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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49 views

Problem with a Brownian motion, Ito's formula and an indicator function

so I have done the first part of this question (it is at the bottom), but I have no clue how to do the second part. I think I understand the theory, but I do not know how to apply it. Any help would ...
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0answers
15 views

Differentiability of a process containing a brownian motion

I have trouble understanding a statement of the paper of Constantinides'1990 "Habit Formation: A resolution to the equity premium puzzle" (for example here ...
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1answer
22 views

Calculating conditional expectation

I have to calculate $E\left(\int_1^4 W_t^3dt |\mathcal F_2\right)$ My solution: $E(\int_1^4 W_t^3dt |F_2)=E(\int_1^2 W_t^3dt |F_2)+E(\int_2^4 W_t^3dt |F_2)=\int_1^2 W_t^3dt+\int_2^4 E(W_t^3 |F_2)dt$ ...
3
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0answers
148 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
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2answers
24 views

Brownian Motion independent increment computation

One can rather easily show that $E\left[\sum\limits_{i = 0}^{i = n - 1}W_{t_i}(W_{t_{i + 1}} - W_{t_i})\right] = -T + W_T^2$. What I'm confused about is why we can't simply say that for each $i$, ...
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0answers
30 views

diffusion- stuck

In a round room of radius R, a large number of coins N of diameter d are randomly dispersed upon the floor. A ladybird starts from the centre of the room, crawling at speed v. Suppose that every time ...
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0answers
16 views

A measure for the “typicalness” of a Brownian path

Suppose I have a continuous function $f:[0,1]\to\mathbb{R}$, and I wish to measure somehow how similar it is, in some sense, to a Brownian motion $\{B(t)\mid t\in[0,1]\}$ (with $B(0)=0$). I was ...
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0answers
24 views

Application of Girsanov theorem

Let $f(t)$, $t \geq 0$ be a smooth function with $f(0) = 0$ and let $B(t)$, $t \geq 0$ be a brownian motion. Let $P$ and $Q$ be two measures on $C[0,1]$ corresponding to respectively, $B(t)$, $t \geq ...
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1answer
39 views

Computing quadratic variation and criteria for Brownian motion

Let $f(t)$ be a nonrandom and continuously differentiable function and $B(s)$ be the brownian motion. a) Computer the quadratic variation of : $X(t) = f(t)B(t) - \int_0^t f'(s)B(s)ds$ b ) For ...
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0answers
14 views

Cardinality of the set of zeros of the solution of an Stochastic Differential Equation

Let $\sigma(x)$ be smooth and bounded above and below from zero. i.e $0 < \alpha^{-1} \leq \sigma \leq \alpha$. Let $X(t)$ be a solution of $dX(t) = \sigma(X(t))\,dB(t)$ Let $A = \{t \in [0,1] : ...
3
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1answer
44 views

What does $s$ and $t$ stand for in this definition of fractional brownian motion?

$$B_H(t_2,\omega)-B_H(t_1,\omega) = \frac{1}{\Gamma(H+1/2)}\Bigg\{\int_{-\infty}^{t_2}(t-s)^{H-1/2}dB(s,\omega)-\int_{-\infty}^{t_1}(t-s)^{H-1/2}dB(s,\omega)\Bigg\}$$ It's taken from Mandelbrot & ...
1
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1answer
46 views

Probability that Brownian Motion hits $t+1$ before $t-1$

Compute the probability that a brownian motion starting at $0$ hits the line $t+1$ before the line $t-1$. Here is what I did: I figured it has to do with optional stopping theorem. The ...
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0answers
55 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
4
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1answer
62 views

Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
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1answer
28 views

Sufficient and Necessary condition for the sum of brownian motions to be a brownian motion

The question has two parts : Part a): Let $(B_1(t), B_2(t), B_3(t))$ be standard a brownian motion in $R^3$. Write down necassary and sufficient condition for $\sum\limits_{i=1}^3 a_iB_i(t)$ to be a ...
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1answer
34 views

Bound probability Brownian motion stays in $[-1,1]$.

Let $T:=\inf\{t>0: |B_t|=1\}$ be a hitting time for standard Brownian motion. I want to show that $$\lim_{t\to\infty} e^{\frac{\pi^2}{8}t}\mathbb{P}[T\geq t]=\frac{4}{\pi}$$ I had a look at A ...
2
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1answer
56 views

Brownian motion and posterior distribution

I am a bit stuck on this question: Suppose that $X_t = W_t + \alpha t$, where $W$ is a standard Brownian motion, and let $\mathcal{F}_t = \sigma ( X_u: 0 \leq u \leq t)$. The drift is constant in ...
2
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1answer
43 views

A good book on Brownian motion

Can you suggest me a good book on Brownian motion, where it is introduced as a limit of measures on polish spaces like $C[0,1]$ and subsequently stochastic calculus is discussed?
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1answer
91 views

Considering stochastic processes as random variables (Brownian motion)

Define process $X$ by $X_{0}=0$ and $X_{t}= tB_{1/t}$ for $t>0$, where $B_t$ is a standard Brownian motion. I want to show that $X$ is continuous in zero. The suggested hint is: "think of $X$ and ...
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2answers
42 views

How can I solve this expected value?

