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38 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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0answers
15 views

About measurability of a stopping time.

If $S,T$ are two stopping time w.r.t. $\mathcal F_t$ define $R=S\wedge T$.Then $R$ is a stopping time .How to prove $R$ is measurable w.r.t $\mathcal F_T$? Is there something wrong with this ...
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1answer
31 views

A step in verifying a stopping time.

Suppose $X$ is a cadlag process adapted to $\{\mathcal F_t\}$ and $H$ is a closed set.Verify $\sigma_H\triangleq\inf\{t\ge0:X_t(\omega)\in H\}$ is a stopping time . The first step is: ...
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1answer
27 views

An equality in stopping time.

In a proof,I need the following equality: Suppose $\tau,\sigma$ are two stopping time and $A$ is a event.Then: $$(A\cap\{\sigma\le\tau\})\cap\{\tau\le t\}=(A\cap\{\color{red}{\sigma}\le ...
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0answers
25 views

How do you solve martingale stopping time problems?

I know this is a rather broad question, so here's a specific example: What is the distribution and expected value of the first time you get X heads during a sequence of coin tosses? Now...is there a ...
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1answer
31 views

Prove of Stopping time

Let $(X_k)_{k\in\mathbb{N}}$ be iid random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$. Let $Z_n=\prod_{k=1}^n(1+X_k)$, so $Z_n$ a martingale. Consider ...
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0answers
17 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
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1answer
41 views

Jumping times of a Lévy Process

If one has a Levy-process, are the times when the process has a jump of size exceeding a positive $\varepsilon$ actually stopping times w.r.t. the canonical filtration? In more detail: Let ...
2
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1answer
32 views

Verifying stopping times…

Let $m$ be a natural number, $$g_m:=\sup\left\{ {n\leq m: S_n\leq 0}\right\}$$ and $$d_m:=\inf\left\{ {n\geq m: S_n \leq 0}\right\}$$ I have to check if they are are stopping times. It's still a new ...
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0answers
42 views

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 ...
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1answer
20 views

Distribution of two-sided boundary stopping time of Brownian motion.

If $B_t$ is a Brownian motion, and a one-sided boundary stopping time is given by: $\tau_a=\inf\{t:B_t=a\}$ the distribution of $\tau_a$ is given by: $f_{\tau_a}(t)=\frac{|a|}{\sqrt{2\pi ...
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1answer
112 views

A martingale with bounded increments either converges or diverges to both infinities a.s.

I am reading page 236 "Probability : theory and examples" by R. Durrett. Theorem 31. Let $X_1, X_2,\ldots$ be a martingale with $|X_{n+1}-X_n|\leq M<\infty$. Let $C=\{\lim X_n \;\;\; \text{exists ...
2
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2answers
168 views

Expectation of a stopping time of a Wiener process

How can we calculate $\mathbb{E}(\tau)$ when $\tau=\inf\{t\geq0:B^2_t=1-t\}$? If we can prove that $\tau$ is bounded a.s. (i.e. $\mathbb{E}[\tau]<\infty$), then we can use the fact that ...
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0answers
873 views

Minimum of two stopping times is a stopping time.

So far I've already shown that the sum and the maximum of two stopping times is a stopping time, but the minimum is giving me some problems which I just can't get around. This is what I've tried: Let ...
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1answer
31 views

Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
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1answer
39 views

Stopping times problem: $ \tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
2
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0answers
17 views

Comparing hitting time of two random walks

There are two random walks, $S^t_i=S^{t-1}_i+ X_i^t$ for $i=1,2$, $X^t_i$ i.i.d they have boundaries $h_1$ and $h_2$ respectively. I'm wondering if it's possible to calculate the probability that one ...
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0answers
55 views

Ito formula proof for bounded functions using stopping time

I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing ...
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2answers
291 views

Stopping time proof

Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider $$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process ...
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162 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...
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0answers
25 views

Stopped strong Markov process again strong Markov?

Following setting: I have a right-continuous strong Markov process X in a right-continuous filtration >$\mathbb{F}=(F_t)$ and a P-a.s. finite stopping time $\tau$. My question is: Is the ...
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3answers
64 views

Optimal stopping in coin tossing with finite horizon

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 ...
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0answers
41 views

Probability of Stopping Time Taking specific value - Random Walk 1d

We are considering a simple random walk $(X_n)_{n\in\mathbb{N}}$ starting at $X_0=0$ with $X_n=\sum_{i=1}^nY_i$ where $Y_i$ are iid and $\mathbb{P}(Y_i=1)=\mathbb{P}(Y_i=-1)=\frac{1}{2}$. We want to ...
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39 views

Why is the Stopping Theorem interesting?

