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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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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18 views

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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18 views

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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21 views

Conditional probability of geometric brownian motion [on hold]

I created paths using GBM to implement The stochastic mesh method. But the method requires the conditional distribution, given some S(t) the probability of S(t+1). I've searched and can't find this ...
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1answer
26 views

What is the difference between these two formulas that price a stock? [on hold]

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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22 views

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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22 views

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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30 views

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 ...
2
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2answers
58 views

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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0answers
22 views

What is the general Taylor Expansion for the following function of a function.

guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: ...
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17 views

System of SDEs and independence

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 ...
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23 views

Gaussian processes and bias

I would like to simulate two Gaussian processes along a time grid. Ideally, I would like to see the average of the samples, for each grid point, to be close to the mean. Using the antithetic method, I ...
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2answers
45 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
2
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0answers
32 views

Survival probability of a biased random walker

A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, ...
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1answer
36 views

How to find the standard deviation from the given information and what is $B(0)$ equal to?

Assume that the risk free rate is $0$ and that the stock price is given by the equation $S(t)=6e^{2t+2B(t)}$ where $B(t)$ is the standard Brownian motion. Determine the price at time $0$ of the ...
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1answer
38 views

Maximum process of Brownian motion

Consider the linear standard Brownian motion $(B_t)_{t \ge 0}$. We define the maximum process $(M_t)_{t \ge 0}$ of $(B_t)_{t \ge 0}$ to be such that $M_t = \max_{0\le s \le t} B_s$. Prove that the ...
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0answers
33 views

How it is shown by the following integral?

Example: Ornstein-Uhlenbeck Process. Let $ dx=-\eta xdt+\sigma dz $ be an Ornstein-Uhlenbeck Process Write the moment-generating function for $x(t)$ as $$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ...
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0answers
22 views

Verifying data came from a Wiener Process

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{} ...
3
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1answer
38 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 ...
3
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1answer
31 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 ...
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33 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} ...
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0answers
46 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 ...
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2answers
29 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 ...
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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, ...
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2answers
134 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 ...
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0answers
49 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 ...
3
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0answers
23 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 ...
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2answers
67 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 ...
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1answer
35 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 $$\left(W_{1},\int_0^1 \operatorname{sgn}(W_s) \, dW_s\right)=:(W_1,X_1)$$ ...
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24 views

Compute the Mean of the Following Process

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 $S(t+1)/S(t)$? ...
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1answer
24 views

Expected location of Brownian motion on the circle

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 ...
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1answer
51 views

Covariance of stochastic integral

I have a big problem with such a task: Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$. I've tried to do this in this way: setting up $t \le r$ $$\text{Cov} \, ...
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0answers
22 views

Probability that Brownian motion falls between two piecewise constant functions

I'll first present the problem, and then describe my motivation: Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \cdots < t_J$ are time points. Let $W_t$ be a standard ...
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1answer
13 views

Show that $(B_t)$ and $(tB_{1_t})$ has the same distribution where ($B_t)_t$ is a brownian motion

Let $(B_t)_{t\geq 0}$ a brownian motion s.t. $B_0=0$. I want to show that $B_t$ and $tB_{1/t}$ has the same law. $$p\{tB_{1/t}\leq x\}\underset{u=1/t}{=}p\{\frac{1}{u}B_u\leq x\}=p\{B_u\leq ux\}$$ ...
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2answers
29 views

Mean and Variance Geometric Brownian Motion with not constant drift and volatility

I have to derive the Geometric Brownian motion (with not constant drift and volatility), and to find the mean and variance of the solution. $\quad \left\{\begin{aligned} & d X_t = \mu(t) X_t d t ...
4
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2answers
54 views

process with integral is martingale

How to show that the process $X_t=tW_t - \int_0^t W_s ds $ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. ...
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1answer
33 views

Basic Question on Definition of Brownian Motion

I am quite new to discrete and continuous stochastic processes. It seems there is something I don`t understand about definition of Brownian motion. Let $\Omega, \mathcal{F}, \mathbb{P}$ be a ...
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0answers
16 views

Understanding of Second Arcsine law for Brownian motion

Ok I'm trying to understand the second arcsine law which states: Let $g_t:=\sup\{s\leq t:W_s=0\}$, then $$\mathbb{P}(g_t\leq s)=\frac{2}{\pi}\arcsin \left(\sqrt{\frac{s}{t}}\right )$$ This won't be ...
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1answer
15 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, ...
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1answer
19 views

Clarification regarding heat kernel for Brownian motion on a manifold

Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. ...
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1answer
13 views

Concerning Covariance and Brownian Motion

Let $\{ X(t), t \ge 0\}$ be standard Brownian motion. How do I find Cov$[X(3) - 2X(2), X(4)]$? The answer is $-1$, but I can't seem to get there no matter what I hit it with. I know that $X(3) \sim ...
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1answer
28 views

Density of Running Maximum of Drifted Brownian Motion Computation

$\textbf{Proposition}$ The $pdf$ of the Maximum of a Brownian Motion with Drift is given by $$ f_{M_t}(m)={\sqrt{\frac{2}{\pi t}}} ...
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31 views

Brownian motion on a manifold

If I have a manifold $M$ and a chart $\left(x,U\right)$, is it possible to simulate Brownian motion on that manifold by solving an SDE in the chart representation $x\left(U\right)$ and then use the ...
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1answer
26 views

SDE for Brownian motion on a circle [closed]

Brownian motion on a circle can be generated by $\left(\cos\left(B_t\right),\sin\left(B_t\right)\right)$ where $B$ is Brownian motion on the real line. My question is what SDE was solved to get this ...
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1answer
25 views

Maximum of standard brownian motion on an interval

I'm trying to find the probability that the maximum of standard Brownian motion on the interval $(t_1, t_2)$ exceeds a value $x$, i.e., $$P(max_{t_1 \le s \le t_2}B(s) \gt x)$$ I initially ...
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0answers
27 views

Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that ...
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1answer
55 views

Simple question on interpreting Geometric Brownian Motion SDE

I'm writing an overview that is more economic than mathematical and I want to explain shortly the stochastic differential equation of Geometric Brownian Motion as simple and clear as possible $$dS_t ...
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1answer
29 views

Probability using Brownian Motion

Assume that $B(t)$ is a Brownian motion and that $S(t)$ is defined as $S(t)=A\cdot e^{B(t)}$ for some positive constant $A$. Calculate the probability of the event ${S(3)>2S(1)}$. How could I go ...
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1answer
29 views

convergence in distribution of exponential of a brownian motion

If $(B_t)_{t≥0}$ is a standard Brownian motion, show that, as $t \to \infty$, $$ \left(\int_0^t e^{B_s} \, ds\right)^{1/\sqrt{t}} \text{ converges in distribution to} \ e^{M_1}, $$ where $M_1 = ...
2
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1answer
25 views

Geometric Brownian motion with exponential of sum of iid's

Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with : $$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$ where $X_i$ are iid. I ...