Question related to Brownian motion, a stochastic process denoted $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.
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1answer
44 views
Stochastic process, Gaussian, with zero mean is a Wiener process
Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
1
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1answer
28 views
How do you make dependent Brownian motions independent?
Can someone explain to me how to take 2 correlated Brownian motions and make them independent? I can't seem to grasp this process.
Just assuming $dB_1(t)dB_2(t) = \rho dt $
From what was explained ...
-1
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0answers
45 views
$P\{X_t=-X_t \}=1$
If we define that $X_t$ is Brownian motion over space $(\Omega,\mathcal F ,\mathcal F_t;P) $,
then why is it true that the fact that $X_t$ is Brownian motion implies that $P\{X_t=-X_t \}=1$ is ...
3
votes
0answers
29 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
1
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1answer
39 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
2
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1answer
45 views
An application of Donsker's theorem.
Let ${X_i}$ be iid with $E[X]=0$ and $Var(X)=\sigma^2$ Let $S_0=0$ and $S_n=X_1+...+X_n$ for all $n \ge 1$. Show that $\lim_{n\rightarrow \infty}P(S_k>0 \space for \space k=n,n+1,\dots,2n)=1/4 $.
...
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1answer
44 views
Product of correlated brownian motions
Consider that the correlation between two standard brownian motions $dB_x$ and $dB_y$ be $\rho$. And we write $\mathtt{Cor} (dB_x,dB_y)$ = $\rho$. Show that $dB_xdB_y$ = $\rho dt$
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1answer
44 views
Let $W$ be a Wiener process and $X(t):=W^{2}(t)$ for $t\geq 0.$ Calculate $\operatorname{Cov}(X(s), X(t))$.
Let $W$ be a Wiener process. If $X(t):=W^2(t)$ for $t\geq 0$, calculate $\operatorname{Cov}(X(s),X(t))$
0
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1answer
34 views
Definition of regular point of a boundary with planar brownian motion
This is an exercise in G.Lawler's book Conformally invariant processes in the plane.
First he defined regular point of a boundary using brownian motion:
Suppose $D$ is a domain in $\mathbb{C}$ with ...
1
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1answer
56 views
The infinity version of Blumenthal's 0-1 law
Blumenthal's 0-1 law states that on the space of continuous maps with domain $[0, \infty)$ with the appropriate (Wiener) measure making the coordinate maps Brownian motions starting at $x$, any event ...
1
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2answers
68 views
Standard Brownian Motion
Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$.
Can anyone give me some hint to start the ...
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1answer
101 views
Show that $M$ is a martingale
Let $B$ be typical Brownian motion with $\mu >0$ and $x \in \mathbb{R}$. $X(t):=x+B(t)+\mu t$, for each $t\geqslant 0$, Brownian motion with velocity $\mu$ that starts at $x$. For $r \in ...
1
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1answer
80 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
...
1
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1answer
44 views
Harmonic Measure & Brownian Excursion
I have a disk of radius R removed from the plane. Brownian excursions are emanated from infinity. I am tasked to find a probability that the brownian motion impacts an arc of the disk boundary. I am ...
1
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1answer
59 views
Using a Brownian martingale to compute the second moment of a hitting time
Prove $ W_t=B_t^4 -6B_t^2t+3t^2$ is a martingale, and compute $E(T^2)$ where $T=\inf(t\ge0,B_t=-a, B_t=b)$ if $a=b$.
Ok, if $0\lt t\lt s$, $W_t$ is a martingale if $E(W_s|[B_r]_{r\le t})=W_t$
So ...
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1answer
23 views
Basic brownian motion computation
Let $B_t$ denote a standard 1-d Brownian Motion. Find $P(B_2 \gt 2)$.
My sol.
$B_2 ~ N(0,2)$ so $P(B_2 \gt 2)=1-P(B_2\le 2)=1-\frac{\int_0^2e^{-\frac{x^2}{4}}}{\sqrt{4\pi}}$, but where do i go from ...
3
votes
1answer
52 views
Distribution of a Brownian motion with respect to $\mathbb{P}^x$
Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space and $(B_t)_{t \geq 0}$ a Brownian motion (started in $x=0$). Then one can define a probability measure $\mathbb{P}^x$, $x \in \mathbb{R}$, on ...
