# Tagged Questions

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 $t-s$.

2k views

### expected value of brownian motion

How can you find this expected value? $$\mathbb{E}[|W_{t}^2 - t|]$$ where $W_{t}$ is a brownian motion.
113 views

### Solve the integral $\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx$

To find the Variance of a Wiener Process, $Var[W(t)]$, I have to compute the integral $$Var[W(t)]=\dots=\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx=\dots=t.$$ I've ...
457 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 ...
3k views

### Expectation of Stopping Time w.r.t a Brownian Motion

How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$\tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\}$$ I understand the optional stopping ...
239 views

### Show that $X(t)=t W(1/t)$ is a Brownian motion if $W(t)$ is a Brownian motion.

I am trying to solve a past exam question for which I have its answers. I've got to the end, but the very last and simplest line has confused me. I've spotted some errors and corrected them, but I ...
158 views

168 views

### Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
3k views

2k views

### Sum of Brownian Motions

I've got a little problem: if $X_{t}$ and $Y_{t}$ are two indipendent Brownian motions, is then $$Z_{t}:=X_{t}+Y_{t}$$ a Brownian motion too? I've got some troubles only with showing that $Z_t$ is ...
132 views

620 views

### Covariance of Brownian-motion-like processes

We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion. What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$? How can I write $B_{f(t)}$ in an Ito ...
2k views

### To show that a given process is Gaussian

Suppose I have given a Brownian Motion $W$, this is a Gaussian process, and I define: $$B_s:=W_{t-s}-W_t$$ for $0\le s\le t$. Clearly this random variable has expectation zero. For the covariance ...
56 views

### process with integral is martingale

How to show that the process $X_t=tW_t - \int_0^t W_s ds$ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. Please,...
34 views

### Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
103 views

300 views

35 views

229 views

### Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
234 views

### Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
78 views

### Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$\tau = \inf\{t \geq 0; B_t < t-2 \}$$ This is a clear ...
1k views

### Expectation of stochastic integrals related to Brownian Motion

I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter: Let B be a standard Brownian Motion ...
919 views

123 views

### Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
118 views

### What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
33 views

### Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$\int_0^T W(t) dt = \int_0^T (T-t) dW(t)$$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ (...
111 views

### A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds$$ is a martingale, where $f\in C^2$ and compactly ...
A (1-dim) Brownian motion $(B_t)_{t \geq 0}$ satisfies the following properties: (B0): $B_0=0$ a.s. (B1): $(B_t)_t$ has independent increments (B2): $(B_t)_t$ has stationary increments, i....