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 $t-s$.

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15 views

$\int_0^t W_s \, ds $ has finite variation

Let $W=\{W_t:t \geq 0\}$ be a standard Brownian motion and consider $$X_t=\int_0^t W_s \, ds$$ We define the variation of a process $X_t$ has $$Var[X]_t=\lim_{n\to\infty} \sum_{i=1}^n |X_{t_i}-X_{t_{...
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1answer
27 views

How to solve this simple stochastic differential equation (SDE) $dX_t=-\frac{X_t}{1-t}dt+dB_t$? (hint included)

Solve the equation $dX_t=-\frac{X_t}{1-t}dt+dB_t$, $X_0=0$ in $[0,1)$. Hint: Apply the Ito folmula to $Y_t=X_t/(1-t)$. How to apply the Ito formula step by step, and how to confirm the solution? $B_t$ ...
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If $t\mapsto X_t$ is continuous almost everywhere and $(X_t)$ has independent increments, then $X_t - X_s$ follows a normal distribution?

The following statements can be found at Glasserman's Monte Carlo Methods in Financial Engineering. Given a stochastic process $(X_t)_{t\in [0,T]},$ if the mapping $t\mapsto X_t$ is continuous ...
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1answer
30 views

Quadratic variation of $ X_t = t W_t $

Let $W = \{W_t : t \geq 0\}$ be the standard Brownian motion and consider $$X_t=t W_t$$ Show that the quadratic variation of $X_t$ is $\frac{t^3}{3}$ I know this question has been answered ...
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0answers
18 views

Infinitesimal generator of the standard Brownian motion

As explained in this Wikipedia page, the infinitesimal generator of the standard Brownian motion is $\frac{1}{2}\Delta$ and for the Brownian motion it has an extra $\partial_t f$ term. Can anybody ...
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1answer
31 views

If $A_t = cos(X_t)$ and $B_t = sin(X_t)$ find the infinitesimal increment for $Y_t = A_t^2 + B_t^2$

If $X_t$ is Brownian motion, I'm not sure how to apply Ito's lemma to get $d Y_t$ for $ Y_t = A_t^2 + B_t^2$ where $A_t = cos(X_t)$ and $B_t = sin(X_t)$ in particular, I get confused because $sin^2(x) ...
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Functions within Expectation of Brownian Motion

I am tasked with the following: $$E[e^{3B_3}|\mathcal{F}_1]$$ For standard Brownian Motion, i.e. the drift ($\mu$) is $0$, and $\sigma^2=1$. What I have done so far is the following: $$E[e^{3B_3}|...
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19 views

The quadratic variation of the Brownian motion almost certainly tends to T

On the segment $[0, T]$, choose $n$ independent points $t_{n,k}$ (each distributed evenly). Prove that the quadratic variation of the Brownian motion on the sequence of random partitions of the ...
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1answer
31 views

Finding the expectation of a Wiener Process

My question is on how to find $\mathbb E[W_t^n]$ where $n= 0,1,2,...$ and $W_t$ is a standard normal Wiener process. Would we need to use a moment generating function. Thanks.
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Question about Ito Process. Stochastic Processes

How to prove, that $W_{t/(1-t)}$ at $[0,1)$ is Ito Process ? (Have stohastic differential)
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Find the Probability density function and prove the solution of the stochastic differential equation

A stochastic process $\{X_t, t \geq 0\}$ satisfies stochastic differential equation $\frac{dX_t}{X_t} = 3μ dt+2σ dB_t$, where $−∞ < μ < ∞$and $σ > 0$ are given constants, and $\{B_t, t \geq 0\...
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38 views

3-dimensional Brownian motion, probability distribution of first hitting time to a sphere

What is the probability density function or probability distribution of the time when 3-dimensional Brownian motion (no drift) starting from origin hits a sphere (ball) centered at the origin for the ...
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1answer
28 views

Distribution of the sum of Brownian motions

I believe that for a standard Brownian motion $W(t)$, $W(t)+W(s)$ has a normal distribution with mean $0$ and variance $s+t$ (because they are two independent normally distributed variables)? But is ...
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1answer
26 views

Radial Brownian Motion

Let $X_i(t)$ be standard Brownian motion in 1D. Define the radial Brownian motion as $\displaystyle R(t) = \sqrt{X_1(t)^2 + \cdots + X_N(t)^2}$. How do we lower bound the probability $\mathbb{P}(R(T)...
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Brownian motion — using limiting distribution to approximate probabilities of random walk

I managed to find these answers just by converting them to normal distributions of large n and using the rules about that I already understand, which I think is similar to using the Brownian motion ...
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1answer
36 views

I'm struggling with finding variance of Ito integral.

