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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Brownian Motion Hitting Time Distribution

Define $\tau_a = \inf \left\lbrace t \geq 0 | B(t) \geq a \right\rbrace $ for some $a>0$. The problem is to show that $ \tau_a \stackrel{d}{=} \sqrt a\tau_1 $. What I've done so far: $$P(\tau_a ...
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1answer
53 views

Brownian motion at infinity

This is probably a standard exercise in stochastic calculus but I haven't been able to come up with a proof that relies only on a given set of results. So my question is about proving the following ...
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1answer
22 views

does brownian motion and poisson random measure have to be independent? [closed]

Suppose a brownian motion $W$ and a poisson random measure $\mu$ are defined on the same filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$, where both $W$ and $\mu$ are ...
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1answer
38 views

Local maximum of brownian motions

Let $B=(B_t)_{t\geq 0}$ be the standard Brownian motion. I want to show that for every $t_0 \geq 0$ $\mathbb{P}$($B$ has a local maximum in $t_0$)=0. I've already shown that for every ...
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0answers
53 views

How to do integration by parts with brownian motion?

I am not sure how to perform integration by parts in the following expression: $$ \left(1-t\right)\left(B_t - B_s + \int_s^t \frac{r}{1-r} \mathrm{d} B_r \right) $$ Can anyone help me to solve this ...
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1answer
21 views

Expected Stopping Time for BM

I'm working on this homework problem for Brownian Motion. Suppose we define a stopping time $\tau_a = inf \left\lbrace t \geq 0 : B(t) = a \right\rbrace$ for some $a>0$. I already showed in a ...
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58 views

Ornstein-Uhlenbeck a Markov process

Consider the Ornstein-Uhlenbeck process defined by $$ X_t = e^{- \alpha t} X_0 + \sigma \int_0^t e^{ \alpha (s-t)} d W_s$$ with $\sigma,\alpha>0$. In many literature I have found they considered ...
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25 views

Brownian Motion Maximum Value Proof

Let $B(t)$ be a Brownian Motion and $$M(t) = max_{s:s \leq t} B(s)$$ and $$\tau_a = min_t{B(t) = a}$$ Then, $P(\tau_a < t) = P(B(t) - B(\tau_a) > 0 \: |\: \tau_a < t) + P(B(t) - B(\tau_a) ...
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1answer
36 views

On random rotational fluctuations in $\mathbb{R}^n$

Imagine first a disk that is mostly stationary, except for random ("thermal" if you like) "rotational fluctuations" around its axis (which is fixed). Something a bit like what's shown in the figure ...
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1answer
43 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
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question about brownian motion and integration

If $X(t)$ is the standard Brownian motion, $0<\alpha<\beta$, and $T$ is the first exit time of $X(t)$ from $[-\beta,\beta]$, then how can I find $E(\int_0^T \mathbb{I}_{(-\alpha,\alpha)} X(t) ...
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46 views

Generator of Wiener process and its running maximum

If we let $W$ be a standard linear Wiener process issued from zero and $M$ its running maximum $$ M_t := \sup \{ W_u: u \leq t \}, $$ then we could show that $(X,Y):=(M,M-W)$ is a Markov process on ...
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0answers
65 views

Absorbed brownian motion is a Markov process

I have been asked to prove that the Brownian motion absorbed at the origin is a Markov process. Formally, let $B_t^x$ be a Brownian motion originating from $x>0$ and let $\tau^x_0 = \inf\{t>0 : ...
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2answers
25 views

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where ...
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16 views

Mutual independence of increments of Brownian motion

Brownian motion has a bunch of different definitions. My question is about showing the property in the title using a certain definition of BM and nothing else. The (partial) definition I am given is ...
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2answers
43 views

Brownian motion and covariance

Show that for $B = (B_t)$ Brownian motion, its covariance is $cov(B_s, B_t) = min(s, t)$. The solution I was given was: For $s ≤ t$, $B_t = B_s + (B_t − B_s)$, $B_sB_t = B_s^2 + Bs(Bt − Bs)$ ...
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41 views

Transition function for absorbed Brownian motion

I need an help with the following exercise. I've already seen this question Prove that Brownian Motion absorbed at the origin is Markov but I don't understand the answer. Also I would like to prove ...
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2answers
30 views

How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
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1answer
45 views

How to solve Stochastic differential equation?

