Question 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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Ito's lemma and Backward evolution operator.

$\Phi$ is the backward evolution operator. $W=\theta+\phi+S$ $d\theta=\mu\theta dt+\sigma_1 dZ_1$ $d\phi=r\phi dt+\sigma_2 \phi dZ_2$ $dS=rSdt$ $dZ_1 dZ_2=\rho$ ...
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2answers
491 views

What is the definition of a sample path of Brownian motion?

My question has been asked before at beginner's question about Brownian motion . There was only one answer, which was not accepted. It was probably incorrect, because nothing was said about ...
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0answers
21 views

$W=\Phi+\theta+S$ where $\Phi, \theta, S$ are geometric brownian motions.

$W=\Phi+\theta+S$ where $\Phi, \theta, S$ are geometric brownian motions. $d\theta=\mu\theta dt+\sigma_1 dZ_1$ $d\Phi=r\Phi dt+\sigma_2 \Phi dZ_2$ $dS=rSdt$ For $t \le s \le T$, ...
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1answer
476 views

Correlation coefficient of Wiener process

First, I'm not majoring mathematics. I'm studying economics and during reading a thesis I can't understand the 'wiener process' well. I read some books about it and understand the main idea and ...
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0answers
25 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
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1answer
138 views

Probability of 2D Brownian motion passing through a particular point.

Let $B_t$ be a two-dimensional Brownian motion at time $t \in [0,\infty)$. Fix a point $p \in \mathbb{R}^2$. Is the probability that $B_t = p$ at some $t > 0$ equal to zero? If so, why?
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1answer
99 views

Integral of absolute value of Brownian motion

I know it is a really stupid question and it should be quite easy, but how can I show that $\int_0^{\infty}|B_t|\mathbb{d}t=\infty$ a.s. with $B_t$ being a standard brownian motion? I just don't get ...
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1answer
48 views

question on Brownian Motion stopping time and end state

I came across this equation in my lecture notes, which states: $P(T_a < t , W_t \ge a) = P(W_t \ge a)$ where $T_a = \min\{t \ge 0, W_t \ge a\}$. I'm really confused by this equation: as far as I ...
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2answers
282 views

Expected stopping time of brownian motion

I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please. Say I define the stopping time of a Brownian motion as followed: $$\tau(a) = \min (t ...
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1answer
32 views

Finding a tail-probability with the momentgenerating function

I wonder if it is possible to estimate $\mathbb{P}(X<t)$ with the moment generating function? This question popped up when I tried to proof this estimate $\mathbb{P}(T_a<t)\leq ...
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37 views

Conditional covariance

Just a simple question when we have $v_{st} = \operatorname{cov}(B_s, B_t\mid Z)$, where $B_t$ is a brownian motion. I know that the answer is $\min(s, t) - E[B_s Z]E[B_t Z]/E[Z^2]$ but i don't know ...
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1answer
44 views

Brownian Motion hitting random point

I got a problem that seems to be quite standard and easy, but I have lots of problems with it. I do already know that $T_a:=\inf\{t\geq 0: B_t=a\}$ is a stopping time for any $a\in\mathbb{R}$ fixed, ...
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1answer
93 views

hitting time of Brownian motion

I'm desperately trying to prove that for a standard BM $B_t$ the stopping time $T:=\inf\{t\geq0: B_t\geq\sqrt{1+t}\}$ is a.s. finite, i.e. $\mathbb{P}[T<\infty]=1$. I actually tried to play around ...
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0answers
25 views

Finding asymptotic behaviour

I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion. The ...
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1answer
145 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
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2answers
71 views

can someone explain this notation to me?

$$ dz_t \sim O\left(\sqrt{dt}\,\right) $$ $z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation ...
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1answer
64 views

Brownian motion at exponential time

I want to find the law of $B_T$, where $B_t$ is a brownian motion, $T$ is exp-distributed with parameter 1, with $B_t$ and $T$ being independent. My idea is to say that the density of $B_T$ is given ...
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2answers
112 views

Question on the proof of the simple Markov property of a Brownian motion

Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof. The theorem states in particular that for $s\geq0$ fixed, the process ...
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0answers
44 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
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1answer
175 views

Why is the Brownian motion a multivariate normal distribution?

I have seen in class that for some reasons I forgot, the Brownian Motion has a Multivariate normal distribution, but I am unable to prove it easily. Could someone tell me why it's true? From what I ...
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0answers
25 views

First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first ...
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1answer
279 views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
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1answer
73 views

Brownian bridge distribution: $\sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}}, 0 < a < b <1 $

If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of \begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation} ...
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1answer
71 views

Basic Martingale question

We know the following is a martingale, and is commonly used to represent Ito's Integral. If $W$ is a brownian motion. $ \int_0^T W(u) dW(u) = \frac{1}{2} W^2 (t) - \frac{1}{2}t, \ for \ \ T \ge t \ge ...
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1answer
96 views

Geometric Brownian motion problem

Here's the question: Let $S(t)$, $t \geq 0$ be a Geometric Brownian motion process with drift parameter $\mu = 0.1$ and volatility parameter $\sigma = 0.2$. Find $P(S(3) < S(1) > S(0)).$ Is ...
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1answer
85 views

Question regarding Ito integral

I have a question regarding Ito integral, in particular, when I am trying to prove the normality of Ito integral, I encountered the following differential equation I need to solve: $$dX_{t} = ...
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1answer
181 views

Joint Distribution of two correlated ito integral

I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in ...
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0answers
73 views

Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
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1answer
107 views

Question regarding Ito Process

I am new to Ito Process, so I have a following question. Consider a standard Ito Process, $$X_t=X_0+\int_0^t\mu_sds+\int_0^t\sigma_sdW_s$$ where W is the m-dimentional Brownian motion and X is a ...
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1answer
47 views

Is filtration necessary for continuous random variables?

