# Tagged Questions

2answers
146 views

### A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
1answer
23 views

### Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
0answers
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### The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
0answers
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1answer
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### $\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
1answer
303 views

### Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
2answers
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### The 1-dimensional Hausdorff measure of a curve in the plane

For a set $X\subseteq\mathbb{R}^2$, let $H^1(X)$ be its 1-dimensional Hausdorff measure. Suppose $X$ is a regular curve (say, a graph of a continuous function $f:\mathbb{R}\to\mathbb{R}$). In that ...
1answer
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### How does the natural filtration of a Brownian motion look like?

I am trying to understand how the natural filtration for a Brownian motion might look like. Definitions: I will start with the definitions for reference. The definition of a natural filtration is ...
1answer
43 views

### Distribution of Difference of Independent Random Variables

Usually in the development of the theory of Brownian motion, one makes the assumption that $X_t$ (the coordinate functions on $(\mathbb{R}^*)^{[0,\infty)}$). have normal distributions with mean $0$ ...
3answers
113 views

### Brownian motion and adapted

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and "adapted" is one of the concept that I could not understand. First, "adapted" is defines at Ch3.1, page 25 (sixth edition): ...
1answer
110 views

### Calculating the probability of following event involving Brownian motion

I have a big time trouble in evaluating the following probability. It is related to brownian motion and measure, so I am asking experts from both fields for help! Denote $B_t$, $t\in [0, T]$ be ...
0answers
161 views

### Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
2answers
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1answer
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### Some preliminaries for the canonical construction of a Brownian Motion, help needed.

I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
1answer
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### Limit of occupation times for Brownian motion

Let $B_t$ be a standard Brownian motion on $\mathbb R$ started at $0$. For $A\subset\mathbb R$ Lebesgue measurable, let $\mu_T(A) = \frac{1}{T} m(t \leq T: B_t \in A)$, where $m$ is Lebesgue measure. ...
1answer
111 views