Find $D \int_{0}^{t}W^{2}_{s}dW_{s}$ My solution is next: variance = $E(\int_{0}^{t}W^{2}_{s}dW_{s})^{2} - (E\int_{0}^{t}W^{2}_{s}dW_{s})^{2}$ Which is equal to (using Ito's isometry principle) = $\...
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1answer
30 views

Solve an Itô Integral by Itô calculus

I saw an example where the following Itô integral was solved by Itô calculus: $\int^{T}_{0}W(t)dW(t)$. They say: let's take the stochastic process $X(t) = W(t)$, which means that $dX(t) = 0 dt + 1 dW(...
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53 views

Find $P(W_{t} < a | W_{2t} > a)$

Here are my thoughts: $P (W_{t} < a | W_{2t} > a) = \frac{P(W_{2t} > 2a, W_{t} < a)}{P(W_{2t} > 2a)} = \frac{P(W_{2t} - W_{t} > a,\, W_{t} < a)}{P(W_{2t} > 2a)}$ Then, taking ...
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Is it a good example for local martingale, but not for martingale?

If $B$ is a Brownian-motion in the $\mathcal{F}$ filtration, then the following process is a good example for being a local martingale, but not a martingale?$$S_{t}=\int_{0}^{t}\frac{1}{1-s}dB_{s},\;\;...
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2answers
24 views

Brownian motion increments - are they random variables or random processes

If $W_t$ is a Brownian motion process and $0 \le t_1 \le t_2$ then is the increment $W_{t2} - W_{t1}$ a random variable or a random process? My lectures say "random variable" but I believe it makes ...
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1answer
36 views

Expected value and variance for Itô Integral

This is my first question and I hope it is ok :) Reading in a book I came along this answer to a question I did not understand and I would like to understand this very much: Problem: Calculate the ...
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30 views

Conditional independence of Brownian Motion in separated intervals

Given that $(B_t)_{t\geq 0}$ is a standard Brownian motion and $0 \leq r < s$ and $x,y \in \mathbb{R}$, how can we show that the three processes $(B_t)_{t \in [0,r)}$, $(B_t)_{t \in (r,s)}$ and $(...
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1answer
49 views

Expectation of Stopping Time for a Brownian Motion with a drift

Let $a,b>0$ and define the stopping time $T_{a,b}$ for Brownian Motion as $$T_{a,b}:=inf\{t>0:B(t)=at-b\}$$ Compute $E[T_{a,b}]$. My idea: I think $E[T_{a,b}]=\infty$. If that was not the ...
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15 views

Given a standard Brownian motion $W_t$, (i) what are the distributions of $|W_t|$ and $tW_t$? (ii) Are they martingales?

For (i), I believe I am correct in saying that $|W_t|$ is a folded normal distribution with mean $\sqrt{\frac{2}{\pi}}t$ and variance $t$, and $tW_t$ is distributed normally with $\mathbb{E}(tW_t)=0, \...
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0answers
12 views

Positive discount factor for geometric Brownian motion

What will the expression below be like if the discount factor is positive? $u(x) = E_x[e^{r \tau}]$ Where $r >0$, and $\tau$ is the first time it hits $X^A$ starting from some point $x>X^A$? ...
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1answer
47 views

Distribution of Brownian Motion with drift

Let $B(t)$ be a BM on $[0,1]$ and let $X(t)=B(t)+t$. Let $P$ be the distribution of $B(t)$ and let $Q$ be that of $X(t)$. Is $Q<<P$? Edit: We have not discussed Girsanov's theorem yet... we ...
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78 views

