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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26 views

Showing that $(X_n)$ obeys the Markov Property. [on hold]

Consider a process $(X_n)_{n\geq0}$ where we define $X_0 = 0$ and for $n \geq 1$: $$X_n = X_{n-1} + Z_n$$ where $Z_n$ for $n \geq 1$ are independent random variables on $\{ -1, 1 \}$ with ...
2
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0answers
23 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?

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?
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1answer
17 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
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1answer
15 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: ...
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13 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
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22 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 ...
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2answers
29 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
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0answers
19 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 ...
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1answer
21 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 ...
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0answers
18 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
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1answer
95 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 $P=\hat{P}$? ...
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0answers
11 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 ...
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2answers
31 views

Reference book for Brownian Motion

I want to know about books for reading Brownian motion. I am aware of measure theoretic probability theory.
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1answer
11 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
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1answer
24 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
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0answers
34 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
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1answer
51 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 ...
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0answers
42 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
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2answers
36 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 ...
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2answers
39 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
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1answer
12 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
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2answers
31 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
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1answer
33 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 ...
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1answer
27 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 ...
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1answer
16 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 ...
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0answers
10 views

Brownian motion, distribution

I have a question about Brownian motion. Let $(B_{t})_{t \in [0,\infty[}$ be a $\mathbb{R}$-valued Brownian motion. I want to know distributions of \begin{align*} \int_{0}^{T}B_{t}dt,\quad ...
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0answers
38 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
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1answer
41 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 ...
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1answer
16 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
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1answer
27 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
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1answer
43 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 ...
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12 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
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1answer
23 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 ...
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1answer
66 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
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1answer
112 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 ...
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1answer
24 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
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1answer
33 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 ...
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1answer
52 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 ...
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0answers
14 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
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0answers
24 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 ...
2
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1answer
31 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
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0answers
17 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
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1answer
75 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 ...
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1answer
50 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 ...
2
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1answer
40 views

The entrance law of a Brownian motion with absorbing boundary

In the article "Construction of Diffusion processes with Wentzell's Boundary conditions by means of poisson point processes of Browninan excursions" one reads: I tried to compute it for $n=1$ ...
2
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1answer
26 views

Compute the expected value of a brownian motion

Suppose $X(t)$ is a brownian motion. Compute $E[X(1)X(5)X(7)]$. I know that the brownian motion has independent increments, so if we could write $X(1)X(5)X(7)$ as such, then we could use the ...
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0answers
46 views

How to calculate hitting probabilities for Brownian motion.

Given a standard Brownian motion with no drift, the PDF is... $${{1} \over {t^{3/2} \cdot \sqrt{2\cdot \pi}}} \cdot e^{-1/{2t}}$$ (Derived from the CDF $\int_{-\infty}^{f(t)/\sqrt{t}} {1 \over {2 ...
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2answers
37 views

Translation invariance of Brownian motion

Beginner here. I'm working through Durrett's textbook's and am just getting into the section on Brownian motion. He gives a 2-line proof for a simple fact but I'm a little stuck understanding the ...
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0answers
18 views

When is an Ornstein-Uhlenbeck process equivalent to Brownian motion up to a given time lag

Background The expected displacement under the assumption of Brownian Motion for time step $\tau$ is given by $$\gamma(\tau) = D|\tau|$$ where $D$ is the diffusion coefficient. If one assumes a ...
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0answers
27 views

Brownian motion needs to be defined continuous for every $\omega$ to be jointly measurable.

Let $B=(B_t)_{t\in[0,\infty)}$ a Brownian motion (BM) and $(\Omega,\mathcal{F},P)$ be the probability on which $B$ is defined. Some define BM as a.s. continuous, e.g., Brownian motion is almost surely ...