The theorem for discrete-time martingales is as follows: Let $X=(\Omega,\mathcal{F},(\mathcal{F}_n)_n,(X_n)_n,\mathrm{P})$ be a supermartingale and $\tau_1,\tau_2$ two a.s. bounded stopping times on ...
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0answers
37 views

Brownian Motion first hitting time distribution

I have a question concerning the distribution of the first hitting time of Brownian Motion $\tau_x = \inf_{t\geq 0}\{W_t=x\}$, where $W_t$ is Brownian motion. Using some calculus, I found out that the ...
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1answer
56 views

Snowplow Problem

A snowplow can remove snow at a constant rate (in cubic feet per minute). One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, ...
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1answer
22 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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1answer
23 views

Expressing units of time

How would you express 8/3 seconds as time after 3pm ? 8/3 = 2.66666 0.66*60 =40 miliseconds = 0.04 seconds so 2.04 seconds after 3 3:00:02:04 pm ? Is this correct?
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2answers
156 views

coupon collector problem for different number of copies of each coupon type

I would like to pose a question on a variation on the classical coupon collector's problem: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number of ...
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3answers
582 views

Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
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1answer
37 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
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1answer
47 views

The law of the iterated logarithm for BM and boundedness of stopping times

My question is regarding the usefulness of the law of the iterated logarithm, and its connection to stopping times. In many answers of this forum, I understand that some people often claim that some ...
2
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1answer
68 views

Hitting time for Brownian Motion Surplus Process

I'm trying to solve this question for a continuous surplus process. The surplus process is $$U_s=U_0+s-B_s$$ where $B_t$ is a Brownian motion representing payouts, $U_0$ is starting capital, $s$ is ...
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1answer
68 views

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 ...
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1answer
38 views

Properties of Sigma Algebras of Information up to a stopping time

first of all i want to ask whether given any two $\{\mathcal{F}_t\}$-stopping times $\sigma, \tau$ is the following properties true: (i) $\mathcal{F}_{\sigma \wedge \tau} = \mathcal{F}_{\sigma} \cap ...
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0answers
57 views

Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$ \tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \} $$ is a stopping time with ...
3
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1answer
55 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
2
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0answers
111 views

Stopping time and filtration

My question is as follow: Let $(\Omega,\cal{F}_\infty,\{\cal{F}_t\},\mathbb{P})$ be the filtred probability space. Further, denote $\cal{F}^*_t$ as the usual augmented filtration. Now, given a ...
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0answers
49 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
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0answers
27 views

Conditioning on $\mathcal{F}_\sigma$ for $\sigma$ stopping time

I'm trying to show that $E[E[\ \cdot\mid \mathcal{F}_\sigma]\mid\mathcal{F}_\tau]=E[E[\ \cdot\mid \mathcal{F}_\tau]\mid\mathcal{F}_\sigma]$ for stopping times $\sigma$ and $\tau$, I've come to the ...
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40 views

Find an example such that $\tau$ is a stopping time and $\mathcal{F}_\tau$ and $\mathcal{F}_\infty$ differ on $\{\tau = \infty\}$.

I need to find an example such that the following is true: $\tau$ is a stopping time and $\mathcal{F}$ is a filtration defined on $\mathbb{R}_+$. Let $\mathcal{F}_\tau$ denote the stopped ...
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1answer
41 views

Stopping time problem - Show that T is bounded

Let $a< 0 < b$ and $W_t$ is Brownian motion $T_a$=inf{$t\ge$0|$W_t\le a$} $T_b$=inf{$t\ge$0|$W_t\ge b$} T=min{$T_a$,$T_b$} $1)$ Show that $T$ $<$ $\infty$ My attempt : ...
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0answers
38 views

Jump time of a previsible process is previsible?

Here is my question: In our setups, the filtration satisfies the usual condition. $V$ is an increasing process with only jumps (between the jumps it is flat). We also know that $V$ is right ...
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24 views

Is the exit time independent of the state jumped to in a Markov chain?

Let $X$ be a continuous time Markov chain on a countable state space $S$, and let $\tau_n$ be the $n^{th}$ time at which the chain jumps out of a set $D$ (i.e. times $t$ at which, for some $\epsilon ...
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2answers
56 views

Show $L$ is not a stopping time

Let $L = \sup\{ n : n \le 10; A_n \in B \}$, $B \in \mathcal B$, $\sup\{\emptyset \}=0$. $(A_n)_{n \ge1}$ is a process adapted by a natural filtration $\{\mathcal F_n\}.$ Show that $L$ is NOT a ...
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52 views

Hitting time Distribution of a Gaussian Random Walk

I am trying to find out the exponential decay rate of the Probability $Pr(T>n)$ where $T$ is the first hitting time of a gaussian random walk with i.i.d random variables i.e. ...
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2answers
154 views

Martingale Stopping Time

Let $(S_n)_{n \ge 0}$ be a $(\mathcal F_n)$-martingale and $\tau$ a stopping time with finite expectation. Assume that there is a $c > 0$ such that, $\forall n, \mathbb E (|S_{n+1} - S_n | | ...
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1answer
42 views

Stopping time problem

I have some difficulty understanding following problem. I need to show any non random time $T$ is a stopping time. I know that we have to show {$T\le t$} is $F_t$ measurable. When $t \le T$ this set ...
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67 views

brownian motion and stopping time

I have an exercise about Brownian motion which I don't understand completely. Let $(B_s)_{s\geq0}$ be a standard real Brownian motion. For $t > 0$, we define the random times $g_t ...
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1answer
51 views

Showing that a nonnegative integer-valued random variable is NOT a stopping time

Suppose that $\left(A_n\right)$ is an adapted process, and that $B\in\mathcal{B}$. Let $L = \sup\left\{n:n\leq10;A_n\in B\right\}$, $\sup\left(\emptyset\right)=0$. Convince yourself that $L$ is NOT ...