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7
votes
0answers
170 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 ...
6
votes
3answers
803 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 ...
5
votes
1answer
127 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 ...
4
votes
1answer
288 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 ...
4
votes
1answer
39 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf ...
4
votes
1answer
36 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
4
votes
0answers
94 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
4
votes
0answers
52 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 ...
4
votes
0answers
223 views

stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. ...
3
votes
2answers
315 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 ...
3
votes
1answer
82 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 ...
3
votes
1answer
120 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: ...
3
votes
2answers
174 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 ...
3
votes
1answer
57 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 ...
3
votes
1answer
138 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 ...
3
votes
1answer
169 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 ...
3
votes
1answer
60 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 ...
3
votes
1answer
248 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 ...
3
votes
1answer
245 views

Stopping time on Wiener Process

Let $W_t$ be a Wiener process and for $a\geq0$ $$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$ Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?
3
votes
0answers
184 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 ...
3
votes
0answers
103 views

Finite stopping times

I've come across two statements in a proof that I don't really understand. Let $X_{i}$ be iid with values in $\{-1,0,1\}$ all with positive probability. Define $S_{n}=\sum_{i=0}^{n}X_{i}$ and the ...
3
votes
1answer
183 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
2
votes
1answer
257 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, ...
2
votes
1answer
31 views

Predictable process with stopping time

I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it. Let T be a stopping time in $\mathcal{F}_t$ for ...
2
votes
1answer
71 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 ...
2
votes
2answers
220 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 ...
2
votes
2answers
382 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 ...
2
votes
1answer
52 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
61 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 ...
2
votes
1answer
198 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 ...
2
votes
1answer
38 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 ...
2
votes
1answer
254 views

Conditional hitting time distribution of a Brownian motion

This problem cropped up in some research I am doing. I imagine it is standard, but I cannot seem to find the answer. Let $W_t$ be a standard Brownian motion. Suppose there are four values $a < 0 ...
2
votes
1answer
455 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: ...
2
votes
1answer
227 views

Doob's stopping time theorem with unbounded stopping time

Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$. ...
2
votes
2answers
187 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 ...
2
votes
0answers
12 views

Increments of a Brownian motion involving stopping times

I don't quite understand a proof involving Brownian motion in my book: Let $B$ be a standard Brownian motion and let $T$ be an a.s. finite stopping time. For some fixed $n \in \mathbb{N}$, let $T_n = ...
2
votes
0answers
56 views

Optimal Stopping for One-Armed Bandit with a Fixed, Known Payout.

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. I have a bandit problem with one ...
2
votes
0answers
121 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 ...
2
votes
0answers
19 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 ...
2
votes
0answers
64 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 ...
2
votes
0answers
118 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 ...
2
votes
0answers
42 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 ...
2
votes
1answer
101 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 ...
2
votes
0answers
54 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 ...
2
votes
1answer
44 views

IID sequence and stopping time

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a ...
1
vote
2answers
1k views

Sum of two stopping times is a stopping time?

Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. Question: Is $\sigma + \tau$ a stopping time? Attempt at a solution: I ...
1
vote
1answer
34 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 $\tau$ ...
1
vote
3answers
106 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 ...
1
vote
2answers
41 views

Stopping time intuition

Let $(X_n)_{n \geq 1}$ be independent and identically distributed random variables with $P(X_n=1)=P(X_n=-1)=\frac {1}{2}$ for all $n \geq 1$ and let $S_n = X_1+X_2+ \cdots +X_n$. If we define a ...
1
vote
2answers
41 views

Is this a stopping time or not?

Let $(\xi_n)_{n\in\mathbb{N}_0}$ be a sequence of independent identically distributed random variables that take values in $\left\{-1,1\right\}$ with equal probabilities. Define ...