Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 $t-s$.

4
votes
1answer
13 views

Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace ...
0
votes
2answers
16 views

Solve SDE for Brownian Bridge

Let $(B_t)$ be a one-dimensional Brownian motion and $y \in \mathbb{R}$. Show that the solution to the SDE $$dX_t^y=dB_t + \frac{y-X_t^y}{1-t}dt$$ with initial value $X_0^y = 0$ on $[0,1)$ is given by ...
1
vote
0answers
15 views

Finding a strong solution to $X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t$

I have an SDE $$X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t,$$ $X_0=x_0$ and $A,a,\sigma,S$ are continuous stochastic processes, $B$ is a BM. Now if I define: $$Y_t:=e^{(\int_0^tA_sds+\int_0^...
1
vote
0answers
27 views

Intersection-exponent for one-dimensional Brownian motion

We let $B^1,B^2$ be independent, one-dimensional Brownian Motions with $B^1(0)=-1$ and $B^2(0)=1$ and $T_n^i=\inf\{t\geq0:|B^i(t)|=n\}$. In Gregory Lawler's: Hausdorff Dimension of Cut-Points for ...
0
votes
1answer
25 views

Stochastic integration rules

I stumbled upon a stochastic integration result in a book and I'm not sure how it is derived. It goes as follows: Suppose the process $Y^{(t,y)}$ has dynamics under the equivalent martingale measure $$...
0
votes
1answer
36 views

How to easily see the time integral of a Brownian motion is normally distributed?

It's well known that $X_t:=\int_0^tB_\tau d\tau$ where $\{B_\tau\}$ is a 1D standard Brownian motion is distributed as $N(0, t^3/3)$. Is there any "immediate" way to see this fact? The easiest one I ...
1
vote
1answer
23 views

What is the quadratic variation of $e^{-t\lambda}B_{\frac{e^{2\lambda t}-1}{2\lambda}}$?

Let us define $$X_t:=e^{-t\lambda}B_{\frac{e^{2\lambda t}-1}{2\lambda}},$$ where $B$ is a Brownian motion and $\lambda>0$ How can I calculate quadratic variation $<X>_t$? The first thing ...
1
vote
1answer
28 views

Lévy process + scaling property $\implies$ Brownian motion

How can I show that if $\xi_t$ is a Lévy process distributed as $\xi_{t+s}- \xi_s$ for all $t,s \in [0,\infty)$ and has independence of increments, and also is distributed as $\lambda\xi_{\lambda^{-2}...
-1
votes
0answers
17 views

Derive a SDE for process X [closed]

Question: Define a process $X$ by $X_t = W_t^2 - 2\int_0^t uW_udu$. Derive a SDE for $X$. I know to solve this problem I need to apply Ito. However, how can I deal with the integral $2\int_0^t ...
0
votes
0answers
29 views

Brownian motion with drift $\mu$ [closed]

Let be $(B_t: t \geq 0)$ Brownian motion with $B_0 = 0$. Let be $S_t = \max_{0 \leq s \leq t} B_s$. First i need to find density for $(B_t, S_t)$. That i solve and i get density of normal ...
0
votes
1answer
30 views

Standard Brownian motion and stopping time

Let be $B$ standard Brownian motion and let $S \leq T$ two stopping times with $E(T) < \infty $ and $E(S) < \infty$. Prove that hold $$ E[(B_T - B_S)^2] = E[B_T^2 - B_S^2] = E(T-S).$$ Please ...
-1
votes
0answers
26 views

Compute $E(\int_0^t(1+e^{B_s})dB_s)$

Asked to calculate $E(\int_0^t(1+e^{B_s})dB_s)$ for $t\geq0$ where $(B_t)$ is standard Brownian Motion. I asked a similar question here: Compute $E\left((B_t−1)^2\int ^t_0(B_s+1)^2 dB_s\right)$, where ...
0
votes
0answers
3 views

Good material on reflecting boundaries for stochastic processes

restricting attention to continuous time, continuous state space ( say $\mathbb{R}$) stochastic processes. Can someone point me in the right direction of how to one imposes reflecting boundary ...
1
vote
1answer
23 views

Proving Brownian motions have bounded quadratic variation

I am looking into arguments which yield the result $(dB_t)^2=dt$ and authors first start by letting P be a partition $P=\{t_0, t_1, ..., t_n\}$ of the interval $ [0,T]$ where $t_i=\frac{i}{n}T$ and ...
0
votes
2answers
38 views

Proving that stopping time is finite a.s.

