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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Exponential of Sums of different times of a Brownian Motion

Let $\{B_s\}_{s\in[0,1]}$ be a Brownian motion, let $t_1 < \dots < t_n \in [0,1]$, I am interested in finding good upper and lower bounds for $$ \mathbb{E}[\exp(B_{t_1}+ \dots + B_{t_n})]. $$ ...
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21 views

What will be the behavior of $R(t)$ if $R(0)=\alpha / \beta$ in Vasicek model

Can someone please help explain what the behavior of $R(t) = \alpha / \beta$ would be if $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ I already know the expression for R(t) and I know the mean and ...
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1answer
34 views

If given the Vasicek Interest rate model $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ how do I use Ito's lemma to find $d(e^{\beta t}R(t))$?

If given the Vasicek Interest rate model $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ how do I use Ito's lemma to find $d(e^{\beta*t}R(t))$ and simplify so it is a solution that does not include R(t). ...
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1answer
45 views

Check that an Ito integral is a martingale.

Before presenting my problem I will introduce some notation. Time index $t\in [0,T]$. $$C_t = \begin{cases} Z_n = B_{t_{n-1}}, & \text{if $t=T$} \\[2ex] Z_i = B_{t_{i-1}} , & \text{if ...
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32 views

Independence of Brownian motions

Let $W_t$ be a Brownian motion at time $t$ and let $t<t_1<T$. I'm trying to find the variance of $$ k\sigma W_{t_1} + \sigma \left(W_T - W_t\right).$$ I started by letting $$ k\sigma W_{t_1} + ...
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1answer
36 views

Variance of integral

I am trying to understand stochastic calculus and got stuck calculating the following. I need the distribution of a zero bond under the black model, so I am deriving the variance using the second ...
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1answer
45 views

Scaling property for Brownian motion [closed]

Define Brownian motion as a continuous process $(B_t)$ with independent increments, such that $B_{s+t}-B_{s}$ has normal distribution with mean $0$ and variance $t$. How do you use the independence of ...
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1answer
34 views

Fubini's theorem for Stochastic Integral, with sum

I am struggling here with part (2), . In usual instances, I've had the question phrased like this but I'm not sure how to deal with the summation?
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2answers
34 views

Prove that Brownian motion $(X_t)$ is such that $P(|X_{t+h} − X_t|> ε)\ll h$ when $h\to 0$

I am facing this problem, but don't have the knowledge to solve it. I need to prove that given a simple Brownian motion $X_t$, I have that $$\frac{P(|X_{t+h} − X_t|> ε)}{h}→ 0 \mbox{ when }h\to ...
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11 views

Question about Brownian motion invariance

We know that if $(W_s)$ is a Brownian motion, then $W_{s+T}-W_T$ is a Brownian motion where T >0. So $W_{s+T}=(W_{s+T}-W_T)+W_T$ is the sum of a Brownian motion and a time-independent process. In my ...
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26 views

First hitting time of an open set

I am trying to go through the problems in Shreve & Oksendal, this is the problem 2.6 Prove that the first hitting time $\tau$ of $A$ an open subset of $\mathcal{B}(\mathbb{R}^d)$ is an optional ...
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1answer
62 views

I've found two different definitions of a cylindrical Brownian motion and don't understand why they are consistent

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space ...
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43 views

Prove of the existence of a cylindrical Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be ...
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48 views

Can we prove/disprove that Brownian motion is nowhere differentiable in $L^2$?

I have read the proof that Brownian motion is a.s. nowhere differentiable. However can we construct a proof that Brownian motion is no where differentiable in $L^2$ for an appropriate definition of ...
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25 views

Representation formula for a Hilbert space valued Brownian motion. Prove independence of the real-valued Brownian motions in the expansion.

