# Tagged Questions

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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### Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
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### Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
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### Independence between the first exit time from an interval and the value of Brownian motion at this first exit time

Suppose you have an arithmetic Brownian motion (or Brownian motion with drift ) called X, started at a level x such that a < x < b, where a and b are two real points . Define tau as the first ...
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### Verifying a Brownian motion through the Laplace transform

Let $X(t)$ be a continuous stochastic process and $\mathcal G(t)$ be the $\sigma$-algebra generated by $\{X(\tau) : \tau\leq t \}$. Suppose that for any $0\leq s\leq t$ and $\lambda\in\mathbb C$ ...
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### Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
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### Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
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### Brownian motion independent RVs

Let $(W_t)_{t\in\lbrack 0,T\rbrack}$ be a standard Brownian motion. Does there hold that $W_s(W_t-W_s)$ and $W_k(W_l-W_k)$ for $0\leq s<t\leq k<l\leq T$ are independent RVs?
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### What is the difference between these two formulas that price a stock? [closed]

What is the difference between these two formulas? They are both related to the price of a stock in the black-scholes model. The fact that the second one uses $t$ as a subscript which means it's not a ...
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### Brownian motion - absolute value

I'm having some trouble integrating the equation in 8.2.5 (I'm trying to do 8.2.6). I need to do some form of u-substitution but I'm unsure of u=?. Also, once I've done the integration, to show that ...
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### Reflected Brownian Motion probability

So I know that R(t) = |5 + B(t)| and that B(25) ~ N(0,25). I was told that P{R(t)>=10} = P{|5+B(25)|>=10} = P{B(25)>=5)+P{B(25)<=-15} but I'm not entirely sure how to get that. And I've been ...
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### Brownian Moment Generating Function and Hitting Times

Here is my question. I've done the first part, but I'm stuck on the second. If I can work out (/be advised) how to do the second, then I hope to be able to do the third similarly. Please note: While ...
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### Crossing of Brownian Motion Sample Paths

I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ...
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From the Wiki article a Wiener Process has the properties that $$E[W_t] = 0$$ $$Var[W_t] = t$$ According to A Standard Wiener Process the Wiener Process is given by: W(t) - W(s) \tilde{} \sqrt{t-... 1answer 42 views ### Limit Brownian Bridge Integral As a solution of the Brownian Bridge SDE, we arrive at the solution \begin{align} X_t = (1-t) \int_0^t \frac{1}{1-s}\ dB_S \end{align} defined for 0 \leq t <1. In order to show that for any g \... 1answer 34 views ### Wiener process and stochastic int Let h:[0,1] \rightarrow \left\{-1,1 \right\}. How to show that X_t=(\int_0^th(s)dW_s)_{t \in [0,1]} is a Wiener process? I know from the lecture that for every h process \int h \ dW_s is ... 0answers 37 views ### n times integrated Brownian motion martingale process According to this post, we found that a n times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ... 0answers 50 views ### Find P(B_3>0,B_6>0) where (B_t) is a Brownian motion Suppose that B_{t} is a standard Brownian Motion. What is the probability that both B_{3} and B_{6} take positive values? This is what I've tried but then I get stuck and I'm not sure how to ... 2answers 32 views ### Integration by parts - Brownian motion and non-random function Let B be a standard one-dimensional Brownian motion. I want to show for a continuously differentiable non-random function \phi that, \begin{align} \int_0^t \phi(s) dB_s = \phi(t) B_t - \int_0^t ... 1answer 22 views ### Derivation of a property of standard Wiener processes I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: W(t) = 0 W(t) - W(s) \sim N(0, t-... 2answers 173 views ### Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion? (First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ... 0answers 60 views ### Brownian Motion Third Power Martingale using Ito Integral Let (B_t)_{t \geq 0} be a standard Brownian motion and M_t = B_t^2 - t. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ... 0answers 24 views ### covariance and expectional in proccess Show that the process X=(W_{\sqrt{t}}I_{(1,2)}(t))_{t \ge 0} \in \mathcal{L}_3^2. (W- Wiener) Additionally calculate, for t,s \in [1,2], EX_t and Cov(X_t,X_s) I have no idea how to start ... 2answers 82 views ### Integral of Wiener Squared process I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from 0 to t results in another Gaussian process with slice distribution that ... 1answer 38 views ### Marginally Gaussian not Bivariate Gaussian - Ito Integral Let (W_t)_{0\leq t\leq 1} be a Wiener process defined up to time 1 on some probability space. Consider the random vector\left(W_{1},\int_0^1 \operatorname{sgn}(W_s) \, dW_s\right)=:(W_1,X_1) ...
Given the following process: $\Delta \ln(St+1)= \mu - (\sigma2/2) + \sigma(\varepsilon(t+1))$ (where both $\mu$ and $\sigma$ squared are of $S$) How does one calculate the mean of $S(t+1)/S(t)$? (...
Intuitively it seems likely that the expected whereabouts of Brownian motion on the unit circle would be the origin $\left(0,0\right)$, at least in the limit as $t\to\infty$. Is this right? Are there ...