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Questions tagged [brownian-motion]

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 $t-s$.

Brownian motion is a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W_t-W_s\sim\mathcal{N}(0,t-s)$, i.e. the increments are normally distributed with $0$ mean and variance $t-s$. Links:

Brownian Motion at Wolfram MathWorld

4494 questions
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Evaluate $E[B(u)B(u+v)B(u+v+w)B(u+v+w+x)] $, where $\{B(t); t\ge 0\}$ is a standard Brownian Motion

Consider a standard Brownian motion $\{B(t); t\ge 0\}$ (zero mean and $\sigma^{2} = 1 $) at times $0 < u < u+v < u+v+w < u+v+w+x $, where $u, v, w, x > 0$ Evaluate the product moment: $$E[B(u)B(u+v)B(u+v+w)B(u+v+w+x)] $$ I tried using the same…
seraphimk
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Product of correlated brownian motions

Consider that the correlation between two standard brownian motions $dB_x$ and $dB_y$ be $\rho$. And we write $\mathtt{Cor} (dB_x,dB_y)$ = $\rho$. Show that $dB_xdB_y$ = $\rho dt$
Carlos Lobato
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Integral of Brownian motion

Is there any formula for the following integral ? $$\int_0^t W_t^n\; \mathrm{d}W_t $$ For $n=1$ the answer is known. What about $n=2,3,\ldots$?
Moh514
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Limit of geometric brownian motion

If a stock price is modelled with a geometric brownian motion process with this definition: $GBM(t)=s_0 e^{X(t)}$ where $X(t)$ is a brownian process $N(\mu - \sigma^2/2,\sigma)$, then doesn't this mean that when $t$ tends to infinite and $\mu=0$,…
mvc
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Is $(B_t,B_t)$ a Brownian motion on the line $\{(x,x)\mid x\in\mathbb R\}$?

I saw today that $(B_t^1,B_t^2)$ is a brownian motion on $\mathbb R^2$ iff $B_t^i$ a independent Brownian motion on $\mathbb R$. So, if for example, $(B_t,B_t)$ won't be a Brownian motion on $\mathbb R^2$ if $(B_t)$ is a Brownian motion on $\mathbb…
Todd
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Brownian Motion at hitting time defined as an infimum

I'm reading a book on Brownian Motion, and they define the hitting time as $$T_x = \inf\{t > 0 : B(t) = x \}$$ Later on they state that $B(T_x)=x$. Why would they use inf instead of min? With inf, if we have infinite amount of crossings/hits at x in…
An old man in the sea.
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Brownian motion square

I need to calculate for Brownian motion $dBt^2$ and $dBt^4$. For $dBt^2$, I think that it is correct to use Ito's lemma. So we get: $$ dBt^2=(1/2)\cdot 2 \cdot dt +2\cdot Bt\cdot dBt$$ Is this correct? What about $dBt^4$?
what_456
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Why the local time $L_T$ of a Brownian motion is not upper-bounded by $T$ a.s.?

Let $L_T$ the local time of a Brownian motion. Roughly speaking, it's the time spend by a BM at $0$, i.e. $$m\{s\in [0,T]\mid B_s=0\}.$$ Now, I know that $$\mathbb P\{L_T\geq x\}=2\mathbb P\{B_T\geq x\}.$$ And since $$2\mathbb P\{B_T\geq T\}>0,$$…
Todd
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Finding the distribution of maximum stock price under Black-Sholes model in a specified interval

Stochastic differential equation of Black-Scholes model is defines as \begin{eqnarray}\label{ref9} dS_t = (r-d)S_tdt+ \sigma S_t dW_t \end{eqnarray} where the interest rate $r$, the dividend yield $d$, and volatility $\sigma$ are assumed to be…
Statistics
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Evalute integral of Brownian motion $E[(\int_0^tB_sds)^2]$

Can someone explain me how to compute this expectation? $E[(\int_0^tB_sds)^2]$ I know that the result is $t^3/3$
user742138
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why does $dX^2 \rightarrow 0$?

In my textbook, we're talking a bit about Brownian Motion and Ito's Lemma. A short note that's in the book that I don't understand is that the Taylor series expansion for our higher order terms (2 or higher) go to zero. I'm just not sure why $dX^2$…
financial_physician
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Independent increment of Brownian motion (mistakes wikipedia definition ?)

I know that Brownian motion has the property that if $0\leq t_1\leq t_2\leq ...\leq t_n$ then $$B_{t_1}, (B_{t_2}-B_{t_1}),...,(B_{t_n}-B_{t_{n-1}})\tag{*}$$ are independents. In wikipedia they say that increment of Brownian motion, and they define…
John
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Expectation of an integral of a function of a Brownian motion

$B_t$ is a Brownian motion and $Y_t:=e^{aB_t+bt}$. For which $a$ and $b$ is $Y_t\in M^2$? I found a theorem that says that sufficient for $Y_t\in M^2$ would be $E[\int_0^\infty Y_t^2 dt]<\infty $ But how can I integrate over a function of a Brownian…
milfor
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Covariance of squares of Brownian motion

$B(t)$ is a Brownian motion. Calculate $Cov[B(s)^2,B(t)^2]$ for $s,t\ge0$. For this I would need $E[B(s)^2B(t)^2]$. Without the squares that wouldn't be a problem but I have no idea how to calculate this expected value. Any suggestions?
milfor
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Covariance and Brownian motion

B(t) is a Brownian motion and $0
milfor
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