Wald meets Weitzman/Gittins

Suppose a searcher is faced with $n$ objects. Each object's quality is an i.i.d. Bernoulli random variable $\Theta_{i}$, $i = 1, \dots, n$, where $\mathbb{P}\left(\Theta_{i} = 1\right) = \mu_{0}$. ...
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1answer
33 views

$\mathbb P(\sup_{t\in[0,1]}|W_t|\le1)$ for Brownian motion

What is $\mathbb P(\sup_{t\in[0,1]}|W_t|\le1)$ for $W_t$ a Brownian motion? Without the absolute value, we have $\mathbb P(\sup_{t\in[0,1]}W_t\le c)=1-\sqrt{2/\pi}\int_c^\infty e^{-x^2/2}dx$ for all $...
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1answer
30 views

Two-dimensional Brownian motion: Distribution of $W_1(\tau_a),$ where $\tau_a$ is hitting time for $W_2$

Let $a>0$ be a real number, let $T=\mathbb R_{\ge0},$ let $(W(t))_{t\in T}=(W_1(t),W_2(t))_{t\in T}$ be a two-dimensional Brownian motion and let $\tau_a=\inf\{t\in T:W_2(t)=a\}.$ What is the ...
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13 views

distribution of the Brownian Bridge for $T=1$

Let $W=\{W_t : t \geq 0\}$ be the standard Brownian motion and $Y = \{Y_t : t \in [0,1] \}$ with $Y_t = W_t-tW_1$ be a Brownian bridge. What is the distribution of $Y_t$? What I got is that $Y_t=...
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13 views

I am trying to construct a process $Y$ in the martingale representation of the following two terms

(i) $M_t = \mathbb{E}[B_t^2|F_t]$ (ii) $M_t = \mathbb{E}[\text{max}\{B_t,0\}|F_t]$ Where $B$ is a one-dim. Brownian Motion and $F$ its $P$-completed canonical filtration. I think in (ii) we can ...
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0answers
12 views

How to prove that $(u(B_t^{(1)},B_t^{(2)}),v(B_t^{(1)},B_t^{(2)}))$ is also a standard Brownian motion?

Suppose $\phi(x,y)=u(x,y)+iv(x,y)$ is a analytic function, i.e $u$ and $v$ satisfy the Cauchy-Riemann condition$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\...
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1answer
22 views

Limit to infinity of Lebesgue-Stieltjes integral with Brownian Motion

Prove that for every fixed $x\in \mathbb{R}$ $$ \mathbb{E}_x \left[ \exp\big(\int_0^t \frac{1}{1+B_s^2} ds \big) \right] $$ goes to $\infty$ as $t\to \infty$.
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16 views

System of Coupled differential equations with Stochastic frequency parameter

I have a second-order differential equation that has been put in the following form: $$dx = y dt$$ $$dy = -ax dt- bydt+ c \sin{(\omega t)} dt - xdB_t$$ where $dB_t = W_t dt$, $B_t$ being the ...
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1answer
28 views

Ornstein–Uhlenbeck stochastic differential equation help

Consider the Ornstein–Uhlenbeck stochastic differential equation: $$dX_t =−aX_tdt+σdW_t, X_0 =x_0 ∈ R$$ where $a$ and $σ$ are constants, and $W = (W_t)$, $t≥0$ is a Brownian motion. i) Using Itô’s ...
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1answer
11 views

Properties of a Brownian motion

Let $W = (W_t)$, $t≥0$ be a Brownian motion. Let $s, t ≥ 0$. Then: i) Show that $E(W_sW_t)=min(s,t)$ ii) Show that $E|W_t−W_s|^{4} =3|t−s|^{2}$ iii) Find the distribution function of the random ...
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1answer
52 views

Is it possible to prove that (almost surely) a path of a scaled random walk does not converge to any function on $C[0,1]$?

