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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Trying to understand Tanaka's example of SDE.

Consider the following Stochastic Differential Equation: $$dX_t = \sigma(X_t) \, dB_t \tag{1}$$ Where $$\sigma(x)= \begin{cases}1 & x \geq 0\\ -1 & x < 0 ...
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1answer
54 views

Show that $\mathbb{P}(\tau_{0}>T)\approx\frac{1}{\sqrt{T}}$ where $\{ B(t) : t\geq 0\}$ is a linear brownian motion started at $B(0)=1$ [on hold]

I'd appreciate if someone could provide me with a solution for the following problem: Let $\left\{ B\left(t\right)\thinspace|\thinspace t\geq0\right\}$ be a linear brownian motion started at ...
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38 views

Normalized hit times of a simple RW converge in distribution to hit times of standard Brownian Motion

I would appreciate some hints or guidance towards solving the following exercise: Let $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,...\right\}$ be a simple random walk on the integers ...
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1answer
24 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
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1answer
32 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
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1answer
50 views

Using Feynman-Kac, compute the following:

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
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14 views

Distribution of hitting time for two border brownian motion

I'm trying to find the distribution of hitting times for two border brownian motion with respect to both the hitting time AND which border is hit. Is this well defined? This is assuming $W_0=0$ with ...
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20 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
2
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1answer
27 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
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16 views

Reflection principle in the proof of the distribution of $M_t - W_t$ (Brownian motion)

Let $W_t$ be the Brownian motion starting at $0$. Consider the following random variables. $M_t = \sup_{0\leq s \leq t} W_s$ and $|W_t|$. We first calculate $$\Bbb{P}(|W_t|>a ) = \Bbb{P}(W_t ...
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1answer
64 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
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1answer
46 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...
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1answer
35 views

The entrance law of a Brownian motion with absorbing boundary

In the article "Construction of Diffusion processes with Wentzell's Boundary conditions by means of poisson point processes of Browninan excursions" one reads: I tried to compute it for $n=1$ ...
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1answer
22 views

Compute the expected value of a brownian motion

Suppose $X(t)$ is a brownian motion. Compute $E[X(1)X(5)X(7)]$. I know that the brownian motion has independent increments, so if we could write $X(1)X(5)X(7)$ as such, then we could use the ...
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0answers
42 views

How to calculate hitting probabilities for Brownian motion.

Given a standard Brownian motion with no drift, the PDF is... $${{1} \over {t^{3/2} \cdot \sqrt{2\cdot \pi}}} \cdot e^{-1/{2t}}$$ (Derived from the CDF $\int_{-\infty}^{f(t)/\sqrt{t}} {1 \over {2 ...
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2answers
35 views

Translation invariance of Brownian motion

Beginner here. I'm working through Durrett's textbook's and am just getting into the section on Brownian motion. He gives a 2-line proof for a simple fact but I'm a little stuck understanding the ...
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16 views

When is an Ornstein-Uhlenbeck process equivalent to Brownian motion up to a given time lag

Background The expected displacement under the assumption of Brownian Motion for time step $\tau$ is given by $$\gamma(\tau) = D|\tau|$$ where $D$ is the diffusion coefficient. If one assumes a ...
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27 views

Brownian motion needs to be defined continuous for every $\omega$ to be jointly measurable.

Let $B=(B_t)_{t\in[0,\infty)}$ a Brownian motion (BM) and $(\Omega,\mathcal{F},P)$ be the probability on which $B$ is defined. Some define BM as a.s. continuous, e.g., Brownian motion is almost surely ...
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1answer
27 views

Showing a process satisfies an SDE

The example of Ito and Watanabe in the following notes http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf is an SDE without unique solutions. $$dX_t = 3X_t^{1/3} dt + 3X_t^{2/3} dW_t$$ ...
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1answer
32 views

what would be power series of $x_t = e^{\beta_t} $ if $\beta_t$ is a Brownian motion process?

In general the power series of $e^x =1+x/1!+x^2/2!+x^3/3!+...$ but because the process is random we can't apply the direct differentiation than how can i write it's power series.In the book stochastic ...
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12 views

Given a geometric Brownian motion, obtain a recurrence relation for moment $\mathbb{E}X_k^q$ and deduce $\mathbb{E}_k^q=\alpha^k(ah^{1/2},q)$

I am really confused about a step in the solution of this problem. I would really appreciate it if someone could explain to me what is in bold below. Part A is OK, but I don't understand something in ...
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1answer
22 views

Distribution of stochastic integral w.r. to brownian motion

Let $B=(B_t)_{t \geq 0}$ be a standard brownian motion, $T > 0$ and $f : [0,T] \rightarrow \mathbb{R}$ a continuous function. I want to determine the distribution of the following integral: ...
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1answer
29 views

