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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 $t-s$.

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Is this stochastic Picard iterator well-defined?

Preliminaries Let $x_0 \in \mathbb{R}^d$. Let $T \in (0, \infty)$. Let $$ \sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$$ and $$\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}$$ be affine ...
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Is the fact that $dW_t\sim (dt)^{1/2}$ come from the $1/2-$holder property of Brownian motion?

I ofter see that if $W_t$ is a Brownian motion, then $dW_t\sim (dt)^{1/2}$. Does it come from the fact that Brownian motion is $1/2-$holder continuous and not better (i.e. $$\sup_{t,s\in [0,1], t\neq ...
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Brownian motion with zero volatility [on hold]

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?
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Proving standard Brownian motion [on hold]

Suppose X_t is a standard Brownian motion. Assuming that Y_t = a^-(1/2) * X_at and that a > 0. Any advice on how I can start this? The concept of Brownian motion is very confusing to me. Thanks!
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Clarification: Brownian Motion Reflection Principle

The Brownian Motion Reflection Principle gives: For $X_t$ BM starting at $a$ and $b>0$ $\displaystyle P(X_s \ge b, 0 \le s \le t) = 2P(X_t \ge b | X_0 = a) = 2\int_b^\infty \frac{1}{\sqrt{2\pi t \...
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23 views

Why is L an interesting random time for a Brownian motion?

Let $B$ be a Brownian motion and define $L=\sup \{ t \leq 1 : B_t = 0 \}$. My question is: Why is $L$ an interesting random time? Durrett's probability book proves something about the ...
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Computing $dY^{-1}(t)$ using the SDE $dY(t)=\mu Y(t)dt+\sigma Y(t)dW(t)$

Let $\mu$ and $\sigma$ be constants and consider the SDE $dY(t)=\mu Y(t)dt+\sigma Y(t)dW(t)$ with $W(t)$ Brownian motion and $Y(0)=y_{0}$. Using the solution to the SDE, $Y(t)=y_{0}\exp[(\mu-\frac{\...
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26 views

Minimal stopping time of brownian motion

Suppose $W$ is a Brownian motion, let $H_B$ be the hitting of $B \in \mathbb{R}$ and let $\tau$ be another stopping time that is taken to be minimal, i.e $(W_{t\wedge \tau})_{t \geq 0}$ is uniformly ...
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Bounded moments for solution of stochastic differential equation

Consider the following SDE: $$\mathrm{d} X_t = - \lambda X_t + \mathrm{d} B_t$$ with initial condition $X_0 = x$, and where $B_t$ is a standard Brownian motion. An application of Ito's formula gives ...
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Reference : Proving something is a Brownian Motion

Given a (standard) Brownian Motion $W_t$ if we do some sort of scaling, inversion or reversal then we also get a Brownian motion. I have seen proofs but the proofs only seem to rely on showing ...
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Independence between Brownian motion and hitting time.

If given a standard one dimensional Brownian motion $B_t$ and stopping time $T = \inf\{ t : B_t = |a|, a \in \mathbb{R}\}$ We will have independence between $B_T$ and $T$ as $P(B_T = a) = \frac{1}{2} ...
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Compute moments of Brownian motion stopped at exit time of $[a,b]$

Given $B_t$ a standard brownian motion and $a < 0 < b$ Set $T = \inf\{ t : B_t = a \vee B_t = b\} $ For any $\alpha \in \mathbb{Z}^+$, find $EB_T^\alpha$. I know I can use optional ...
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Finding the joint density distribution of $B(1),B(1)+B(2),2B(3)$ where B is the standard Brownian motion

I am trying to figure of the way to find the joint density function $f_{B(1),B(1)+B(2),2B(3)}(x_1,x_2,x_3)$ of $B(1),B(1)+B(2),2B(3)$ where $B(t)$ is the SBM. I know that $$B(1)=x_1\implies B(2)=x_2-...
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Bitcoin price Distribution: GBM or Not?

Is the Geometric Brownian Motion (GBM) a suitable model to describe the Bitcoin price over time? In my opinion it is NOT and a distribution which changes over time is more appropriate model (Btc is ...
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Stochastic Integration $\int_0^T \exp[W(t)-t/2]\,\mathrm d W(t)$ [on hold]

Using Ito formula with time dependence integrate or any way you want $$\int_0^T \exp[W(t)-t/2]\,\mathrm d W(t)$$
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Continuous-time Martingale and Brownian Motion Supremums

I am reading through Le Gall's book on Brownian Motion, Martingales, and Stochastic Calculus. I just read through the chapter on optional stopping of martingales, but I cannot solve the first exercise:...
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Why is Brownian Motion so Big in the Theory of Stochastic Differential Equations?

I am reading some introductory material on stochastic differential equations at the moment. In almost all cases, the equations which are presented are of the form $$ dX_t = \mu(t,X_t) dt + \sigma(t, ...
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Brownian motion to model future asset values

https://quant.stackexchange.com/questions/45104/brownian-motion-for-modelling-future-asset-values Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is ...
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Levy modulus of continuity for a martingale

Given a Brownian motion $B(t)$ then (Levy, 1937) \begin{equation} \mathbf{P}\bigg(\lim_{h\rightarrow 0}\frac{\sup_{0\le t\le 1-h}|B(t+h)-B(t)|}{ \sqrt{2hlog(1/h)}}=1\bigg)=1 \end{equation} Can the ...
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38 views

evaluate conditional probability in brownian motion

Let $W_t$ be a standard brownian motion, and let $0 < x < y$. We want to calculate: $\mathbb{P}(W_y > 0 \vert W_x > 0)$. I am pretty stuck on how to do this. The only insight I have is ...
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Expected Solution of a Stochastic Differential Equation as a Conditional Expectation (this is a tough one).