Let $$\tau_{a} = \inf\{t>0 : W_{t} + at = 5\}.$$ Prove that $\mathbb{P}(\tau_{a}<\infty) = 1$ for $a\ge0.$ My solution: We know that $W_{0} +a*0 < 5$. Furthermore, because $W_{t} \sim \...
0
votes
1answer
44 views

Wiener process equality

I came across such a problem that I can't solve Prove that with probability $=1$ the equality $$W_{t}^{4} = W_{t}^{3} - 3W_{t}^{2} +1$$ holds for infinite number of $t \ge 0$ but it doesn't hold for ...
0
votes
1answer
35 views

Compute $E\left((B_t−1)^2\int ^t_0(B_s+1)^2 dB_s\right)$, where $(B_t)$ is a standard Brownian motion

Compute $E((B_t−1)^2\int ^t_0(B_s+1)^2 dB_s)$ for $t≥0$ given that $(B_t)_{t≥0}$ is a Standard Brownian Motion. Presume we will need to compute $E((B_t+B_s)-(B_s-1))^2$ to get some independent terms ...
0
votes
1answer
23 views

Compute $E(M_σ)$ when $σ = \inf(t ≥ 0: |B_t| = 1)$ and $M_t = 4B^2_t +e^{4B_t−8t}−4t$

Given $M_t = 4B^2_t +e^{4B_t−8t}−4t$ for $t ≥ 0$ and a Brownian motion $(B_t)_{t \geq 0}$. Compute $E(M_σ)$ when $σ = \inf(t ≥ 0: |B_t| = 1)$. I have tried to show that $E|M_σ|\leq K$ to apply Doob's ...
1
vote
0answers
25 views

measurable functions commute with conditional expectation

Let $W_t = W (t)$ be a (one-dimensional) Wiener process, and fix an admissible filtration $\mathbb{F}.$ An adapted process $V_t$ is called elementary if it has the form $$V_t = \sum_{j = 0}^{k} \xi_j \...
3
votes
1answer
52 views

How to compute $\mathbb{E}[X_{s}^{2}e^{\lambda X_{s}}]$ where $(X_s)$ is a Brownian motion with drift $\mu$?

I'm working on a problem and at a certain point I ran into the problem as described in the title. We have that $\{W_t,t\geq 0\}$ is a Brownian motion and $\mathscr{F}_t$ is the corresponding ...
2
votes
1answer
21 views

Reflexion principle equivalence of statements

$B_t$ is a Brownian motion, $S_t$ is defined as $$ S_t := \sup_{0 \leq s \leq t} B_s $$ I want to show that $$ P(S_t\geq b , B_t \leq a) = P(B_t \geq 2b-a) $$ for $a \leq b$ implies $$ P(S_t \geq a) =...
1
vote
1answer
46 views

Expected value of Brownian motion at a time decided by a rate one Poisson process.

Situation: We have that $\{W_{t},t\geq 0\}$ is a Brownian motion and $\{N_{t},t\geq 0\}$ is a Poisson process such that $N_{t}$ follows a Poisson distribution with parameter $t$. This process ...
-1
votes
1answer
34 views

Brownian motion, stochastic process

Question: Let $W$ be a Brownian motion with $W(0) = 0$. Determine $E[\cos{W(t)}+\sin{W(t)}]$. I let $Y=\cos{W(t)}$ $dY=-\sin{W(t)}dW(t)-\frac{1}{2}\cos{W(t)}(dW(t))^2=-\sin{W(t)}dW(t)-\frac{1}{2}\...
0
votes
0answers
19 views

Squared Brownian motion and its moments

I have the following $X_t$ which satisfies: $X_t=a \cdot t+b \cdot W_t$ where $a,b \in R$ and $W_t$ is a Wiener process such that $W_t$ is normally distributed with $N(0,t-s)$ for $t>s$. Suppose ...
1
vote
1answer
27 views

Inferring observation time from a Brownian motion

This might be a bit lengthy question. So let me proceed in steps. General description: I have some observations, based on which I want to infer their occurring time. Specific setting: Let $W(t)$ be ...
4
votes
1answer
28 views

Expectation value of the average displacement squared in a random walk

Consider a simple 1D random walk, with equal probability of going to the right (toward positive x) by one unit of distance and to the left (toward negative x) with one unit of distance. Let x=0 be the ...
2
votes
1answer
28 views

If $ S_t $ follows a log-normal Brownian motion, what SDE does the square of $ S_t $ follow?