I do not have a clue on how to solve out this type of question, and how to deal with integration with a combination of brownian motion and linear function. Can anyone help me out please?
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2answers
18 views

Using Ito's formula, write down a stochastic diferential equation satiesfied by $Y_t:=X_t^2$, given both $Y_t$ and $X_t$

I am trying to solve this exercise and I am stuck in the third part of it. I checked the solution and it makes no sense to me, so I would really appreciate it if someone could explain to me how Ito's ...
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1answer
87 views

Existence of the Brownian Motion using the Kolmogorov extension theorem

Kolmogorov extension theorem: Let $T$ denote some interval (thought of as "time"), and let $n \in \mathbb{N}.$ For each $k \in \mathbb{N}$ and finite sequence of times $t_{1}, \dots, t_{k} \in T$, ...
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1answer
35 views

Applying Picard-Lindelöf iteration to a stochastic integral equation

Suppose we have the following stochastic integral equation (we can make it an SDE) where $W$ is a standard Brownian motion $$ X_t = 1 + \int_0^t X_s d W_s. $$ I want to show that by using Picard ...
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Brownian motion proof of Dirichlet problem

I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let $D$ be a bounded domain in $\mathbb{R}^d$ such that every boundary point satisfies the Poincare cone ...
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1answer
34 views

If $B$ is a Brownian motion and $B'_t:=B_{T+t}-B_T$ for a fixed $T$, then $(B'_t,t\ge 0)$ and $(B_s,0\le s\le T)$ are independent

Let $B=(B_t,t\ge 0)$ be a Brownian motion and $$B'_t:=B_{T+t}-B_T\;\;\;\text{for }t\ge 0$$ for some $T\ge 0$. Especially, $B$ has independent increments, i.e. ...
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1answer
203 views

Convergence of Ornstein-Uhlenbeck process as a scaled Brownian Motion

Let $W$ be a standard Brownian motion. Let $\alpha,\sigma^2 >0$, and let $X_0$ be a $\mathbb{R}$-valued random variable with distibution $\nu$ that is independent of $\sigma(W_t,t\geq 0)$. Now ...
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1answer
15 views

Ito's formula for this stochastic differential - please explain this step?

Referring to those two lines, can someone please explain how those results were obtained? My understanding is, the following formula is being referenced: $$dV_t = dV(S_t,t) = \frac{\partial ...
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1answer
39 views

Itô integral of an elementary process

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t,t\ge 0)$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t,t\ge 0)$ be a stochastic process on ...
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2answers
214 views

Is the reflected Brownian Motion a Markov process

Let $W$ be a Brownian Motion (BM). The reflected BM is defined by $X=|X_0+W|$. We need to show that this process is a Markov process w.r.t. its natural filtration and we need to compute its ...
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0answers
44 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
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1answer
44 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even tough its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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1answer
18 views

Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such ...
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Liminf Brownian Motion question

For this assignment I'm working on, I was able to prove that: $$\limsup_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = \infty$$ where $B_t$ is a Brownian Motion. I'd like to be able to prove: ...
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0answers
42 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
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2answers
45 views

Standard Brownian Conditional expectation

(Given the process $(B(t))t≥0$ of Brownian motion, define the random variables $$Y=\int_0^{1}B(s)\,ds $$ $$X=B(1) $$ Determine the quantities $E(Y|X)$, $Var(Y−E(Y|X))$ and the conditioned density ...
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1answer
39 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
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1answer
71 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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1answer
71 views

Local martingale but not martingale

On wikipedia there is an example of a local martingale which is not a martingale, but I do not understand why it is a local martingale. We have the process $ X_t = \begin{cases} ...
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1answer
95 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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1answer
41 views

Strong Markov Property and Product of Expectations

Let $(B_{t})_{t\geq0}$ be a Brownian motion and let $\tau=\inf\left\{ t\geq0:B_{t}\leq-4\right\} $ be a stopping time. Then the strong Markov property ensures that e.g. $A:=\left\{ ...
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0answers
31 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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1answer
26 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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1answer
16 views

Brownian Motion Finding M(t)

If I have that {$B(t); t >=0$} is a standard Brownian motion, with $B(0)=0$, and I let $M(t)$ = max{$B(u) ; 0 \leq u \leq t$} and I am supposed to: a) Evaluate Pr{$M(4) \leq 2$} and b) Find the ...
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1answer
43 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
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0answers
30 views

Augmentation of a Filtration

In class, we showed that Brownian Motion is a martingale with respect to the filtration $F_t = \sigma(B(s): 0\leq s \leq t) $. For a HW assignment, I need to show it's a martingale with respect to a ...
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1answer
19 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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1answer
43 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
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1answer
42 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
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39 views

What is the resulting stochastic process of divided Geometric Brownian motions

Let $W_{1,t},W_{2,t},...,W_{n,t}$ be $n$ independent geometric Brownian motions. Now let's say I construct the following processes: $$ X_1 = \frac{W_1}{\sum_i^n W_{i,t}} $$ $$ X_2 = ...
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2answers
61 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
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24 views

Levy process of argument in the complex plane

I am stuck on this question: Let $B$ be a Brownian motion in $\mathbb{C}$ started at $1$. Let $\theta_t$ be a continuous determination of the argument of $B_t$, i.e. $B_t = |B_t| e^{i \theta_t}.$ ...