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20 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
36 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
37 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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0answers
37 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
32 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
40 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
19 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
22 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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1answer
33 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 ...
2
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1answer
43 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
57 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
32 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 ...
2
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0answers
50 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
52 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
102 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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1answer
63 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 ...
3
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0answers
41 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
20 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 ...
1
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0answers
37 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
38 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
33 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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0answers
23 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
54 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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0answers
43 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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1answer
37 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 ...
2
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1answer
50 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 ...
1
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1answer
55 views

Adaptedness of random variables

Suppose we have an RCLL adapted process $(X_t)$. Moreover we are given a stopping time $T$. We define $\mathcal{F}_T=\{A\in\mathcal{F}\mid A\cap\{T\le t\}\in \mathcal{F}_t, \text{ for all }t\ge0\}$. ...
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0answers
66 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 ...
2
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2answers
115 views

Wald's equation example controversy

I'm trying to get a grip of Wald's equation, applying it to the following example. Suppose, we have a simple sequence of fair coin flips, where heads wins us a dollar, while tails means loss of a ...
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1answer
146 views

Wald equality, expectation of a stopping time

Let $(X_n)$ be a sequence of iid random variables such that: $$\mathbb{P}(X_k=-1)=q \\ \mathbb{P}(X_k=1)=p=1-q$$ (two points distribution) Let $\tau$ be the first moment when number of successes ...
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0answers
15 views

Stopping time and right-continous filtration

I have to prove that if $T=[0,+\infty)$ and $(F_t)_{i\in T}$ is a right continous filtration, then: $\tau$ is a stopping time $\iff \forall t \in T :\{\tau<t\}\in F_t $ My attempt: The most ...
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0answers
47 views

Stopping times and stopped sigma algebras

Let $\tau$ and $\rho$ be stopping times with respect to filtration $\{F\}_i$ I have to show that: $[\tau<\rho]$ is in both $F_\tau$ and $F_\rho$ Is this ok (for $F_\rho$): $$ ...
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0answers
79 views

Solution of the problem 1.2.2 from “Brownian Motion and Stochastic Calculus” of Karatzas & Shreve

Does anybody have the solution of that problem, please? I don't understand the relation between random variables $X$ and $T$. Regards Edit : Thank you for the comments. Let me first apologize for ...
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1answer
94 views

Expectation of brownian motion at hitting time

Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
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1answer
44 views

Expectation of stopping times

Let B = (Bt)t¸0 be a standard Brownian motion started at zero, let $X_t$ be a non negative stochastic process solving: $dX_t=1/X_tdt+dB_t$ Compute $E[\sigma]$ when $\sigma=\inf \{ t\ge 0 : X_t= 1 ...
2
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1answer
82 views

Show that this is a stopping time

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 ...
0
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1answer
43 views

equality of value implies equality of stopping time

Question: Let X be a stochastic process and T a stopping time of ${\mathcal{F}^{X}_{t}}$. Suppose that for some pair $\omega$, $\omega$' $\in$ $\Omega$, we have $X_{t}(\omega)=X_{t}(\omega')$ for all ...
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1answer
63 views

If AC is false , is this statement about the halting problem true?

Assume AC is false. (AC = axiom of choice ) Let $n,m$ be positive integers. Let $f: \Bbb N \rightarrow \Bbb N$ and $f(m)=m$. Let $g(n,m)=1$ if the iterations $f(n),f(f(n)),...$ converges to $m$. ...
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0answers
33 views

New stochastic calculus

I am interested in Kagi and Renko approach and hope I can use it for a random walk process. I searched for it on internet but I couldnt find any basic material to read about it. Can someone please ...
3
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0answers
147 views

Essential supremum of a conditional expectation

Given the function \begin{equation} P(x,t) := \sup\limits_{t \le \tau \le T} E\left( g(X^{t,x}_{\tau}) \right) \end{equation} where $X^{t,x}$ is the unique solution to the SDE \begin{equation} X_u ...
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1answer
46 views

Probablity and Expected value

Suppose you are playing a fair coin game and you win a dollar if a head shows up and lose a dollar if tail. what is the expected value of rounds you played before you lose the first dollar from your ...
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1answer
78 views

how to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale.and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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1answer
89 views

Stopping times of Markov chains

I have the following problem: Consider a state space $E$ and a Markov chain $X$ on $E$ with transition matrix $Q$ such that for every $x \in E$, $Q(x,x)<1$. Define: $\tau:=\inf\{n\geq 1:X_n\neq ...
2
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1answer
154 views

Conditional Expectation of martingale at stopping time

I am trying to understand the implications of the optimal stopping theorem, which is why I tought of the following problem. Consider the continuous-time Martingale $X = (X_t)_{t \geq 0}$ and the ...
1
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1answer
58 views

Strong Markov Property Brownian Motion Question

If $\tau$ is a stopping time and $\omega(t)$ is Brownian Motion then the Strong Markov Theorem states that $Z(t)=\omega(t+\tau) -\omega(\tau)$ conditioned on $\{\tau <\infty\}$ is distributed as ...
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0answers
50 views

Lower bound for stochastic process

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the ...
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0answers
39 views

Time after last jump and waiting time before the next jump of Poisson process

Consider $N =(N_t)_{t\geq0}$ a Poisson process of intensity $\lambda > 0$ and $(T_n)_{n\geq 1}$ its jump instants. Then consider for all $t \geq 0$, $Z_t = t- T_{N_t} \mathbb 1 _{\{ t \geq ...
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0answers
43 views

Conditional distribution of compounded Poisson process

Consider a Poison a process $N = (N_t )_{t\geq 0}$ of intensity $\lambda >0$ whose instants of jumps are $(T_n)_{n\geq0} $ $(T_0 =0)$ and a process $\tilde{N} =(\tilde N_t )_{t\geq 0}$ defined as ...
3
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1answer
114 views

find the Law of probability Stopping time $T=inf\{n\ge 0: R_{n}\gt a\}$ for fixed number $a\gt 0$.

suppose $R_{n}=\sum_{i=1}^{n} X_{i}$ for $n\ge 1$ and $R_{0}=0$ , that $X_{i}\gt 0$ Random variables Are independent and distributed.find the Law of probability Stopping time $T=inf\{n\ge 0: ...
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vote
2answers
86 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

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 ...