Mean Squared Displacement of Biased Random Walk [closed]

If $x_t=x_{t-1}+\mathcal{N}(\mu,\sigma)$ and $x_0=0$ what's the value of $\langle x_t^2\rangle$?
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1answer
23 views

Why does dZ(t)dt=(dt)^2=0 [duplicate]

I am having trouble with this question: $$\mbox{Let Y(t)}=\begin{cases} 0 & -1\le t \le 0\\ Z(t)-tZ(t) & 0 < t < 1 \\ 1-t & 1 \le t \le 2\end{cases}$$ Find the quadratic ...
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0answers
10 views

relation between 3-dimensional bessel process and the SRW on Z conditioned on never hit zero?

What is the relation between 3-dimensional bessel process and the SRW on Z conditioned on never hit zero? From Exercise 11 of Link: http://www.math.bme.hu/~gabor/Feladatok3.pdf
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1answer
40 views

Show that $\lim_{|P|\to 0}\sum_{k=0}^{n-1}\frac{W(t_{k+1})+W(t_k)}{2}\left[W(t_{k+1})-W(t_k)\right]=\frac{W^2(T)}{2}$

I have this problem which I am stuck in because it seems very obvious to me that the result is correct, but I don't know how $|P|\to 0$ can be used in the proof. Thanks a lot! QUESTION: Show that ...
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2answers
48 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 ...
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1answer
74 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
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0answers
23 views

Integration of Brownian Motion with a function of t

What would be integration of Brownian Motion with $t$ and is there a general formula for $f(t)$ , i.e. What is $E[\int_{0}^{t} t W_t dt]$? Is it $0$? Any general rule or result for ...
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1answer
43 views

Expectation of a product involving Brownian motion

I would need to verify if this solution is fine. Let $W_t$ be a Brownian motion and $\lambda > 0, \text{ } \lambda \in \mathbb{R}$. Calculate $\mathbb{E} \left[W_t e^{(\lambda W_t)}\right]$. ...
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26 views

Moment Generating Function for Brownian motion's exit of interval.

Let $B(t)$ be a standard BM. Consider the stopping time $T = \inf\{ t > 0: |B(t)| = a\},$ the usual first exit time of the interval $(-a, a).$ We can see that $\mathbb{E} e^{tT} < \infty$ for ...
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26 views

The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

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 ...
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0answers
26 views

harmonic functions and ito formula

I am trying to prove the mean-value property for harmonic functions in $R^k$ by ito calculus. given $G$ bounded domain and $u$ harmonic function on $G$ then $u(a)=\int_{\partial B_r} u(y)ds(y)$ ...
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1answer
35 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
3
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1answer
42 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
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1answer
22 views

Notation in stochastic integrals

There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts: $$E^x\left[u(W_0)\right]=u(x)$$ What is the notation ...
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1answer
54 views

Itô integral with respect to a diffusion

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 ...
4
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1answer
28 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 ...
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1answer
23 views

Prove that the Itô integral for elementary predictable processes builds a martingale

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 ...
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1answer
34 views

Distribution Stopping time under Brownian motions

Considering $W$ the canonical process on $C([0,1],\mathbb{R})$ and the row filtration generated by the coordinate process of $W$, I want to prove that ...
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2answers
45 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
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1answer
22 views

Fubini's Theorem for Stochastic Integral

Probably a bit trivial, but I was curious about the validity of interchanging the following integrals (where $W_t$ is Brownian Motion): $\mathbb{E}[\int^{t}_{0} W^2_s ds] =? \int^{t}_{0} ...
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1answer
31 views

Why a change of a brownian motion does not depend on the past values of it? [duplicate]

$(B_t)_{t \in \mathbb R_0^+}$ are random variables on $(\Omega,\mathcal A,P)$. $\forall r \le s, t > s, B_t-B_s,B_r$ are independent (i.e. $\sigma(B_t-B_s)$ and $\sigma(B_r)$ are independent, ...
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2answers
29 views

If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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1answer
28 views

Motivation behind the definition of the Itô integral for elementary predictable processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$. A real-valued stochastic process $H=(H_t)_{t\ge 0}$ is called elementary ...
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1answer
27 views

Solving the Geometric Brownian Motion on a general interval.

I know that the Geometric Brownian Motion, with the expression $dX_t = v X_t dt + \sigma X_t dW_t$ has the next solution $$X_t = X_0 e^{\sigma W_t+ (v-\frac{\sigma ^2}{2})t}$$ on the interval $[0,T]$. ...
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1answer
41 views

(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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1answer
32 views

Prove that the increments of the Brownian motion are normally distributed

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
1
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1answer
35 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
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0answers
29 views

Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...