If $ S_t $ follows a log-normal Brownian motion, what SDE does the square of $ S_t $ follow? I have found two possibles ways of solving it. But, they diverge with respect to the drift. First ...
2
votes
0answers
51 views

Diffusion equation for Brownian motion with a systematic drift component

I know, that BM formula $$X_{i+1} = X_{i}+\sqrt{\delta t}Z_i,$$ $$Z_i\sim N(0,1),$$ with $X_0\equiv0$ leads to Diffusion equation $$u_t=-\frac{1}{2}u_{xx}$$ for marginal distributions in $X_i=X(t)...
0
votes
1answer
39 views

PDE with terminal condition, Feynman-Kac

I have the following PDE: $$2u_t+9u_{xx}-2u_x=0$$ $$u(x,T)=e^x$$ and I get that $$dX(s)=-dS+3dW(s)$$ $$X(t)=x$$ But how do I get the expected value of $e^x$? I tried substituting $Y(s)=e^x$, but I ...
0
votes
0answers
35 views

Bessel process time change

Let us consider a Bessel process $$dX_t = \frac{\mu - 1}{2 X_t} dt + dW_t$$ where $\mu > 1$ is constant (dimension) and $W_t$ is a standard Wiener process. Then I denote the additive functional $Y ...
1
vote
0answers
32 views

Find constants $a$ and $b$ such that $X(t)$ is a Brownian Motion

Let $B(t)$ be a Brownian Motion. Find all constants $A$ and $b$ such that $X(t)=\int_0^t(a+b\frac{u}{t})dB(u)$ is also a Brownian Motion. First we know that if $f \in L^2[a,b]$ then $\int_a^bfdB(u)$...
0
votes
1answer
44 views

What is a white noise ? What is the derivate of the Brownian motion? [duplicate]

Could someone explain me what is a whit noise ? In my course it's written that it's the derivate of a Brownian motion, but how can it be the derivative of something that doesn't exist ?
0
votes
2answers
14 views

Why is $E([\Sigma_{i=0}^{n-1}((\Delta_i^nW)^2-{T\over n})]^2) = \Sigma_{i=0}^{n-1}E[((\Delta_i^nW)^2-{T\over n})^2]$ in this solution?

I'm studying Wiener process (Brownian motion) with the book: Brzezniak, & Zastawniak. "Basic Stochastic Processes." Springer. I don't understand the solution for one of the exercises in the book. ...
0
votes
0answers
23 views

How to evaluate $Var$ $\int_0^t (B_s + s )^2 d B_s $?

I am stuck trying to find $Var$ [$\int_0^t (B_s + s )^2 d B_s $] where $B_t$ is brownian motion. I want to using Ito isometry. Here is my attempt to use Ito isometry. $\int_0^t (B_s + s )^2 d B_s $ ...
1
vote
1answer
47 views

Show that stochastic integral w.r.t. Brownian motion is normal distributed

I want to show the following claim: Let $B$ be a one-dimensional Brownian motion and let $$I(\phi):=\int_0^1 \phi(s) \text{d}B_s.$$ Show that $\mathbb{E}(I(\phi))=0$ and $\mathbb{V}(I(\phi))=\int_0^1(...
0
votes
2answers
54 views
0
votes
0answers
20 views

Integral representation of fractional Brownian motion and covariance function

It is well known that a gaussian process is identified by its covariance function and its mean function. Given the covariance function: $$ R \left( t , s \right) = E \left[B_t^H B_s^H \right] = \frac{...
1
vote
0answers
71 views

Hitting time distribution with exponential growth

Let $A_0=A>0$ and let $$dA_t = (rA_t - x)dt + \sigma dB_t,$$ where $B_t$ is standard Brownian motion and $r,x$ and $\sigma$ are positive constants. Let $T= \inf \{ t: A_t = 0 \}$ and $$G(A)=\Bbb{E}[...
2
votes
1answer
31 views