Define $(\mathscr{F}_t)_{t\geq 0}$ being the natural filtration induced by the Brownian motion $(B_t)_{t\geq 0}$. That is $$\mathscr{F}_t=\sigma(B_s\mid 0\leq s\leq t), \forall t\geq 0,$$ i.e. ...
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3answers
98 views

Brownian motion and adapted

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and "adapted" is one of the concept that I could not understand. First, "adapted" is defines at Ch3.1, page 25 (sixth edition): ...
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151 views

How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
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1answer
101 views

Calculating the probability of following event involving Brownian motion

I have a big time trouble in evaluating the following probability. It is related to brownian motion and measure, so I am asking experts from both fields for help! Denote $B_t$, $t\in [0, T]$ be ...
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0answers
39 views

Bounded Brownian Bridge

I am trying to calculate the following expectation value of $$ E[exp(-\int_{t_0}^{t_1} X_s ds)] $$ in which Xs is a bounded Brownian bridge, which means $X(t_0)=a$, $X(t_1)=b$ and $A<X(t)<B$. ...
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1answer
106 views

Local maxima and minima of Brownian motion

I have trouble understanding why the Brownian motion is nowhere differentiable, and I found somewhere that after showing the total variation of the Brownian motion is $+\infty$, the author claims that ...
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1answer
144 views

Mean and variance of a brownian bridge

I am trying to compute mean and variance of the stochastic process $X_t$, which is a Brownian bridge from x to y, in the time-interval $[t,T]$. $$X_t = y + ...
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1answer
75 views

On the quadratic variation

I understand that the Quadratic Variation of Brownian Motion $B_t$ is $[B_t,B_t]=t$ and I know that the equality is under the meaning of $\mathcal{L}^2$ convergence. Yet I saw in some book saying that ...
6
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0answers
123 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
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0answers
101 views

Brownian motion with drift

I need help with the following problem: Let us denote the water level in a dam at time $t$ by $X(t)$, where $t$ is measured in months. We will assume that, at least until the first time that the dam ...
2
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2answers
169 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
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1answer
80 views

an issue with expectation

in book's Bernt.Øks SDE i read that book and i have some serious issues :( page 21 Example 7.4.2 ) Consider n-dimensional Brownian motion $W=(W_1, \ldots ,W_n)$ starting at $a=(a_1,\ldots,a_n) \in ...
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1answer
58 views

Strong markov property Brownian motion

I have a question regarding an argumentation, which is not clear to me: Let $B_t$ be the standard Brownian motion and $\tau$ a respective stopping time with finite mean. Fix $\epsilon>0.$ We can ...
2
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2answers
95 views

Progressive measurability of a specific set related to Brownian motion

Let $\{W_t: t \in R_+\} $ be a standard Brownian motion process on a given probability space. I am interested in assessing the progressive measurability of the following set: $Z(\omega) := \{t: ...
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0answers
114 views

Conditional expectation with three random variables

We have $N_1, N_2, N_3$ normally distributed random variables with $µ_i =E[N_i]$, $σ_{ij}=Cov(N_i,N_j)$. We also have $\tilde{µ}_i=E[N_i|N_2 = x] $, $\tilde{σ}_{ij}=Cov(N_i,N_j|N_2 = x)$ and $v^2 ...
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1answer
106 views

What is the expected value of the product of 3 Ito Integrals?

How can I calculate the expected value $\mathbb{E}(I_{110}^2 * I_{10})$, where $I_{110}$=$\int_{t_0}^T \int_{t_0}^{s_3} \int_{t_0}^{s_2} 1\, \, dW(s_1) dW(s_2) ds_3$ and $I_{10}=\int_{t_0}^T ...
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0answers
56 views

Expected value of the product of multiple Stratonovich Integrals

I want to calculate: $\mathbb{E}(J_1* J_{10}* J_{10} *J_{110})$ where $J_1=\int 1 dW$, $J_{10}=\int\int 1 dW dt$, $J_{110}= \int \int \int 1 dW dW dt$ are multiple Stratonovich Integrals over the ...
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1answer
36 views

Inequality brownian motion

I have the probability $P(e^{σB_t+ αt}>Kf_t)$, where σ, α, K constants, f is a function of t and $B_t$ brownian motion. This probability must be independent of t. So why is $f_t$ chosen such as ...
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1answer
113 views

Conditional expectation with brownian motion

I want to find $E[e^{σB_t}|∫_0^1B_s ds]$. I make the notation $∫_0^1B_s ds = z$, and I know that: $E[B_t|Z]= 3t(1 - t/2)z $, $Var [B_t|Z] = t - 3t^2(1 - t/2)^2$ and $Var(∫_0^1B_s ds) = 1/3$. The ...
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0answers
48 views

Contradiction on equality with stochastic integrals

I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like: $$\begin{align} & E\left[(tB_t- \int_0^tu ...
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1answer
78 views

Computing expectation of a stochastic integral

I need to compute the expectation $$E\left[\int_0^tu \, dB_u \int_0^s u \, dB_u \right].$$ Being that is my first question, how can I initialize MathJax if I have it on my hard drive.