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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$\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum _{i=0}^{n}|B_{\frac{i+1}{n}}-B_{\frac{i}{n}}|^{p}=c_{p}$

This is a Homework question, so please do not answer it. Find real constants $a_{p},c_{p}$ s.t. $\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum ...
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Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book Probabilità e Modelli Aleatori of Enzo Orsingher, at page 134, it shows that the transition function of an absorbing Brownian ...
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30 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
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20 views

Exclusion-Inclusion principle for Hitting times of two disjoint sets

Consider disjoint sets A, B and Brownian motion $B_{t}$ with $B_{0}\notin A\cup B$. Let $T_{A}:=inf_{t>0}\{B_{t}\in A\}$. Then, do we get $P(T_{A\cup B}<\infty)=P(T_{A}<\infty)+P(T_{ ...
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64 views

Simple question about the definition of Brownian motion

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition: ...
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1answer
46 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
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129 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
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1answer
28 views

Solve parameter from stochastic integral

how can I solve $\rho$ from the following: $\int_0^T dV_t = \int_0^T \kappa (\theta - V_t) dt + \int_0^T \sigma \rho \sqrt{V_t} dW_t + \int_0^T \sigma \sqrt{1-\rho^2} \sqrt{V_t} dZ_t$, where $W_t$ ...
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55 views

$E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t )ds$

I was trying to compute $E[e^{\lambda X_t}|\mathcal{F_s}]$, where $X_t=\int_0^t(W_s-\frac{s}{t}W_t) ds$, $\mathcal{F}$ is associated to $W$. I tried the following. 1) Splitting the integral ...
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56 views

$E[W_s\int_s^t W_sds]$, $W_s$ is a brownian motion

Let $W_s$ be a brownian motion, I found $E[W_s\int_s^tW_sds]$ in a much longer exercise but I don't know how to compute it. Any suggestion?
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1answer
63 views

For a Poisson process prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
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3answers
77 views

Is $W_{2t}-W_t$ a brownian motion?

Is $W_{2t}-W_t$ a brownian motion? $(W_t)_{t\geq 0}$ is a brownian motion, I have to show that $X_t:=W_{2t}-W_t$ is a brownian motion as well. $$W_{2t}= 1/\sqrt{2} W_t$$ (by scaling property) then ...
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1answer
52 views

Distribution of two-sided boundary stopping time of Brownian motion.

If $B_t$ is a Brownian motion, and a one-sided boundary stopping time is given by: $\tau_a=\inf\{t:B_t=a\}$ the distribution of $\tau_a$ is given by: $f_{\tau_a}(t)=\frac{|a|}{\sqrt{2\pi ...
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94 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
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70 views

Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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1answer
63 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
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240 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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1answer
54 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
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1answer
44 views

Stopping times problem: $ \tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
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Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
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32 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
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1answer
37 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
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1answer
219 views

Quadratic Variation of Diffusion Process and Geometric Brownian Motion

I'm looking to find out the stochastic differential equation satisfied by the quadratic variation of Geometric Brownian Motion, Diffusion Process. For example, for a diffusion process that ...
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136 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
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49 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
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55 views

Independence of increments of a pair of independent Brownian motions

Suppose I have two Brownian motions $X$ and $Y$, which are independent. In other words, for any finite set of times $0 < t_1 < t_2 < \cdots < t_n$ the random vectors $(X(t_1),\ldots , ...
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2answers
75 views

independence two stochastic processes

being $X, Y$ two continuous processes, $\theta \in R$ $U_t=\sin{(\theta)}X_t+\cos{(\theta)}Y_t$ $V_t=\cos{(\theta)}X_t-\sin{(\theta)}Y_t$ I have to show that U and V are independent brownian ...
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Is there a modern iteration of Einstein's Brownian motion theory?

I was arguing with my friend that Brownian motion, in the sense of a pollen moving in the fluid, could be explained by physics laws (such as $F=ma$) and statistics laws. To check it out I found ...
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59 views

Discontinuous Lévy-Processes with normal increments

Does there exist a Lévy-Process with normal increments but with paths that aren't even continuos when modified on null sets? I'm asking because when defining Brownian motion as Lévy-Process, ...
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exercise 1.21 of chapter 1 of Revuz and Yor's

This is the exercise 1.21 of chapter 1 of Revuz and Yor's: Let $X=B^+$ or $|B|$ where $B$ is the standard linear BM, $p$ be a real number $>1$ and $q$ its conjugate number ($q^{-1}+p^{-1}=1$). ...
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1answer
41 views

Proving that a process has the Markov property

Let $X_t=xe^{ct+aB_t}$ where $B_t$ is one dimensional Brownian motion. How would I prove this is a Markov process using the expectation definition of a Markov process, i.e., ...
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63 views

Law and Brownian Bridge

Let $Z_{t}= W_{t}-tW_{1}$ and $Y_{1}=\sup_{0\leq t\leq 1}Z_{t}$, $(W_t, t \geq 0)$ standard Brownian motion Find the law of $Y_{1}$ I know that $\textbf{P}(\sup_{0\leq t\leq 1}W_{t}\geq x , ...
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1answer
72 views

Exercise 1.13 of chapter 1 of Revuz and Yor's

This is the exercise 1.13 of chapter 1 of Revuz and Yor's. Let $B$ be the standard linear BM. Prove that $\varlimsup_{t\to\infty}(B_t/\sqrt{t})$ is a.s. $>0$ (it is in fact equal to ...
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87 views

Expectation of e^(cX) if X is a geometric Brownian motion

(Edit:) The short version: Calculate $$E[e^{cY}]$$ if $c < 0$ and $Y$ is lognormally distributed, i.e. $\log(Y) \sim N(\tilde\mu, \tilde\sigma^2)$. The long version: I want to calculate ...
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567 views

Deriving Geometric Brownian Motion's solution?

The Black Scholes model assumes the following underlying dynamics, known as Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: ...
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The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
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1answer
88 views

Expectation of exponential of Brownian motion

I want to compute the following expectation: $\mathbb{E}[\int_0^\infty-e^{-\mu t+\sigma W_t}dt]$ where $W_t$ is a brownian motion, $\mu$ and $\sigma$ constant. I am already stuck at computing the ...
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1answer
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Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
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1answer
81 views

Ito formula applied to $\frac{1}{t}\int_0^t W_s ds $

I got this expression and I have to calculate its differential by the Ito formula, $W_t$ denotes the Brownian motion: $$\frac{1}{t}\int_0^t W_s ds $$ I calculate the derivative of ...
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Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
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1answer
71 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
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32 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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0answers
60 views

Branching Brownian Motion and KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
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1answer
54 views

What is the distribution of this random variable? [closed]

Find the distribution of this random variable: $$X_t=\exp\left(t \int_0^t sdW_s\right)$$ knowing that $W$ is a Brownian motion in the filtered space $(\Omega, \mathcal{F},P,(\mathcal{F}_t)_{t\geq0} ...
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1answer
58 views

Solve Itô integral with power

$$\int_0^t e^{Ws} W_s^r dW_s$$ where $W_s$ is Wiener process and r> in $\mathbb{Z}$ My first approach would be to use Ito's lemma, however, coming up with the function $g(t,x)$ is difficult The ...
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1answer
55 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
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2answers
137 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
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Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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20 views

Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
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89 views

Brownian Motion calculation

I am reading the following statement and cannot justify the upper limit. Any help would be greatly appreciated. Thanks. $$ E\left[\exp\left(2\int_{0}^{T}B_{s}^{2}\,{\rm d}s\right)\right] < \infty ...