = \inf \left\lbrace t \geq 0 | B(t) \geq a \right\rbrace $ for some $a>0$. The problem is to show that $ \tau_a \stackrel{d}{=} \sqrt a\tau_1 $. What I've done so far: $$P(\tau_a ...
1
vote
1answer
45 views

Gambling Game martingale

State the optional sampling theorem for martingales and bounded stopping times. You start with a capital of £100 and bet repeatedly on the toss of a coin. On each toss you may bet any whole number of ...
0
votes
1answer
21 views

Expected Stopping Time for BM

I'm working on this homework problem for Brownian Motion. Suppose we define a stopping time $\tau_a = inf \left\lbrace t \geq 0 : B(t) = a \right\rbrace$ for some $a>0$. I already showed in a ...
3
votes
1answer
76 views

Tower Property for Expectations and Stopping Times

Let $(\Omega,(\mathcal{F_t})_{t\geq0},P)$ be a filtered proability space with $X\in L^1(P)$ and two stopping times $S$ and $T$. Show that \begin{equation*} ...
2
votes
1answer
28 views

First hit of a martingale

I came across this result somewhere and I don't grasp its proof in its entirety. Let $M$ be a continuous martingale such that $M_0 = 0$. Define $\tau_x = \inf\{t\geq 0: M_t =x \}$. Then, $$P\{\tau_a ...
3
votes
1answer
41 views

Brownian motion proof of Dirichlet problem

I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let $D$ be a bounded domain in $\mathbb{R}^d$ such that every boundary point satisfies the Poincare cone ...
1
vote
1answer
19 views

Showing that a set is included in a filtration at a stopping time

The title may sound strange. Sorry for that but the question is short and easy to understand. I have a set $A \in \mathcal{F}_t$ where $(\mathcal{F}_t)$ is a filtration on some probability space. For ...
1
vote
1answer
18 views

Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such ...
0
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0answers
27 views

Cramér Lundberg Risk Model - exponential distribution of claim sizes

I am studying the classical ruin model, which express the insurance company free surplus at time $t$ as $C_t=u+ct-\sum_{i=1}^{N(t)}Y_i $ where: $ct$ is the premium income up to time t $u$ is the ...
2
votes
1answer
42 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
0
votes
1answer
29 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
23 views

Expectation of a Wiener process at a Stopping Time - 2

I am working through an answer to the following question and I do not understand a statement given towards the end of the solution, specifically why $\tilde{W}(\sigma) = 1$. (This question is related ...
0
votes
0answers
24 views

Levy process of argument in the complex plane

I am stuck on this question: Let $B$ be a Brownian motion in $\mathbb{C}$ started at $1$. Let $\theta_t$ be a continuous determination of the argument of $B_t$, i.e. $B_t = |B_t| e^{i \theta_t}.$ ...
0
votes
1answer
29 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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)] = ...
1
vote
1answer
70 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
1
vote
1answer
53 views

Justifying a step in proving $M_{S\wedge T} = \mathbb{E}[M_T | \mathcal{F}_S ]$

$S,T$ are stopping times and $M$ is a (right) continuous martingale. My lecturer set this as an exercise and I am given a solution(essentially split $M_T = M_T \mathbf{1}_{S≤T} + M_T ...
1
vote
1answer
64 views

Martingales and stopping times question

Let $X_n$ be iid r.v.s such that $P(X_n=1)=P(X_n=-1)=1/2$, and $S_n=\sum_{k=0}^{n}X_k$. Define $S_0=0$ a.s. . Prove that for all $k,n \in \mathbb{N}$, $\mathbb{E}[S^2_{n \wedge T_k}]=\mathbb{E}[{n ...
0
votes
1answer
43 views

Proof of Optional sampling theorem

In the proof of the optional sampling theorem they define for a stopping time $\tau$ the sigma algebra $\mathcal{G}=\sigma(\cup_n \mathcal{F}_{\tau\wedge n})$. Then they use the fact that for the ...
0
votes
1answer
73 views

Prove that discrete first hitting time is a stopping time

I have problems with the proof that a first hitting time is a stopping time: Let $\tau$ be the first hitting time into the set A, for a process $\{ X_n \}$ adapted to a filtration $\mathcal F_n$. I ...
2
votes
1answer
64 views

Proof of stopping theorem for bounded stopping times

Let $\tau$ be a bounded stopping time and $X=X_n$ a martingale. Then $X_\tau$ is integrable and $E(X_\tau)=E(X_0)$. I need help with the proof at discrete time, at one step I am not sure I ...
0
votes
0answers
30 views

Optimal stopping strategy

I try to solve the following problem : Given a series of random variables : X1,X2,... such that each one can get either -1 or 1 with probability 0.5, give a strategy to maximize the expected value of ...
1
vote
0answers
37 views

Expected value of Brownian Bridge evaluated at a stopping time

Denote by $B$ a Brownian bridge process, by $B(\omega)$ a realization of it and by $B_t$ the projection to the time point $t \in [0,1]$. Now let $c < 0$ and $$t^*(\omega) = \sup\{t \in [0,1]: ...
1
vote
1answer
36 views

Upper bound for martingale at a stopping time

This seems like a simple question, but I cannot figure out the following. Let $\{M_i\}_{i\geq 0}$ be a martingale adapted to a filtration $\mathcal{F}_i$, with the following conditions: ...
1
vote
1answer
49 views

Distribution of Brownian motion before stoping time.

Let $B_{t}$ be a standard Brownian motion. Stopping time $\tau_{a} = \inf \{t \ge 0: |B_{t}| = a\}$. How to find $E[B_{\frac{\tau_{a}}{2}}]$? Or where is it possible to read about it? Thanks in ...
2
votes
1answer
510 views

Quadratic variation and stopping time.

Let $X_t$ be a strict local martingale and let $S$ and $T$ be stopping times with $S \leq T$. Prove that $[X]_S=[X]_T$ implies that $X_t$ is almost surely constant on $[S, T]$, where $[X]_S$ and ...
1
vote
1answer
37 views

Are the following Stopping Times?

I've been working through the following list of stopping time questions. I am have problems with the final two (e and f). I appreciate any assistance offered. $\textbf{Question:}$ Let $S,T : ...
0
votes
1answer
44 views

Stopping times and typing monkeys

This is a question about the "standard solution" in this question: Let $(X_t)_{t\in\mathbb{N}}$ be the stochastic process modeling a monkey who types a random letter (uniform distribution) of the 26 ...