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 $t-s$.

2,583 questions
66 views

Probability of Brownian Motion hitting -2 before 1?

Why is the probability of Brownian Motion hitting -2 before 1 is equal to 1/3? This is an interview question asked for Quant roles. I found a similar question was previously asked: Brownian motion ...
92 views

What is the distribution of minimum of Brownian motion on arbitrary interval?

We know that $P(\min_{0 \leq s\leq t} B_t \leq x)=2P(B_t\leq x)$. This can be found in any standard stochastic calculus textbook. However I am curious about instead of the interval $[0,t]$ if we ...
14 views

Is there a statistical test to test whether a time series behaves like a Brownian motion?

The observed data that I am dealing with is pictured above. It is generated by some mathematical algorithm (simulations) described in section 4.1.(c) in my article, here (data available in spreadsheet ...
12 views

Understanding a statement about Brownian motion processes

A book I'm reading says, "an important result about Brownian motion is that, conditional on the value of the process at time $t$, the joint distribution of the process values up to time $t$ does not ...
46 views

Why does proving $\lim_{h\to 0} (X(t + h) - X(t)) = 0$ prove continuity?

Why does proving $\lim_{h\to 0} (X(t + h) - X(t)) = 0$ prove continuity? I don't think it matters, but more specifically, $X(t)$ is a Brownian motion. I thought that a function $f$ is continuous at ...
31 views

Covariance of Gaussian Processes based on Brownian Motion

I know that the covariance of two Brownian Motions (Bs,Bt) is min(s,t), but I don't know how to use that to find the covariances of Gaussian processes that are based on Brownian Motion t>0. Yt := e^(...
151 views

Black-Scholes model with time-dependent volatility

We consider the Black-Scholes model with time-dependent volatility $\sigma(t)$: $$dS_{1}(t)=rS_{1}(t)dt+\sigma(t)S_{1}(t)dW(t)$$ The question: what constant $\hat{\sigma}$ one needs to apply such ...
215 views

Probability that Brownian motion is negative in $[1, 2]$, given endpoints are positive

Let $B_t$ be a Brownian motion. Compute $P(\inf_{t \in [1,2]} B_t < 0 \mid B_1 > 0, B_2 > 0)$. This is a practice interview question I found here. My attempts are below, and I would ...
36 views

56 views

72 views

Proving $\omega\mapsto B(\cdot, \omega)$ is measurable.

Let $(B_t)_{t \geq 0}$ be a Brownian motion on the probability space $(\Omega, \mathcal{A},P)$, and $\pi_t$ the canonical projection at time $t$, and $C(\mathbb{R}^+_0,\mathbb{R}^d)$ is the set of ...
1k views

Explaining Ito's Lemma

Find $$\int_{0}^{T}W(t)dW(t)$$ using Ito's Lemma. Now, I know that the answer to that question is: $\int_{0}^{T}W(t)dW(t)= \frac{W^2(T)}{2}-\frac{T}{2}$ but can somebody explain the idea behind the ...
24 views

Construct Correlated Wiener Processes

Construction Hello.. I am trying to better understand how one can correlate independent Wiener processes given a correlation matrix. Please see the attached notes. This method uses the Cholesky ...
29 views

We know that for a submartingle $A(t)$, $A(t)-\langle A\rangle_t$ is a martingale where $\langle A\rangle_t$ is its quadratic variation. For processes like $W^3(t)$ ($W(t)$ being standard Brownian ...
46 views

Prove that $\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0$, for each $t\geq0$, where $B$ standard Brownian motion.

Let $B(t)$, $t\geq0$, be a standard Brownian motion. I would like to prove that $$\lim_{L\rightarrow\infty} P\left(\sup_{0\leq s\leq t}|B(s)|>L\right)=0,$$ for each $t\geq0$. In my class notes, ...