1
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1answer
30 views
Normal probability and Brownian motion
Let $X_t$ be a Brownian motion with parameter $\sigma$. Find the probability in terms of $$\Phi(x)= \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^x e^{- \frac{ \alpha ^2}{2}}d\alpha$$
How would I do this for ...
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1answer
61 views
Local Maximum of Brownian motion.
Given two positive rational number $a,b$. How to show that almost surely Brownian motion
attains a local maximum at some time in $(a,b)$?
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0answers
48 views
Drift equation / Girsanov's Theorem
Define $$\frac{dS_t}{S_t} := \mu \, dt + \sigma^B \, dB_t +\sigma^W \, dW_t,$$ $$dS_t^{(0)} := S_t^{(0)}r \, dt,$$ where the constants $\mu, \sigma^B,\sigma^W$ and $r$ all positive and $\mathcal{F}_t$ ...
1
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3answers
34 views
Scaling and time inversion for Brownian motion basically the same?
Let $B(t)$ be a Brownian motion. For $a>0$, we have the scaling relation
$$\hat{B}(t)=aB(t/a^2) \sim B(t)$$
and $\hat{B}(t)$ is also a Brownian motion.
The time inversion formula states that
...
5
votes
2answers
75 views
Construction of Brownian Motion
In Wiener's construction of Brownian Motion, it is assumed that there exists a probability space $(\Omega,\mathcal F,\mathbb P)$ and random variables $X_n:\Omega\rightarrow\mathbb R$ for $n\in\mathbb ...
1
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1answer
65 views
Approximation of stochastic integral
Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
1
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0answers
38 views
Rate of increase of maximum process of Brownian Motion
Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely?
Thanks!
1
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1answer
67 views
Laplace Transform of a Brownian motion
If $v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$ is a Standard Brownian motion, then for what values of $s,\omega$ does the Laplace transform $l(\omega,s) = \int_0^\infty e^{-st} v(\omega,t) ...
1
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0answers
44 views
Expected value of brownian motion for all positive paths
I've got this question but I can't figure it out.
Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$?
Thanks
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0answers
53 views
Quadratic variation process of $G$–Brownian motion
I would like to prove the inequality
$$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$
where $\langle B ...
5
votes
1answer
131 views
Ito's Lemma and Brownian Motion
Show by using Ito's Lemma, for $k \geq 2$ the following result hold.
$$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$
where $W(t) = N(0,t)$ is standard Brownian motion.
I think ...
1
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1answer
50 views
Backward martingale property of quadratic variation
Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
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1answer
75 views
Master equation for a Brownian Bridge
Anyone has an idea how to constract a Master equation for the Brownian bridge?
In the form of: W[i+1]=W[i]+Pr*DeltaX-Pl*DeltaX
where Pr (Pl) is the prob. to jump right (left) and DeltaX is the ...
2
votes
1answer
89 views
Brownian Motion Conditional Expectation Question
I have a real number $x$, and $W$ is a standard Brownian motion. Let $0 < s < t$. How to find
$$
\mathsf E[W_s | W_t = x]
$$
Please provide me with a step by step answer as I want to ...
1
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1answer
37 views
Lack of right-continuity of the filtration adapted to Brownian motion
Let us consider the standard Brownian motion and the natural filtration $(\mathcal{F}_t^B)$. It is known that $(\mathcal{F}_t^B)$ is not right-continuous at $t=0$. But what about $t>0$? Is it true ...
4
votes
1answer
76 views
Application of central limit theorem for triangular arrays
A (1-dim) Brownian motion $(B_t)_{t \geq 0}$ satisfies the following properties:
(B0): $B_0=0$ a.s.
(B1): $(B_t)_t$ has independent increments
(B2): $(B_t)_t$ has stationary increments, ...
3
votes
1answer
41 views
Interpolation result for Brownian Motion in Donskers Theorem
Suppose we have an increasing sequence of stopping times $\{\tau_n\}$ such that $\tau_n-\tau_{n-1}$ are iid. Furthermore let $B$ be a Brownian Motion and we define $S_n:=B(\tau_n)$ which gives a ...
