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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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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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} ...
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0answers
44 views

Mistake in using Monotone convergence theorem

I am getting a contradictory result and can't find my mistake. I hope you can help. We have the following result by Spitzer (see (1) or Port) $\lim_{t\to ...
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1answer
19 views

Covariance of Wiener Processes on the same Brownian Motion

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & ...
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0answers
28 views

Strong markov property in two dimensional Brownian motion

I don't understand the following claim from my book: Let $(B_t)$ be a standard Brownian motion. Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, ...
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1answer
49 views

Continuous in probability of hitting times

Let $(B_t)$ be a standard Brownian motion. How can we show that the process $$ \tau_t := \inf \{ s \geq 0 : B_s >t \}$$ satisfies continuity in probability? $$\bigg( \text{i.e. } \quad \lim_{h ...
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0answers
37 views

Independence of increments of some processes

I am stuck on this question: Let $(B_t)$ be a standard Brownian motion. Define $$ (\tau_1)_t := \inf \{s \geq 0 : B_s = t \} ; \quad (\tau_2)_t := \inf \{s \geq 0 : B_s > t \}. $$ Any ideas how ...
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1answer
51 views

Equality relating $L^2$ convergence and martingales

I am baffled with this question: Let $(B_t)$ be a standard Brownian motion. For any $n \in \mathbb{N}$, let $(f_n)$ be a sequence of functions defined by $$ f_n(x) = \left\{ \begin{array}{lr} ...
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0answers
40 views

Is this stochastic differential equation wrong?

The following is an old exam question I think might be misstated. Consider the SDE $$dX(u)=(a(u)+b(u)X(u))\,du+(\gamma(u)+\sigma(u)X(u))\,dW(u)$$ where $W(u)$ is a brownian motion relative to ...
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1answer
40 views

Expectation of a function of Brownian motion at a stopping time

I do not understand the following claim from my book: Let $(B_t)$ be a Brownian motion on $\mathbb{R}^d$ starting at $x$. Let $\tau = \inf \{ t>0 : B_t \in \partial B( x, r) \}$. Also, let $u ...
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2answers
76 views

For every $\epsilon>0$, the probability of $W_t>(1+\epsilon)\sqrt{t\log(t)}$ tends to $0$ as $t\to\infty$

Can anybody give a hint to show for all $\epsilon>0$ $$\lim_{t \to \infty} P \left( \frac{W_t}{\sqrt{t\log(t)}}>1+\epsilon \right) = 0$$ with $W_t$ Brownian Motion? (Or W(t), a Brownian motion ...
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1answer
26 views

Hitting time of Brownian Motion on a line

Given a 3-dimensional Brownian motion $B_t$, we know that it is transient. But how can we show that if it starts outside a straight line, it will remain outside forever with probability $1$ ? Any ...
2
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1answer
65 views

Path properties of Brownian Motion: relation between its maximum and hitting time

Let $B(t)$ be a Brownian motion. $$T_a=\inf\{t>0,B(t)=a\}$$ $$M(t)=\max_{0\le s\le t} B(s)$$ There is a statement in Durrett's textbook (3rd last line in page 318, 4th edition): ...
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1answer
32 views

independence of stopping time and a sigma algebra

Let $(B_t)$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. For a stopping time $\tau$, we know that $\{B_{\tau + t} - B_{\tau}\}_{t \geq ...
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0answers
50 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
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1answer
28 views

Sign of differentiable function near critical point

Background : We know that Brownian path oscillates infinitely often changing signs in any neighbourhood of $0$. I was trying to understand if this property holds because that Brownian paths are not ...
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1answer
54 views

Hitting time process of Brownian motion [closed]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
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1answer
62 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
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1answer
49 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
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47 views

Conditional expectation involving Brownian Bridge

I have no ideas on this problem: Let $(B_t, 0 \leq t \leq 1)$ be a standard Brownian motion in $1$ dimension. Let $Z^y_t = yt+ (B_t -tB_1)$. We call $\{Z^y_t\}_{0 \leq t \leq 1}$ a Brownian Bridge ...
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1answer
66 views

Position of Brownian motion at exit time from the upper half plane

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion. For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the ...
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1answer
81 views

What is the explicit obstruction to almost sure convergence in stochastic integrals?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
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1answer
57 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
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1answer
63 views

Verifying that a certain process is not a Brownian motion

Let $B$ be a standard Brownian motion in $1$ dimension. Define \begin{equation} \tau = \inf \bigg\{ t \geq 0 : B_t = \max_{0 \leq s \leq 1} B_s \bigg\}. \end{equation} We want to show that $(B_{t+ ...
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2answers
45 views

probability of a brownian motion being equal to the running maximum

Let $B$ be a standard Brownian motion on $\mathbb{R}$. I would like to show that $$ \mathbb{P} \bigg\{ B_1 = \max_{t \in [0,1]} B_t \bigg\} =0 .$$ I argue that since $\max_{t \in [0,1]} B_t $ has the ...
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2answers
37 views

Prove that lim sup of a function belongs to a certain sigma algebra

I am so baffled with this problem: Let $B$ be a standard Brownian motion, $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. I would like to show that for any $k>0$, ...
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1answer
87 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
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154 views

A simple characterization of the Brownian Motion

A well-known characterization of the Brownian Motion says that it is the only continuous process $X_t$ (defined on $[0,\infty)$) such that $P(X_0=0)=1$, $E[X_t^2]=t$, $E[X_t]=0$ for any $t\ge 0$ the ...
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0answers
52 views

continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
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1answer
105 views

Variance of integrated squared wiener process

So I'm trying to figure out the mean and variance of $X = \int_{0}^{1} W^2(t) dt $ where $W$ is the Wiener process. The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having ...
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1answer
39 views

Problem 4.2 (p. 60) in Karatzas and Shreve

I'm looking at problem 4.2 in "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve. The goal is to show that on $C[0,\infty)$, the Borel sigma algebra generated by "topology of local ...
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1answer
37 views

Brownian brigde, brownian motion and independence.

