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 $t-s$.

learn more… | top users | synonyms

3
votes
1answer
63 views

Standard Brownian motions?

Suppose $A_t$ is a standard Brownian motion and $x \in \mathbb{R} \setminus \{0\}$. Let$$B_t = xA_{t/x^2}, \text{ }C_t = tA_{1/t}.$$What is the easiest way to see that $B_t$ and $C_t$ are standard ...
0
votes
0answers
22 views

Time integral of Brownian motion's running maximum

Let $\mu \geq 0$ and consider $B_{\mu}(t) := B(t) + \mu t$ a one-dimensional BM with drift $\mu,$ and let $M_t := \max_{0 \leq s \leq t} B_{\mu}(t)$ be its running maximum. My question involves two ...
0
votes
0answers
46 views

Law of iterated logarithm for $d$-dimensional Brownian Motion

Let $\left\{ B(t)\mid t\geq0\right\} $ be a $d\text{-dimensional}$ Brownian Motion, show that almost surely $$\lim_{t\to\infty}\frac{\left\Vert B\left(t\right)\right\Vert }{\sqrt{2t\log\log ...
1
vote
1answer
71 views

What probability topics can be read without Measure Theory

I know Intermediate Probability Theory and Statistics (distributions, convergence concepts, characteristic functions, etc.) and am pretty good at these. I would love to know more about Probability ...
0
votes
0answers
38 views

$\sup B_{1 + t} - B_1$ is independent of $B_1$ for $B$ a brownian motion

Is $\sup_{t \ge 0} B_{1 + t} - B_1$ is independent of $B_1$ for $B$ being a brownian motion? For all $t$ $B_{1+t} - B_1$ is independent of $B_1$ but this isn't enough. I also can't use that $\sup_{t ...
0
votes
0answers
23 views

An SDE involving two brownian motions

Consider the following SDE - $dX_t = \mu(X) dt + \sigma_1(X) db_{1t} + \sigma_2(X) db_{2t}$ $0< a < \mu(x) < b$ $ 0< m < \sigma_1(x), \sigma_2(x) < M$ That is $\mu, ...
0
votes
0answers
63 views

Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
0
votes
0answers
11 views

How to compute this integral using Ito isometry? [duplicate]

I am trying to evaluate the following integral: $E\Bigg[\Bigg(\int^{t}_{0} \frac{B_s}{e}1\big(B_s\in(-e,e)\big)\Bigg)^2\Bigg]$ I cannot figure out how to apply Ito isometry when the indicator ...
2
votes
0answers
34 views

Example of a bounded simple process $A_t$ that changes value only once s.t. $\int_0^t A_s dB_s$ doesn't have normal distribution? [closed]

As the title of the question suggests, what is an example of a bounded simple process $A_t$ that changes value only once such that$$\int_0^t A_s\,dB_s$$does not have a normal distribution?
1
vote
1answer
38 views

Expected value of distance between independent Brownian motions

Suppose $\{W^{(1)}_t, t\geq 0\}$ and {$W^{(2)}_t, t\geq 0\}$ are two independent Brownian motions. If I recall correctly, the distance between the two at a given time has the following property: ...
0
votes
1answer
25 views

Brownian motion conditional expectation

I need to solve for the following in my model: $E[X_t^i|X_s < K_1, X_t > K_2]$ where $X$ is Brownian motion and $i$ is a real number. any suggestion? I already know about the simpler case: ...
2
votes
0answers
53 views

Definition of Standard Brownian Filtration

I am trying to learn about stochastic calculus for my research, so self study, and I came across the notion of a Standard Brownian Filtration. I cannot find a good definition of what the Standard ...
0
votes
0answers
24 views

Does a ratio of PDFs have any usable meaning?

I'm calculating the probability that a standard Brownian motion path will cross a boundary. I have $A$ and $B$ representing the PDFs for the Brownian motion going above a boundary function $a$ and ...
1
vote
2answers
33 views

Given a Brownian motion $W$, and $k \in (a,b)$, I'm trying to find the distribution of $W(k)$ in terms of $W(b)$, $W(a)$, and $k$

I'm trying to perform this "interpolation" because I ultimately am trying to write a small library to simulate stochastic processes. I realized I might need to figure out what is the distribution of ...
3
votes
0answers
30 views

What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
1
vote
1answer
97 views

Expectation of exponential of integral of absolute value of Brownian motion

Sorry about all the "of"s in the title... here's my problem: I want to compute the expected value of $$ \exp\bigg\{ C \int_0^t |W_s|ds\bigg\} $$ where $W$ is a Brownian motion and $C$ is a positive ...
1
vote
1answer
41 views

Can someone, please, suggest some books for Stochastic Processes with exercises?