2
votes
0answers
43 views
Negative moments of a functional of Wiener process
At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
2
votes
1answer
53 views
Some preliminaries for the canonical construction of a Brownian Motion, help needed.
I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
1
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1answer
97 views
Distribution of integral with respect to Brownian motion
Let $B$ be a Brownian motion and define the complex sequence $(X(n))_{n\in \mathbb Z}$ as
$$ X(n) := \int^\pi_{-\pi} e^{inx} dB(\pi + x)$$
What is the distribution of $X(n), n\in \mathbb Z$?
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votes
1answer
55 views
Specific use of reflection principle for Brownian motion
Let $B$ be a Brownian motion with $B(0)=0$, $x,y>0$ and $B^*$ the Brownian motion reflected at $-x$.
I came across the following:
$$ \mathbb P_0(\inf_{s\in [0,t]}B(s)<-x, B(t)\geq y-x) = ...
2
votes
1answer
108 views
Doob's stopping time theorem with unbounded stopping time
Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$.
...
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votes
2answers
39 views
Identity for exponential of Brownian motion using scaling relation
Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$.
I stumbled over the following identity:
$$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))]
\\=\mathbb ...
0
votes
0answers
35 views
Intuitive meaning of the generator of a Brownian motion $L=\frac{d}{\lambda(dx)}\frac{d}{dx}$
For a standard Brownian motion $B_t$, the generator is
$$
L_B=\frac12 \frac{d}{dx} \frac{d}{dx},
$$
we say that $B_t$ is a diffusion with canonical scale the Euclidean space, and speed measure the ...
4
votes
1answer
126 views
Stopping time and Brownian motion (specific example)
Let $B$ be a Brownian motion. I want to show that
$$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$
is not a stopping time w.r.t. the standard filtration.
How can one intuitively see that this ...
0
votes
1answer
114 views
Checking for Martingales on Stochastic processes
I am confused about how to check whether a process is Martingale. I know, I have to check for clear drift but a bit confused about to approach this problems. I need to apply Ito's first i think. For ...
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0answers
43 views
Quadratic Variation of a Brownian Martingale
I would like to show that the quadratic variation of the square integrable martingale $(W_t)^2-t$ where W is a Wiener process is $\int_{i=0}^t (W_s)^2ds$ . Any hints?
thank you
0
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1answer
47 views
Expectation of a stochastic exponential
In class a while ago we used the following simplification:
$$ \mathbb E \left[ \exp\left(\langle \boldsymbol a,\mathbf W_t\rangle \right) \right] \quad =\quad \exp\left(\frac12 |\boldsymbol a|^2 ...
0
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0answers
137 views
Are these processes martingales?
Determine and prove if the following processes $ Y(t) $ are martingales. Assume that $ X(t) $ is the standard Brownian Motion
$$ Y(t) = e^{\sigma X(t)-0.5\sigma^2t} $$
$$ Y(t) = e^{0.5t}\Bigg(1 - ...
0
votes
0answers
135 views
Analysis of Brownian Motion
The following tasks consider transformation an analysis of Brownian Motion.
For the proces $ Y(t) = -\theta \mu t + \sigma X(t) $ design an algebraic substitution to $ X(t) $ that removes the drift ...
0
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0answers
117 views
Geometric Brownian Motion
Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$
$$dS = \mu Sdt + \sigma SdX$$
Show by the application of Itô's Lemma to function $\log S$ ...
0
votes
1answer
63 views
Squared increments of Brownian motion in $L^2$
I have encountered an $L^2$ limit, which I am not sure of:
Let $B_t$ be a Brownian motion and $a=t_0<t_1<\dots <t_n=b$. Show that
$$\lim_{max(t_{i+1}-t_i)\mapsto 0} \sum_{0\leq i \leq n-1} ...
0
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0answers
44 views
how to recognize stochastic process among wiener process, log-normal, normal, and mean-reversion
Wiener process (brownian motion), normal distribution, log-normal, and mean-reversion are 4 most frequently used stochastic processes in modelling.
i wonder, given 30-50 sample points, is there a ...