What's the distribution of $\int tW(t)dt$ for Brownian motion $W(t)$

Let's say a standard Brownian motion is written as $B(t)$. Then, define $W(t) = \sigma B(t)$. I know that $\int_0^1 W(t)dt $ has the distribution Normal(0,$\sigma^2/3$). But, what about $\int_0^1 tW(t)...
1
vote
0answers
37 views

Differential of Stochastic process

Given a stochastic process $$Z_t = e^{4t} \int_0^{t} e^{-2s} \, \,dB_s$$ where $B$ denotes the standard Brownian Motion. Determine $dZ_t$. I tried to make use of Ito's rule, seeing that $Z$ is a ...
0
votes
0answers
22 views

Is the following stopping time finite: $T:=\inf\{t\geq 0:B_t\geq \sqrt{t}+1\}?$

We have a Brownian motion process $B$ and a stopping time defined like this: $$T:=\inf\{t\geq 0:B_t\geq \sqrt{t}+1\}.$$ Is this stopping time almost surely finite, eg. $T<\infty$, and why? My ...
1
vote
0answers
19 views

Show $(B_t )^2$ i.e. square of a Brownian motion is a Markov process. [duplicate]

Problem: Show $(B_t )^2$ i.e. square of a Brownian motion is a Markov process. To do this, I want to show $$P( B_t ^2 | B_{t_1} ^2 , ... , B_{t_n} ^2 )= P( B_t ^2 | B_{t_n} ^2 ) $$ where $0<...
1
vote
0answers
39 views

Show that this processes are martingales

I have to show that the following two processes are martingales $M_t=g(t)B_t-\int_0^tg'(s)B_sds$ $X_t=\exp\left(e^tB_t-\int^t_0e^sB_sds-\frac{e^{2t}}{4}+\frac{1}{4}\right)$ Where $g(t)$ is a real ...
0
votes
0answers
51 views

Python monte carlo value at risk with non normal distribution

What is the right way to implement a monte carlo method on a currency portfolio with non normal distribution? I am using a geometric brownian motion with normal random variables but I would like to ...
0
votes
2answers
22 views

Proving alternative notation of quadratic variation of Brownian motion

Setting: Let $\{W_t,t\geq 0\}$ be a Brownian with respect to the standard filtration $\{\mathcal{F}_t,t\geq 0\}$. Problem: We fix $t>0$ and must prove that: $\left \langle W,W \right\rangle^{(n)...
1
vote
0answers
15 views

Proving Kolmogorov's continuity condition holds for Brownian motion?

Let $B(t)$ be $n$-dimensional Brownian motion. I want to show that $E|B(t)-B(s)|^4 = n(n+2)|t-s|^2$ so that Kolmogorov's continuity theorem can be used to show that Brownian motion has a continuous ...
1
vote
1answer
21 views

Why is $P^b(B_{T-T_b} \in (-\infty,b))=1/2$

Why is $P^b(B_{T-T_b} \in (-\infty,b))=1/2$ on the set $\{T_b<t\} $ where $T_b=\inf\{t \ge 0 :B_t=b\}$ and $T=t 1_{\{T_b<t\}}+\infty 1_{\{T_b \ge 0\}}$. I am trying to understand Proposition 2....
1
vote
1answer
22 views

Why is $T:=\inf\{t\geq0:B_t\leq at^p-b\}$ a stopping time?

How can I prove that $T:=\inf\{t\geq0:B_t\leq at^p-b\}$ is a stopping time w.r.t. a natural filtration of $B$, where $B$ is a $BM$, $p>1/2$ and $a,b>0$? I can introduce a new random process, $...
1
vote
0answers
14 views

Brownian motion in random landscape

I have some trouble with an exercice: Take $(W_t)_{t\ge0}$ and $(B_t)_{t\ge0}$ two independent standard Brownian motions started from 0 and define the process $$X_t=\int_{\mathbb{R}}\sqrt{L_t^x(B)}...
2
votes
1answer
90 views

Probability on first hitting time of Brownian motion with drift

I am struggling with the following problem: Let $B$ be a one dimensional Brownian motion and $a,b>0$. Show that $$P[B_t=a + bt \text{ for some } t\geq 0] = e^{-2ab}.$$ The following hint is ...