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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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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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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26 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
42 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
66 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
16 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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21 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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25 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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20 views

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
26 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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25 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
30 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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13 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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27 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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40 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
27 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
25 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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39 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
22 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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2answers
30 views

Expectation equation with wiener process [closed]

Can somebody help me with working out $E((W_{t}^2-t)(W_{s}^2-s)$ where $W_{t}$ denotes a Brownian motion. I tried it with rewriting $W_{t}=W_{t}-W_{s}+W_{s}$ but it doens't work yet. Many thanks!
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37 views

Showing equality Wiener process [closed]

Let $M_t=\max W_s$ over $0 \leq s \leq t$ with $W_s$ a Wiener process. Can somebody help me with showing out: $P(M_t>a, W_t<b)=P(M_t>a,W_t>2a-b)$ with use of the reflection principle.
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36 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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28 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
37 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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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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33 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
20 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
38 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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25 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
54 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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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)= ...
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How to evaluate the expectation of the exponential of reflected brownian motion

How do you compute this expectation $\mathbb{E} \left [ e^{\varepsilon|W_t|} \right] $ where $W_t$ is a Brownian Motion Do I need to expand the absolute value? Can I use the standard Taylor series ...
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57 views

Showing that $P(W_{t}/\sqrt{t \log(t)}>1+\epsilon)\to0$ when $t\to\infty$, where $(W_t)$ is a Wiener process

I have a question about the martingales $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$. With use of this martingale I want to show that $P(\dfrac{W_{t}}{\sqrt{t log(t)}}>1+\epsilon)$ goes to $0$ if $t$ ...
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1answer
19 views

Mean time for the trajectory. Find mean

What is the mean of time when the trajectory of the wiener process, $W_t$, is over the line $y=t$? We need to find $\Bbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
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1answer
54 views

Quadratic variation of semi-martingale

$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. ...
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25 views

Brownian motion and sup of a Brownian motion

I am stuck with the following problem: let $B_t$ be a standard Brownian motion and let $S_{t}:=\sup_{0 \leq s \leq t} B_s$. Prove that for every $\lambda \geq 0$ and $\mu \leq \lambda$, ...
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Proof finite stopping time and Wiener process bounded

Let $T_{-a,b}=\inf\{t\geq 0: W_{t} \notin [-a,b]\}, a,b>0$. I want to show that this is a finite stopping time ($P(T_{-a,b}<\infty)=1$) and that $|W_{\min(T_{-a,b},t)}|$ is bounded by a ...
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modulus of continuity of Ito process

We know from Levy's (uniform) modulus of continuity that for Brownian Motion, almost surely any sample path is locally Holder continuous for any $\rho <\frac{1}{2}$, i.e. $$ |W_t - W_s | \leq ...
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Why is Wiener measure on $C[0,1]$ strictly positive?

The question is as stated. I have thought about this for a while and can't really get anywhere. Here strictly positive means non-zero on non-empty open sets (in this case with a finite interval we are ...
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Why does $1 \leq \sup \limits_{0\leq t \leq 1}( C|B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold?

Why does $1 \leq C\sup \limits_{0\leq t \leq 1}( |B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold ? I am trying to show by contradiction that the Burkholder-Gundy ...
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1answer
29 views

Proving a simple equality involving integrals and a brownian motion

I'm trying to prove the following equality $$ \int_0^T W(t) dt = \int_0^T (T-t) dW(t) $$ where $W(t)$ is a standard brownian motion. I'm been trying to make use of the fact, that $dt = dW(t) dW(t)$ ...
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1answer
29 views

Why does $(W_t)^2$ have mean $t$?

This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...
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31 views

Can we apply the Itō formula to find an expression for ${\rm d}\eta_t(X_t)$ where ${\rm d}X_t=v_t(X_t){\rm d}t+\xi_t(X_t){\rm d}B_t$?

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

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this ...
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1answer
72 views

Showing martingale for a Brownian motion $(W_t)_{t \geq 0}$

I want to show that $\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$ is a martingale with respect to $F_{t}$. We can use that $$E(e^{\alpha ...
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5answers
109 views

Solve the integral $\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx$

To find the Variance of a Wiener Process, $Var[W(t)]$, I have to compute the integral $$ Var[W(t)]=\dots=\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx=\dots=t. $$ I've ...
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41 views

Show that the solution to a stochastic differential equation is satisfied by the following

I am confused on how to get from the first statement to the second. Getting from the second statement to the third would just a simple case of substituting s=0. The solution sheet says to use ...
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1answer
43 views

What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...
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1answer
52 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...