# Questions tagged [brownian-motion]

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

2,535 questions
17 views

### Lévy processes, Brownian motion and Lie groups

We know that Brownian motion describes any (non-deterministic) Lévy process with continuous sample paths. The above statement is true in Euclidean space. My question is: does it stand for other ...
30 views

### Gaussian process is a Brownian motion.

I am referencing from the book Probability from Dava Khoshnevisan. Thats my definiton of a brownian motion: 1) $W(0) =0$ and for all $t > 0$ is W(t) normal distributed with mean 0 and variance t. ...
18 views

### Let $\{B_t\}_{t \ge 0}$ a Brownian Motion, prove that for all $a \gt 0$: $P(B_t \ge a$ for some $t \ge 0 )=1$

Let $\{B_t\}_{t \ge 0}$ a standard Brownian Motion, prove that for all $a \gt 0$: $P(B_t \ge a$ for some $t \ge 0 )=1$ Use the principle of reflection Thoughts: The reflection principle is: ...
20 views

### Density and Probabilities of Geometric Brownian Motion

Let $\{ X_t \}_{t \ge 0}$ defined by $X_t := 10+3t+3B_t$ where $\{ B_t: t \ge 0 \}$ is a standard Brownian Motion. Let $\{ S_t \}_{t \ge 0}$ defined by $S_t := e^{X_t}$ a) Write explicitly the ...
20 views

### SDE with absolute value

Find the unambiguous strong solution of the equation: $dX_{t} = dt+ |X_{t} - t|^{\frac{1}{2}}dW{t}, X_{0} = 0$. It is easy to tell that the solution exists but I don't know how to find it. Any help ...
82 views

18 views

### Difference of normal random variables

I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that... $$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$ I understand ...
33 views

13 views

### Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
21 views

### $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
42 views

### Calculating $E(2W(s)+W(u)|W(u)=2)$

Let $W(t)$ be standard Brownian motion and let $u<s$. I know that $W(s)\sim\mathcal{N}(0,\sqrt{s}), W(u)\sim\mathcal{N}(0,\sqrt{u})$ and $2W(s)+W(u)\sim\mathcal{N}(0,\sqrt{8s+u})$. How should I ...
9 views

28 views

### Why does calculating the quadratic variation of a Brownian motion in this way not work?

This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to ...
39 views

20 views

### How do I make sense of this expectation?

I am having some difficult in making sense of ${\rm{E}}\left[ {\int\limits_0^t {{W_u}du} } \right]$ where W is just your standard one dimensional brownian motion. Is interchanging the order of the ...
104 views

### “Conditional distribution” of Brownian sample paths

I would like to consider the "conditional distribution" of the Brownian sample paths conditional on certain sample path functionals, in a similar way that one considers the Brownian bridge. For ...
49 views

### Brownian motion, compact interval

Let $(W_t)_{t\geq0}$ denote a standard Brownian motion and $I=\left[a,b\right]$ a compact interval. Show that $P\left[\frac {W_{t+h}-W_t} {h} \in I\right] \rightarrow 0$ as $h\rightarrow 0$. What does ...
74 views

### Distribution of last exit time of Brownian motion with drift

If $X_t = \mu t + \sigma W_t$ with $W_t$ a Wiener process, I would like to know if the distribution for the last time $X_t = a$ is known - and if so, what it is. My googling has turned up a bunch of ...
45 views

### Find the quadratic variation process of $\int f(s) \, dB_s$
Let $f \in L^2[a,b]$ and let $\displaystyle M(t)=\int_a^tf(s)dB(s)$. Find the quadratic variation process, $[M]_t$ , of $M(t)$. Here the quadratic variation process is the limit in probability of \$\...