Can someone, please, recommend me some books about Stochastic Processes,Martingales and Brownian Motion with many exercises? (I would be very happy if some of them are for beginners :D) Thank you!
5
votes
1answer
116 views

Can we have a Brownian motion under two different probability space?

Is it possible to construct a stochastic process $B_t$ such that $B_t$ is a Brownian under $(\Omega, F, P)$ and $B_t$ is a Brownian under $(\Omega, F, \hat{P})$? If not, how to argue that $P=\hat{P}$? ...
5
votes
1answer
109 views

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

My search results keep bring up planar Brownian motion on the unit disk. However, I am specifically referring to $e^{jW_{t}} = [\cos(W_t),\sin(W_t)]^{T}$ where $W_t$ is Brownian motion. I am at a ...
2
votes
2answers
61 views

Reference book for Brownian Motion

I want to know about books for reading Brownian motion. I am aware of measure theoretic probability theory.
0
votes
1answer
13 views

Solution to sde with specfic mean

Goal: I'm attempting to work backwards to recover an SDE as follows: Example: $e^{\mu t}$ is the mean of the geometric Brownian Motion, which solves the SDE: \begin{equation} dS_t = \mu S_t dt + ...
2
votes
1answer
35 views

Independence of a hitting time and the underlying stochastic process

While I was playing around with the Girsanov's Theorem I stumbled upon the following absurdity and I couldn't resolve it with the current knowledge of stochastic analysis that I have. $B$ being ...
3
votes
0answers
62 views

What can you tell me about backward Brownian motion?

I'm trying to understand "backward Brownian motion" and how it relates to standard Brownian motion. In this paper, they construct a solution to Burgers Equation (transformed via Cole-Hopf) with ...
1
vote
1answer
72 views

Calculation with Ito processes, what is $ds \, dt$, $dW_t \, ds$ and $dW_s \, dW_t$?

I am working on an exercise and I am not sure how to deal with these 3 cases... For example, is $ds \, dt=0$? I know $(dt)^2=0$, but I am not sure when it is 2 different variables. And what about ...
1
vote
0answers
47 views

Stopped Brownian motion proof

I'm trying to work through a proof in Durrett's textbook of a martingale convergence theorem via an embedding of the martingale in Brownian motion, and am stuck verifying a detail as usual. I'm ...
2
votes
2answers
42 views

Show that $X_tY_t=X_0Y_0+\int_0^tX_sdYs+\int_0^tY_sdX_s+[X,Y](t)$, where $X_t,Y_t$ are Ito processes

So I have done this exercise and the proof holds, but I really don't believe it can be correct because the proof is worth twice as much as other exercises. I am also not 100% sure if $d_s ...
0
votes
2answers
54 views

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ...
0
votes
1answer
18 views

Brownian motion exit time

I don't understand why $ \{a<B(s)<b, \forall s\in[0,1] \}\subset\{a < B(1)< b\}$. I'm almost certain that this must be a typo in my *book. But, I thought I would confirm it on the math ...
3
votes
2answers
36 views

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

so I know the result is trivially correct, but I am being asked to prove it. I tried using a theorem, but it seems rather contradictory. Thanks in advance! Question: Show that ...
2
votes
1answer
39 views

Calculate the distance $d_{\mathcal{H}}(X_n,X):=\mathbb{E}\left(\int_0^{\infty}(X_n(t)-X(t))^2dt\right)$ for all $n\ge 1$

I have done this exercise but I have done something wrong because I don't get the correct result for the next part of this exercise (this is part B). I posted something earlier that is related to ...
1
vote
1answer
96 views

Joint distribution of Brownian motion and its running maximum

$B$ being standard Brownian motion, its running maximum is defined as $M_t = \sup_{0\leq s\leq t} B_s$. I am trying to follow the proof of the following result but I don't understand some of the steps ...
0
votes
1answer
18 views

