0
votes
0answers
36 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 ...
0
votes
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 ...
1
vote
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
votes
1answer
42 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
votes
1answer
56 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 ...
0
votes
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 ...
0
votes
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
votes
0answers
40 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 ...
0
votes
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
vote
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 ...
0
votes
1answer
37 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 : ...
2
votes
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
vote
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 $\mathcal{F}_T=\{A\in\mathcal{F}\mid A\cap\{T\le t\}\in \mathcal{F}_t, \text{ for all }t\ge0\}$. ...
0
votes
0answers
65 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
votes
2answers
109 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 ...
1
vote
1answer
139 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 ...
0
votes
0answers
45 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 $F_\rho$): $$ ...
2
votes
1answer
81 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 ...
3
votes
1answer
77 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 ...
1
vote
1answer
86 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
votes
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
vote
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
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 ...
0
votes
0answers
82 views

First hitting time on a element of $\mathcal B ( \mathbb R^d) $ a (right, left) continuous path stochastic process

It's known that, given $\Gamma \in \mathcal B (\mathbb R ^d)$ and $X = > (X_t)_{t\geq 0}$ with right-continuous path, the random time $$T_{\Gamma} = \inf \{ t\geq 0 : X_t (\omega) \in ...
1
vote
0answers
51 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
2
votes
2answers
232 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
1
vote
0answers
156 views

Hitting times and stopping times for cadlag processes

Let $X$ be a cadlag stochastic process. If $X$ is continuous, then I already know that $\inf\{t\geq 0: X_t \in C\}$ is a stopping time whenever $C$ is closed in $\mathbb{R}$. What if $X$ is only ...
1
vote
2answers
53 views

Optional sampling

Let $(X_i)_{i\in\mathbb{N}}$ be iid random variables with $\mathbb{E}|X_1|<\infty$ and let $S_n \stackrel{\rm{}def}{=} X_1+\cdots+X_n$ for all $n\in\mathbb{N}$. If $T$ is a stopping time with ...
2
votes
1answer
339 views

proof that a stopped martingale is a martingale?

Defenition. $\mathcal{F}_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\}, F\cap(\tau\leq n)\in \mathcal{F}_{n}$} is a sigma-algebra. Defenition. $\forall \omega \in \Omega: ...
1
vote
1answer
114 views

Optional Sampling Theorem in discrete setting

I have a question about proving the optional sampling theorem in discrete setting. I dont know if what I am doing is mathematical justified. Can someone help me with this? Defenition: Let $\tau$ be a ...
0
votes
2answers
135 views

Stopping times, Filtration, Martingales,

I am new here and I have a question. Defenition: Let $ \tau$ be a stopping time, then $\digamma_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \digamma_{n}$} is a ...
1
vote
1answer
70 views

Probability that all events occur equally often in finite time

An experiment with $n$ equally likely events is repeated until all events have occurred equally often. What is the probability that the stopping time is finite ? The probability could be denoted by ...
2
votes
1answer
224 views

Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$

Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$? With added assumptions on how $X_n$ behaves, ...
1
vote
2answers
176 views

$(S_n^2-n)_{n\ge 0}$ martingale and bounded stopping times

Consider the random walk $$S_n=\sum_{k}^{n}X_{k}$$ Where $X_k$'s are iid, $$\mathbb P(X_1=1)=\mathbb P(X_1=-1)=\frac{1}{2}$$ and $\mathcal{F}_{n}=\sigma(X_i,0\leq i\leq n)$. How do I prove that ...
0
votes
2answers
146 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 | | ...
1
vote
1answer
221 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 ...
1
vote
0answers
51 views

Expectation related to renewal measure

Let $X_1,X_2,\ldots$ be i.i.d. random variables, and $S_n=X_1+\cdots+X_n$. Assume that $0 < \mathbf{E}(X_1) < \infty$ (but don't assume that the $X_i$ are $>0$). Let $N$ be the almost surely ...
1
vote
2answers
202 views

Stopping time for a martingale

Let $X_1,X_2,\ldots$ be iid random variables where $X_i\in\{-1,0,1,2,...\}$, $P(X_i=0)<1$ and $E(X_1)=\mu$. Let $S_n=1+X_1+\cdots+X_n$ and $T=\inf \{n:s_n=0\}$. Show that $E(T)=\infty$ if $\mu=0$ ...
4
votes
1answer
258 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
3
votes
1answer
208 views

Favourable modification of “Double or Nothing”

I am working through 'Great expectations: the theory of optimal stopping' by Y.S. Chow, H. Robbins, D. Siegmund and cannot fill in the gap in reasoning regarding the existence of optimal stopping ...