Questions tagged [local-time]

Local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level.

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Countably many discontinuities in a function of two variables

This question concerns an object from probability theory, but it should require only analysis to answer. The local time process $(t,a) \mapsto L_t^a$ of a continuous semimartingale is a real-valued ...
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Supremum of Brownian local time

Exercise 1.22 in Chapter 6 of Revuz and Yor's "Continuous Martingales and Brownian Motion" states that if $L$ is the local time for a standard Brownian Motion then $$\sup_{s \le t} L_s^{B_s} ...
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What's the proof for the formula for the local time of a Semimartingale?

I'm currently studying Local times of Semimartingales and I came across the following formula for the local time at $a$ of $X \in S_c$: $$ L_t^a (X) = \lim_{\varepsilon \to 0} \frac{1}{2\varepsilon} \...
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Infimum of Brownian motion local time

I'm working on a revision exercise that is as follows: For Brownian motion B, find a function $f:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that $$ L_t^0 (B) = \inf_{s \geq t} f(B_s, L_s^...
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How can I find the time when two people agree to meet if both of their watches have an offset?

The problem is as follows: Jenny and Vincent agreed to meet at the library at $6\,p.m$. Both synchronized their watches at midnight ($\textrm{0 hours}$). We know that Vincent's watch is not ...
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Simple Time Pattern [closed]

I am given the times: 1:38, 2:44, 3:49, 4:55, ? , 7:05, 8:11, 9:16, 10:22, 11:27, 12:33 I am asked to find which time is missing and what the times signify? Thanks!
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Proof local time of BM only increasing on 0

Let $B$ be a standard Brownian motion and the local time of $B$ at $0$ defined by Tanaka's formula \begin{equation} L_t = |B_t| - \int_0^t sgn(B_s) dB_s \ . \end{equation} Now I want to prove that $...
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Switching the hands of the clock [duplicate]

Let us switch the positions of the minute hand and the hour hand of a calassic clock with only 12 hours, not 24. Note that most of the time we will end up with an absurd configuration. But are there ...
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Changing my Clock

I was given a problem in class. Let's say I bought a clock with no numbers on it (has only the hands). Because of the recent time change, I had to change my clock one hour back. In order to do this, I ...
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Proof of Non-differentiability of Brownian motion using local times

I was reading Chapter 3 of Brownian Motion and Stochastic Calculus by Karatzas and Shreve and Exercise (6.6) asks us to prove the following(I also include the screenshot of definition 6.3) I tried ...
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How to calculate mean value on cycle space?

Maybe I am not very familiar with terms, but I will try. Imagine loop(or cycle) space, where the first element of space is the next to the last element of the same space. The simples example is time....
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Differentiation of local time in spatial variable

Consider a continuous martingale $(X_t)_{t≥0}$ starting at 0 and define the associated Local Time at $a$: $$L^a_t=\lim_{ϵ→0} {1 \over 2ϵ}∫_0^t 1_{[a−ϵ,a+ϵ]}(X_s)ds.$$ Is it possible for the local ...
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Continuity of a certain integral/additive functional

Let $L_x^t$ local time of a standard Brownian motion on $(\Omega,\mathcal{F},\mathbb{P})$. Furthermore let $L_x^t$ the jointly continous version. Assume $m[0, \infty) \rightarrow [0,\infty)$ is ...
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Working with ratios - how to get math correct for this question?

On TVTropes, Webcomic Time (link) is when real-world time does not reflect the passage of time in the comic, basically, real-time and time in the comic book are asynchronous. On the link above, it ...
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Time derivative of Local time

Consider a Ito Process,not necessarily Brownian Motion, $(X_t)_{t\geq 0}$ starting at 0 and define the associated Local Time at $0$: \begin{equation} L_{t}=\lim_{\epsilon \rightarrow 0}\frac{1}{2\...
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2 pipes working to empty a tank.

A pipe can empty a tank in 40 min. A second pipe with diameter twice as much as that of the first is also attached with the tank to empty it. The two together can empty the tank in ? I am attaching ...
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1 Gigasecond is also a Teramilisecond? [closed]

I'm reasoning about some time parsing thing in JavaScript which uses millisecond-precision for Date objects and Unix System time does use seconds. Putting problems with precision loss for conversion- ...
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Figuring time since epoch given different time scale

In programming, we use a start date and count the milliseconds since that date, to calculate the 'time since epoch' (which is 1970 Jan 1st 00:00:00 I believe). A class called DateTime does all the ...
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time, rate and distance

a group of storm chasers is following behind a tornado in order to film it. The tornado is moving a straight line at 53 km per hour, and the storm chasers are driving in a van in its wake, travelling ...