Show that $X_n\in\mathcal{H}$, where $\mathcal{H}:=\{h(t):h(t)\text{ is an adapted process, }\mathbb{E}[\int_0^{\infty}h^2(t)dt]<\infty\}$

I am not sure if I got this exercise right... I have 2 questions: Have I obtained the final result correctly? If so, I used Wolfram Alpha to obtain the value of the series, but how else can I obtain ...
1
vote
0answers
53 views

Process convergence of sum of i.i.d. random variables

I'm interested in the convergence of a stochastic process. Let $(X_i)_{i \geq 1}$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Furthermore, assume $0 < t < T$ for some ...
2
votes
1answer
45 views

Compute $\mathbb{E}[\tilde{X}_t]$, where $\tilde{X}_t=X_t=(1-t)\int_0^t\frac{1}{1-s}dW_s$ for $0\le t<1$ and $\tilde{X}_t=0$ for $t=1$

I have the following exercise and I don't really understand the answer. I am going to write my professor's answer first, then a question about what I don't understand about my professor's answer and ...
0
votes
1answer
42 views

Application of Ito's isometry in deduction of Wiener Ito Chaos expansion

I am trying to learn about the Wiener Ito Chaos expansion and starting reading Oksendal's notes on Malliavin calculus where it is treated in Chapter 1. For a link to the notes, please see ...
3
votes
1answer
28 views

Evaluate $\mathbb{E}\left(\left[W\left(\frac{k}{n}\right)-W(t)\right]^2\right)$ for all $t\in\left(\frac{k}{n},\frac{k+1}{n}\right]$

I am trying to do a past exam paper to practice, but I don't know if I have answered this question properly... I would really appreciate it if someone could double check it. Thanks a lot! QUESTION: ...
2
votes
1answer
50 views

Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
1
vote
0answers
27 views

Continuity of the Loewner flow (SLE theory).

In the SLE paper "Basic Properties of the SLE" from Rohde and Schramm, it is mentioned on page 898 that the map $$(y,t)\mapsto g_t^{-1}(iy+\xi(t))$$ is clearly continuous on $y>0,t>0$, where ...
2
votes
1answer
42 views

Trying to understand Tanaka's example of SDE.

Consider the following Stochastic Differential Equation: $$dX_t = \sigma(X_t) \, dB_t \tag{1}$$ Where $$\sigma(x)= \begin{cases}1 & x \geq 0\\ -1 & x < 0 ...
-1
votes
1answer
79 views

Show that $\mathbb{P}(\tau_{0}>T)\approx\frac{1}{\sqrt{T}}$ where $\{ B(t) : t\geq 0\}$ is a linear brownian motion started at $B(0)=1$ [closed]

I'd appreciate if someone could provide me with a solution for the following problem: Let $\left\{ B\left(t\right)\thinspace|\thinspace t\geq0\right\}$ be a linear brownian motion started at ...
4
votes
1answer
134 views

Normalized hit times of a simple RW converge in distribution to hit times of standard Brownian Motion

I would appreciate some hints or guidance towards solving the following exercise: Let $\left\{ S\left(j\right)\thinspace:\thinspace j=0,1,\ldots\right\}$ be a simple random walk on the ...
1
vote
1answer
27 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
4
votes
1answer
43 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
0
votes
1answer
60 views

Using Feynman-Kac, compute the following: [closed]

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
1
vote
0answers
23 views

Distribution of hitting time for two border brownian motion

I'm trying to find the distribution of hitting times for two border brownian motion with respect to both the hitting time AND which border is hit. Is this well defined? This is assuming $W_0=0$ with ...
2
votes
0answers
35 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
3
votes
1answer
141 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
2
votes
0answers
33 views

Reflection principle in the proof of the distribution of $M_t - W_t$ (Brownian motion)

Let $W_t$ be the Brownian motion starting at $0$. Consider the following random variables. $M_t = \sup_{0\leq s \leq t} W_s$ and $|W_t|$. We first calculate $$\Bbb{P}(|W_t|>a ) = \Bbb{P}(W_t ...
4
votes
1answer
89 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
2
votes
1answer
64 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...