Let $\{W(t)\}_{0 \le t \le 1}$ a Brownian motion. Then $\{B(t)\}_{0 \le t \le 1}$ with $B(t)=W(t)-tW(1)$ is a Brownian brigde. My goal is to prove that $B(t)$ and $W(1)$ are independent. Since their ...
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0answers
22 views

Solution of $dX_{t}=(sin(X_{t})+2)dB_{t}$

I am curious if $dX_{t}=(sin(X_{t})+2)dB_{t}$ has a solution i.e $X_{t}$=(stuff in terms of $B_{t}$). What about for $dX_{t}=\sigma(X_{t})dB_{t}$, where $0<\gamma^{-1}\leq \sigma\leq ...
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1answer
26 views

Independence of random variables involving Brownian motion

I am reading a book on stochastic analysis and I don't understand the following (i.e. don't know how to prove it rigorously): Let $B$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the ...
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0answers
12 views

density of $X_{t}$ satisfying $dX_{t}=dB_{t}-V'(X_{t})dt$

find density of $X_{t}$ satisfying $dX_{t}=dB_{t}-V'(X_{t})dt$ where $V(x)=\frac{x^{2}}{2}+W(x)$ and $x_{0}$ has density $\frac{e^{-V(x)}}{\int e^{-V(y)}dy}$ and W(x) is smoothly compactly ...
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1answer
56 views

How to show the expected value of a hitting time Brownian motion?

We have $W_t$ as a Brownian motion and $$T_{−a,b} = \inf \{t ≥ 0 : W_t \not\in [−a, b]\}\qquad a, b > 0$$ How do you show $\mathbb{E} (W_{T_{-a,b}}) = 0$?
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1answer
35 views

Density of $\int_{0}^{t}W'(B_{s})ds$ where $W'$ is smooth and compactly supported.

Only hints please Density of $\int_{0}^{t}W'(B_{s})ds$, where $B_{s}$ is 1-d Brownian motion. The density of $Y_{s}:=W'(B_{s})$ is $g_{Y}(y)=p_{B_{s}}((W')^{-1}(y))|\frac{d(W')^{-1}(y)}{dy}|$. How ...
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41 views

Question about calculate expected value

Assume $X(t)$ is a Brownian motion. Find $E[X(u)X(u+v)X(u+v+w)]$, where $0<u<u+v<u+v+w$ I have an idea to solve this problem, as follows: ...
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1answer
59 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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1answer
151 views

Solving a PDE with Feynman-Kac Formula

I'm trying to solve this PDE using Feynman-Kac formula Now i follow the regular steps Here is where I don't know how to proceed. How do I calculate this expectation?
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2answers
83 views

Sample path of Brownian Motion within epsilon distance of continuous function

Given a continuous function $f:[0,1]\rightarrow\mathbb{R}$, $f(0)=0$, how can one show that $P(\underset{0\leq t\leq1}{\sup}\left|B_{t}-f(t)\right|<\varepsilon)>0$, where $P$ is the probability ...
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0answers
34 views

Understanding simulation of Brownian Motion

I am trying to understand the simulation of Brownian Motion given at http://www.math.uah.edu/stat/applets/BrownianMotion.html. There are four boxes in this simulation. For the purpose of this question ...
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1answer
59 views

3-dim Brownian motion, harmonic function and its expectation

Given $f(x)=\frac{1}{|x+z|}$, a function from $\mathbb{R}^3\backslash \{z\}$ to $\mathbb{R}$, $z \in \mathbb{R}^3\backslash \{0\}$ and $B$ a 3-dim Brownian motion. I had succes showing that this ...
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1answer
106 views

perfectly correlated processes

I am really stuck in this question: Let $\{S_t\}$ and $\{S'_t\}$ be two stochastic processes, satisfying \begin{equation} dS_t = S_t ( \sigma_t \,dB_t + r_t \,dt), \quad dS'_t = S'_t (\sigma'_t ...
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1answer
198 views

Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a ...
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1answer
145 views

Covariance of two geometric Brownian motions

Assume we have two geometric Brownian motions $$ dX_t = \mu X_t dt + \sigma X_t dW^1_t, \qquad \qquad dY_t = \mu Y_t dt + \sigma Y_t dW^2_t $$ where the Wiener processes are correlated with $E[dW^1_t ...
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0answers
162 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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1answer
32 views

Is $(\int_0^t W_s ds, W_t)$ Markov?

Approximating $I_t = \int_0^t W_s ds$ by Riemann sums I have convinced myself that it is not Markov, but I have been met by the claim that $(I,W)$ is and I cannot figure out why. Do you guys have any ...
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1answer
39 views

Hitting time of Integrated Brownian Motion with drift

In Mckean's article A winding problem for a resonator driven by a white noise, there's a passage that I can't seem to understand. What arguments do I use to prove this equality in law: $$ ...
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0answers
62 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...