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be ...
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25 views

If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

For my study, as a part of a Matlab exercise, the following question is asked: Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ ...
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43 views

$ \lim\limits_{n\rightarrow \infty} {\frac{B_{t}}{t}}$ Brownian Motion

I want to prove in two ways that $ \lim\limits_{n\rightarrow \infty} {\frac{B_{t}}{t}}\rightarrow 0$ almost surely, where $B_{t}$ is a standard Brownian Motion. 1) in $L^2$ Can we say ...
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13 views

Why is there periodicity in the output of Richard Voss' fractional Brownian motion?

I am trying to figure out why the output of fractional Brownian motion (fBm) as described by Richard Voss (Random fractal forgeries. In: Fundamental Algorithms for Computer Graphics, R. A. Earnshaw ...
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1answer
20 views

Martingale $c^{W_t}$ where $W$ is Brownian motion

I have the process $c^{W_{t}}$ where $c$ is a constant and $W$ is Brownian motion. I would like to check if $\mathbb E[c^{W_{t+1}}|F_t]=c^{W_t}$. Dividing the right site yield $\mathbb ...
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113 views

How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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0answers
45 views

Correlated Brownian motions

Let $V$ and $W$ be Brownian motions such that $\mathbb{E}W_tV_t=\rho t$. Let $$R_t=\sup_{u \le t} V_u \mbox{ and } Z_t=\sup_{u \le t}W_u .$$ Show that $$\mathbb{E}R_tZ_t=tf(\rho) .$$ Can you find ...
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0answers
51 views

Generalised arcsine law of brownian motion

It is well known that for a standard brownian motion, the time spent above $0$ follows an arcsine distribution (whose density function is U-shaped). Can anyone tell me how to generalise this result to ...
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37 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
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1answer
98 views

What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
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2answers
73 views

Compute integral: $\int_0^{+\infty}\int_{-\infty}^{-x}\frac{1}{2\pi}e^{-\frac{1}{2} (x^2+y^2)}dx dy $

I would like to resolve this exercise: Let $W$ be a Brownian motion with $T_1=1 \text{ year}$ and $T_2=2 \text{ years}$. I want to compute the probability that $W_{T_1}$ be positive and $W_{T_2}$ ...
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1answer
34 views

Probability Brownian motion is positive at two points

Let $0<s<t$ and $(B_r)_r$ is Brownian motion. Does anybody know what $P(B_s>0,B_t>0)$ is? I think I remember it was some $arctan$-law but I don't know the exact form. So I do not need a ...
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29 views

How can we evaluate the material derivative of the velocity of an particle by means of an Itō formula?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a $\mathbb R^d$-valued Brownian motion with respect to ...
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1answer
27 views

Let $X(t) = e^{r(T-t)}/S(t)$. Find the SDE of $X(t)$ provided that $S(t)$ satisfies the BSM model.

This is the last part to a 3 part question! I am nearly done going through the questions I had difficulty with while studying, again, anyone's help would be greatly appreciated! Let $X(t) = ...
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1answer
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If $S(t)$ is the stock price that satisfies BSM model in SDE form how can I derive an SDE for $S^n (t)$ for some positive integer n

If $S(t)$ is the stock price that satisfies BSM model in SDE form where $dS(t) = \mu S(t) dt + \sigma S(t) d W(t)$ where $\mu >0$ and $\sigma>0$ are two constants. how can I derive an SDE for ...
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Use Ito's Lemma to compute $d(\log S(t)$ and use this to find the closed form solution of S(t)

I am having issues with this practise problem. If someone could help me solve it that would be greatly appreciated! Let $S(t)$ be the stock price that satisfies the BSM model in SDE form $dS(t) = ...
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2answers
33 views

derive integration by parts for a stochastic integral

The question is to show the following identity: $\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$ This can be done quite easily with ito's however the question explicitly says to show the identity ...
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38 views

How can we prove that the derivative of a generalized Hilbert space valued Brownian motion is a Gaussian white noise?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of ...
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2answers
44 views

How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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16 views

Covariance functional of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle ...
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33 views

Distribution of “range” of a process

Let $X_t$ be a stochastic process, for example a brownian motion (i.e. $X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$). The difference between now's value $X_t$ and a past value $X_{t-100}$ is $$M_t ...
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0answers
45 views

Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
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1answer
30 views

Expectation of a generalized real-valued Brownian motion

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $\mathbb R$ and $$\langle ...
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1answer
27 views

Is $\phi B(\omega,\;\cdot\;)$ Lebesgue integrable over $[0,\infty)$ for a real-valued Brownian motion $B$ and $\phi\in C_c^\infty(\mathbb R)$?

Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $\lambda$ be the Lebesgue measure on $\mathbb R$. Is ...
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45 views

Distance between Brownian Motion and scaled Gaussian random walk

I'm currently reading this paper: http://user.math.uzh.ch/barbour/pub/Barbour/SteinDiffusion.pdf and in equation (2.26) the author uses the following fact: If $Z(t)$ is a standard Brownian Motion and ...
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1answer
24 views

how to show that definition for stochastic process in continuous time applies to stock prices

I know that the formal definition of a stochastic process is: {$X(t,\omega)\,\,t\ge0$} is a stochastic process if: For any fixed $t\ge0$, $X(t,\omega)$ is a random variable For any fixed $\omega$ ...
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38 views

Why is $E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t$?

Why is $E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t$ ? Does this always hold In an exercise I have to show that $E(X_t|B_t)\neq X_t$, where $X_t=\int_0^t B_s ds$, I think the definition of $X_t$ ...
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1answer
35 views

Compute $ \mathbb{E} [W(t_1)W(t_1 + t_2)W(t_1 + t_2 + t_3)] $ when $W$ is a Brownian motion

Let $(W(t))_{t \geq 0}$ be standard Brownian motion, and let $t_1, t_2, t_3 \in \mathbb{R}_{> 0}$ with $t_1 < t_2 < t_3$ be arbitrary. Compute: $$ \mathbb{E} [W(t_1) * W(t_1 + t_2) * ...
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1answer
49 views

Computing expectation of brownian motion

I need to compute the following: $E\left[ B_t \int_0^tB_s^2 \, ds \right]$ for $t≥0$ Where $B_t$ is a standard Brownian motion. I'm thinking this is really obvious, But I cannot get my head round ...
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24 views

Wiener's construction of the Wiener Measure

I am writing an essay about Norbert Wiener and I already have sufficient info about him in general and his history, but now I would like to know how he constructed the Wiener measure. I found some ...
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42 views

Brownian motion, harmonic functions and the Dirichlet problem

I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ...
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1answer
28 views

An issue of dependent and independent random variables involving geometric Brownian motion.

Let $X(t)=X(0)e^{\mu t + \sigma Z(t)}$ be a geometric Brownian motion (GBM) where $Z(t)$ is the standard Brownian motion with drift $0$ and the variance rate per unit of time is $1$. Now, let $s<t$ ...
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1answer
51 views

Iterated logarithm law for difference (supremum(W) - infimum(W) ) is it 2srt(2/pi) sqrt(t loglog(t))?

Law of iterated logarithm says that $$\sup(W(t)) \sim \sqrt{2 t \log(\log(t))}.$$ Consider $\sup(W(t)) - \inf(W(t))$ my guess based on numerics that it should be $$2\sqrt{\dfrac 2\pi} \sqrt{t ...
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48 views

Simulation of brownian motion and fractional brownian motion

It's easy to simulate a path of a brownian motion with the method explained in Wiener process as a limit of random walk: ...
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1answer
71 views

Exercise 8.12 Introduction to stochastic processes Gregory Lawler [closed]

Let $X_t$ be a standard Brownian motion starting at 0 and let $T=min \{t:|X_t|=1\}$ and $\hat{T}=min \{t:X_t=1\}$ (a) Show that there exist positive constants $c$, $\beta$ such that for all ...
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30 views

Construction of Wiener Process using integral of covariance multiplied by a function

I read in the notes of Stochastic Processes that there is a construction of Wiener Process (knowing that $Cov(W_s, W_t)=min(s,t)$ ) which going like this: consider operator $Q$ on $C([0,1])$ $$Qf(t)= ...