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New answers tagged brownian-motion

1 vote
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• 890

Limit for Brownian local time

I think I have an answer: can you please let me know if you spot any mistake? Here we can write for every $n\in \mathbb{N}$, $L^{(n)}(t)=\frac{L(nt)}{\sqrt{n}}$ and $L^{(n)}(t)\overset{(d)}{=}L{(t)}$, ...

• 146

The almost sure event in the law of the iterated logarithm for the Brownian motion: what it looks like

The event you have captured in more explicit terms is not $\{\limsup_{t\to 0}W_t/h(t)=1\}$, but rather $\{\limsup_{t\to 0}|W_t/h(t)-1|=0\}$. The placement of the absolute value is crucial!
• 25.8k
1 vote
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$W_{10t}^2 - W_{5t}$ is not gaussian

Set $$W_{10t }-W_{5 t} =a, W_{5t}=b$$ and determine the moments of $$(a+b)^2 - b$$ with $a,b$ independent Gaussians with the speciality, that the 2n-moments are powers of the second moment with ...
• 2,172
1 vote

Using Ito's Lemma to take a stochastic integral

Please avoid to ask too many questions at the same time in future posts. Let's tackle the side questions first. First side question. I mean, why not ? If you have a stochastic variable $X_t$ itself ...
• 8,318
Accepted

How can I show that the first exit time by a planar Brownian motion is a.s. finite, i.e. $\mathbb{P}_z(\tau_D<\infty)=1$?

My first question is, where do i need that $W_t$ is a complex Brownian motion, I mean why can't I only work with $B_t$ instead of $W$? You could do it with just using neighborhood-recurrence for 2d-...
• 3,611
Accepted

What is the solution to the SDE $X^x_t = B_t + \int_0^t \frac{x − X^x_s}{1-s} ds$

Setting $Z_t$ as what you did in your work, and then writing (2) in Differential form gives us that $$dX^x_t = xdt + dB_t - \frac{B_t}{(1-t)}dt + Z_tdt \tag{4}\label{eq4}$$ And (2) also gives us that ...
• 48

Markov Property of a Ito Process

Rather than bringing in $B_{t+h}-B_h$, consider using $B_{t+h}-B_t$: Because $X^x_{t+h} = X^x_t\cdot\exp(ch+\alpha(B_{t+h}-B_t))$, and $B_{t+h}-B_t$ is independent of $\mathcal F_t$ with the same ...
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• 3,611
1 vote

• 8,196

• 25.8k

Joint law of Brownian motion maximum and its values at different points

It seems that Thomas Kojar's suggestion does help to solve the problem. Let $0 < u < v < T$. Then, since  M_T = \max \{ M_u, B_u + \max_{t \in [u, T]} ( B_t - B_u ) \} = \max \{ M_u, B_u + ...
• 258

Proving the process $(Y_t)_{t\in[0,\infty)}$ is a standard Brownian motion

First, lets clarify what independent processes means Independent stochastic processes and independent random vectors. The definition for the two processes to be independent is given by PlanetMath: ...
• 3,611
Proving the process $(Y_t)_{t\in[0,\infty)}$ is a standard Brownian motion
The quadratic variation of $Y$ is $t\,.$ (The rest of the things you need to apply for Levy characterisation I leave to you.) Proof. We only have to consider partitions $0<t_1<\dots<t_n$ of ...