Consider the interpolated and scaled random walk generated by the independent random variables $(\xi_i)_{i \ge 1}$ with mean zero and finite variance: $$ X^{(n)}_t := \sum_{i=1}^{\lfloor tn \rfloor } ...
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1answer
49 views

Proof of Expected value of Brownian Motion

Consider the following exercise: Let $T_{[-a,a]} = \inf \{t: B_t \notin [-a, a] \}.$ Show that $E[T_{[-a,a]}]$ $=$ $a^{2} \times E[T_{[-1,1]}]$. Please tell me if this reasoning is correct: $T_{[-a,...
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1answer
27 views

Properties of Brownian Motion (Expected value)

Consider the following exercise: Let $T_{[-a,a]} = \inf \{t: B_t \notin [-a, a] \}.$ Show that $E[T_{[-a,a]}]$ $=$ $a^{2} \times E[T_{[-1,1]}]$. Please tell me if this reasoning is correct: $T_{[-a,...
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1answer
37 views

Itô's formula and Integration by parts

Could someone please help with this question: By applying the generalized Itô’s formula to the 2-dimensional process $ {\{(Xt,Yt),t \ge 0 }\}$ with the function $ F(x,y) = xy $, show the integration ...
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0answers
39 views

Shifted process with Brownian motion

We're starting with a probability space $(\Omega,\mathcal F,\mathbb P)$. We have the process $V_t=V_oe^{(\mu-\delta)t+\sigma W_t}$, where $W_t$ is a standard Brownian motion. Having the shift ...
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17 views

Distribution of Itô Formula (Brownian Motion) [duplicate]

can you please help me with this question: Let $f: [0,T] \to \mathbb R$ be a deterministic function, with $\int_{0}^{t} {f^{2}(s)ds < \infty}$. Prove that $\int_{0}^{t} {f(s)dW_{s}$ has normal ...
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1answer
22 views

Brownian motion and expectation

I'm having trouble solving the following exercise: Let $T_{[-a,a]} = \inf \{t: B_t \notin \{-a, a\} \}.$ Show that $E[T_{[-a,a]}]$ $=$ $a^{2} \times E[T_{[-1,1]}]$. I don't see how I can solve this. ...
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1answer
18 views

Geometric Brownian Motion and Stochastic Calculus

I am trying to calculate $d(X_t^2)$ for Geometric Brownian Motion. I know that for GBM we have $dX_t=\mu X_tdt + \sigma X_tdW_t$, where $W_t$ is a Wiener process. I am trying to work towards the ...
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1answer
43 views

Find the density function of sum of Brownian motion

The first passage time for Brownian motion $\tau_{a}=\inf\{t>0:B(t)=a\}$ is a random variable with density $$f_a(t)=\frac{|a|}{(2\pi\,t^3)^{1/2}}\exp(-a^2/(2t))$$ for $t>0$. Show that if $a,b>...
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1answer
23 views

Negatively Correlated Diffusive Processes

I am trying to generate 2 diffusive process that are correlated. One way that was recommended was to set up the process such that the changes are correlated - both processes have 0 drift and ...
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1answer
29 views

Stochastic process, why this is that?

Let $W_t$ be a process such that: $dW_t = (r+\pi* (\mu-r)) ) * W_t * dt + \pi * \sigma * W_t * d_{z_t}$ where $r, \pi, \mu, \sigma $ are constant, $Z_t$ is the Geometric Brownian motion. Applying ...
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1answer
26 views

calculating expectation of standard Brownian motion $W_t$

let $W_t$ be a standard Brownian motion, how can I calculate $E(W_t)$, I know the standard Brownian motion is symmetric, so this would be $0$ but how could I prove it via normal distribution? and ...
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0answers
20 views

Proof check: Show that $B_{t} \sim \mathcal{N}(0,t)$ is integrable

Let $B_{t} \sim \mathcal{N}(0,t)$, and show that $E[\lvert B_{t}\rvert ] < \infty$ My idea: $E[\lvert B_{t}\rvert ]=\frac{1}{\sqrt{2\pi}t}\int_{-\infty}^{\infty}\lvert y\rvert e^{-\frac{y^2}{2t}}...
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1answer
68 views

Deriving stochastic integral $ X+\frac{1}{2}\int_t^T Z_s^2 ds - \int_t^TZ_s dB_s$

If $X,\eta<\infty$ where $$ \exp(X) = \mathbb{E}[\exp(X)]+ \int_0^T \eta_s dB_s$$ Then let $$ \exp(Y_t) = \mathbb{E}[\exp(X)|\mathcal{F}_t]$$ Prove that for some $t\in [0,T]$, that there is ...

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