Good evening, how can I solve this expected value? $$ E \Bigl[ B_1 \int_0^{x} B_u du\ \Bigr] $$ where $B_t$ is a standard Brownian Motion and x > 0.
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0answers
14 views

Sufficient condition of boundedness of diffusion process

I came across the following statement in Sebastian Bossu's book "Advanced Equity Derivatives", page 27. He says that the time-homogeneous diffusion process $dX_t=a(X_t)dt+b(X_t)dW_t$ (coefficients ...
2
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2answers
70 views

Product of stochastic integral and brownian motion

I am trying to compute the following expectation: $$ M_T = \mathbb E\left[W_T\int_0^T\,t\,d W_t \right] $$ where $0<t<T$ and $W = (W_t)_{t\geq 0}$ is a standard Brownian Motion started at $0$. ...
4
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1answer
60 views

Show that $E[X_t^2]<\infty$

Show that $E[X_t^2]<\infty$, where $$ X_t=e^{3W_t-\frac{3t}{2}}-3e^{W_t-\frac{t}{2}}\underbrace{\int_0^te^{2W_s-s}ds}_{A_t},\quad. t\geq0, $$ where $t$ is a fixed number and $W_t$ is Brownian ...
6
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1answer
70 views

Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
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1answer
45 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
1
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1answer
55 views

Why is Brownian Motion B_t distributed as N(0,t)?

Almost all textbooks define a Brownian Motion ($B_t$)using three / four points: $B_0 = 0$; it has stationary independent increments; for every $t>0$, $B_t$ has a normal $N(0,t)$ distribution; it ...
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1answer
33 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...
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0answers
17 views

Showing Brownian motion is measuable

How can I prove Brownian motion is measurable with respect to the corresponding product sigma algebra? I am struggling to extend the measurability from holding for rational times to all times using ...
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0answers
24 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
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0answers
32 views

How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
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1answer
50 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
3
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1answer
75 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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1answer
56 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
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2answers
70 views

Brownian motion: first-hitting-time with double barrier [closed]

Let $(B_t)_t$ be a standard ($B_0=0$) Brownian motion , and $$ T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\} $$ where $a<0<b$. What is the expected first-passage time $\mathbf{E}[T_{a,b}]$?
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2answers
29 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
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1answer
46 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
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0answers
24 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
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0answers
39 views

Deriving mean and variance of a function of Gaussian process

Suppose $\mathbb{G}$ is a tight zero mean Gaussian process and $F$ is an absolutely continuous CDF $$Y=\int_a^b\frac{d\mathbb{G}}{1-F}-\int_a^b\frac{\mathbb{G} \, dF}{(1-F)^2}$$ I know that $Y$ is a ...
2
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0answers
47 views

Distribution of $(\sup_{0\leq s\leq t} W_s -W_t)$

I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion. I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density ...
2
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0answers
20 views

Intersection of two independent 1-d Brownian motions.

I am interested in the first intersection of two independent 1-d Brownian motions. More precisely, what is the joint distribution of the intersection point and intersection time? Any help is ...
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1answer
59 views

Martingale property of Brownian motion with resprect to a different filtration

Let $W$ be a Brownian motion on $(\Omega,\mathcal F,\mathbb P)$ and let $N$ be a Poisson process on the same probability space. Denote by $\mathbb F$ the filtration that is generated by $(W,N)$. Now ...
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1answer
29 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
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1answer
40 views

Is square of Wiener process an orthogonal process?

I'm trying to prove: Let $t_1 < t_2 \leq t_3 < t_4$ and $(X)_t$ is the square of Wiener process. Then $E(X_{t_2} - X_{t_1})(X_{t_4}- X_{t_3}) \neq 0.$ Progress Maybe the fact $E(X_{t_2} - ...
1
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2answers
59 views

Law of large numbers for Brownian Motion (Direct proof using L2-convergence)

In “Brownian Motion” by Schilling and Partzsc, they give a HINT to prove the Law of Large Numbers for Brownian Motion (not in their solutions, fyi) by (1) Noting that ...
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1answer
34 views

How to identify the future distribution of a stochastic variable from its SDE

I would like to know some common practice to identify the future distribution of a random variable modelled by an arbitrary SDE. Would you study it empirically (like generating Monte-Carlo ...
2
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1answer
35 views

$L^{2}$ -limit of expression involving Brownian Motion

Let $(B_{t})_{t\geq0}$ be a Brownian Motion. I would like to prove that $\max_{n\leq s\leq n+1}\left|\frac{B_{s}-B_{n+1}}{n}\right|=\frac{1}{n}\max_{n\leq s\leq n+1}\left|B_{s}-B_{n+1}\right|$ ...
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0answers
16 views

Why is $l(t,x,\omega)=\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon}\int_{0}^t1_{[x-\varepsilon,x+\varepsilon]}(X_s(\omega))ds$

Currently I am reading the book "Brownian motion and stochastic flow systems" (Harrison) and in chapter 1 paragraph 3 he states the following deep theorem about Brownian motion: Theorem Let ...
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2answers
57 views

Brownian Motion and Continuity

Consider a Brownian Motion $(B_{t})_{t\geq0}$. In my lecure notes it says, without proof, that $\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} ...
1
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1answer
26 views

A question on integration wr.t to a local martingale

In a lemma in my graduate level course on financial mathematics uses the fact that integral of a progressive portfolio process(which is almost surely lower bounded i.e it is admissible) $\theta_t$ ...
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0answers
28 views

Existence of local time of Brownian motion

Suppose we define the local time $L_0(t, \omega)$ of the standard Brownian motion $B(s, \omega): [0,t] \times \Omega \rightarrow \mathbb{R}$ by $$ L_0(t, \omega) = \lim_{\epsilon \rightarrow 0} ...