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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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Brownian motion exit time

I don't understand why $ \{a
user93511
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Brownian motion: Why $p_x\{B_{T_{a,b}}=b\}=p_x\{\tau_a<\tau_b\}$?

Let $(B_t)$ a Brownian motion. I denote $\tau_a=\inf\{t\geq 0\mid B_t=a\}$, $T_{a,b}=\tau a\wedge \tau b$ and $p_x\{A\}=p\{A\mid B_0=x\}$. Why $$p_x\{B_{T_{a,b}}=b\}=p_x\{\tau_a<\tau_b\}\ \ \ ?$$ To me,…
idm
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Brownian motion: Why $p\{-x\leq B_t\leq x \mid B_{t_n}=\pm x_n,...,B_{t_1}=\pm x_1\}=p\{-x\leq B_t\leq x\mid B_{t_n}=x_n,...,B_{t_1}=x_1\}$

Let $(B_t)$ be a Brownian motion. Why: $$p\{-x\leq B_t\leq x \mid B_{t_n}=\pm x_n,...,B_{t_1}=\pm x_1\}=p\{-x\leq B_t\leq x\mid B_{t_n}=x_n,...,B_{t_1}=x_1\}\ \ \ ?$$
idm
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Brownian Motion Maximum Value Proof

Let $B(t)$ be a Brownian Motion and $$M(t) = max_{s:s \leq t} B(s)$$ and $$\tau_a = min_t{B(t) = a}$$ Then, $P(\tau_a < t) = P(B(t) - B(\tau_a) > 0 \: |\: \tau_a < t) + P(B(t) - B(\tau_a) < 0 \: |\: \tau_a < t)$ So this says that the probability…
jj238sdfaaa
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Brownian Motion Finding M(t)

If I have that {$B(t); t >=0$} is a standard Brownian motion, with $B(0)=0$, and I let $M(t)$ = max{$B(u) ; 0 \leq u \leq t$} and I am supposed to: a) Evaluate Pr{$M(4) \leq 2$} and b) Find the number c for which Pr{$M(9) > c$}=0.10 I'm having a…
user193429
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Proving that a process is a Brownian motion by covariance and mean functions

The steps to showing that a process $(W_t)_{t \geq 0}$ is a Brownian motion (BM) are as follows: (1)$W_0 = 0$ (2) $ \forall t ~~~W_t$ is continuous (3)$W_t \sim N(0,t)$ (4)$W_{t+s}-W_{s} \sim N(0,t)$ (5)$W_{t+s}-W_{s} \bot \mathscr…
WeakLearner
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Problem with a Brownian motion, Ito's formula and an indicator function

so I have done the first part of this question (it is at the bottom), but I have no clue how to do the second part. I think I understand the theory, but I do not know how to apply it. Any help would be really appreciated, thanks a lot! :) Let…
s1047857
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Brownian Motion independent increment computation

One can rather easily show that $E\left[\sum\limits_{i = 0}^{i = n - 1}W_{t_i}(W_{t_{i + 1}} - W_{t_i})\right] = -T + W_T^2$. What I'm confused about is why we can't simply say that for each $i$, $W_{t_{i}}$ is independent of $(W_{t_{i + 1}} -…
user7348
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Show that $\mathbb{E}[\tau] = b^2$, where $\tau = \inf[t>0: |W_t| >b]$.

Let $\tau = \inf[t > 0 : |W_t| > b]$. The author says that $\mathbb{E}[\tau] = b^2$, and I want to verify it. I am aware of $T_b$ having the same distribution as $b^2/Y$, where $T_b = \inf [t > 0: W_t >b]$, and $Y$ is a standard normal. My attempt…
shk910
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Brownian Motion for Price of a Stock

Suppose that the current ( t = 0 ) price of a stock is 1, the drift µ = 1 and the volatility σ = 0.5. I am willing to sell you the option to buy from me at a price 2 at time t = 1. What would be the fair price to charge for this option? your…
user427820
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standard Brownian motion. Calculate the probability that W(3) > W(2) > W(1).

Assume that W(t) is the standard Brownian motion. Calculate the probability that W(3) > W(2) > W(1). Hi I am really bad with BM so can anyone please help me here?
Master Saan
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Brwonian Process ITO Integral

Briefly I am on a calculus course and right now we are learning Brownian Motion, its properties and proofs. And I have a question such as below in my study set that I cant find a solution. I would apreciate any help. Show that, $d(W(t)^n) =…
skatip
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Finding Moments of Brownian Motion

I am trying to calculate the K-th moment for Standard Brownian Motion: $$Z(t) \sim N(0,t)$$ I'm given that the second moment is $t$, but I'm having trouble seeing how that was arrived at. I thought to use $M_X(t) = E[e^{tX}] = e^{\mu t…
mathhelp
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Brownian motion with zero volatility

is it possible for Brownian motion to deal with zero volatility? and if it does, does it mean that the fund experiencing deterministic increment in value?
Raihana Nasir
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