On all you geniusses out there: this is a tough one. Preliminaries and Rigorous Technical Framework Let $T \in (0, \infty)$ be fixed. Let $d \in \mathbb{N}_{\geq 1}$ be fixed. Let $$(\Omega, \...
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1answer
26 views

Finding the twice differentiable function in Itos Lemma

(I am an undergrad in Econ and new to this forum, so I'm sorry if this will be easy for you guys) Im currently struggling with Stochastic Calculus, resp. Itô's Lemma. I understand that once we have ...
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57 views

Solving Stochastic Integral with Ito's lemma

I want to solve the following: $\int\limits_{0}^{T} exp[S(t)-t/2] dW(t)$ where $dS=µSdt+\sigma S dW$ is the Brownian motion. The Ito's formula I need to use reads: $df=(\frac{\partial f}{\partial ...
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1answer
37 views

Hitting time of a sphere by a Brownian particle

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let us consider a sphere of radius $r$ in $\mathbb{R}^n$ centered at the origin. We know that the probability that the particle ...
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Suppose 𝑊1(𝑡) and 𝑊2(𝑡) are two independent standard Brownian motions. What is the probability that both processes are larger than 0.667 at t=1.0?

Suppose $W_1(t)$ and $W_2(t)$ are two independent standard Brownian motions. What is the probability that both processes are larger than 0.667 at t=1.0? $\textbf{Attempt:}$ $W_i(t) \sim N(0,t)$, at ...
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Probability of a sequence of events converging to 0, has an event whose probability is actually 0 in the context of Stochastic processes (BM)

Why is this true? It appears in the proof of a Holder condition for Brownian motion in Kenneth Falconer Fractal Geometry 2014 edition at page 283 (proposition 16.1) For $X$ beaing a random process (...
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Is there an alternative approach to using Itô's lemma for computing this stochastic differential?

We define $$M_t=(B_t+t)^{-(B_t+\frac{1}{2}t)}$$ Where $B$ is a Brownian motion. I must compute the stochastic differential of $M_t$, i.e. $dM_t$. I figured this should be possible with Itô's lemma, ...
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the continuity in the proof of Ito integral

This is in regards to constructing the Ito integral, specifically the second step of approximating bounded functions by bounded and continuous functions. Let $(\Omega, \mathcal{F}, P)$ be a ...
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Showing log(Wt) is a martingale using Ito's Formula

Let d > 1 and let $W_t$ denote a standard d-dimensional Brownian motion starting at $ x \neq 0$. Let $M_t = log|W_t|$ if d = 2 and $ M_t =| W_t|^{2-d}$ if d > 2. Show that $M_t$ is a martingale. So I ...
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1answer
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Filtering Sum of Brownian Motions

Let us assume that there exist two independent Brownian Motions $B_t$ and $W_t$, and consider their sum $Y_t=B_t + W_t$. Next, define the filtration generated by the sum, $\mathcal{F}_t^{Y}=\sigma(Y_u)...
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Brownian motion independence from stopping time

Let $X_t$ be a standard one dimensional Brownian motion. Let $T = \inf\{t : X_t \in\{ 1,-1\} \} $ and $S = \inf\{ t : X_t \in\{ 1, -3\}\}$ a) Explain why $X_T$ and $T$ are independent. ...
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Function of d-dimensional brownian motion is a martingale.

Let $B(t)$ be a d-dimensional Brownian motion such that $B(0) \neq 0$ where $d > 1$. Let $M(t) = \log\lvert B(t) \rvert$ for $d= 2$, otherwise let $M(t) = \lvert B(t) \rvert^{2-d}$. In either case,...
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Brownian Motion Proof

Let $W_t$ be a Brownian motion. Show that $\displaystyle\int_{0}^{T} W_t \, \text{dt} \sim {\cal N}(0,T^3).$ Currently I have tried using a sum that approximates the integral, however I couldn't get ...
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How can I simulate the Stochastic integral $\int X_sdW_s$ when X is a stochastic process and W is a Brownian motion?

How can I simulate the Stochastic integral $\int_0^1 X_sdW_s$ where $X$ is strong solution of of an SDE driven by a Brownian motion independent of $W$(the integrator above). I have already computed $...
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Proof of $\mathbb E[u(B_{t+s})\mid \mathcal F_s]=\mathbb E[u(B_t)\mid B_s]$.

I reading the proof of R. Schilling in his book : Brownian motion, introduction to stochastic process, and his proof is really unclear for me. I want to prove that if $u$ is bounded, then $$\mathbb E[...
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1answer
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Estimation of integral of stochastic process(Krylov estimation)

Let $X_n$ be a sequence of Ito diffusions $$dX_n(t)=b_n(t) \, dt+\sigma_n(t) \, dW(t), \qquad 0\leq t\leq T$$ with $b_n$ uniformly bounded and $\sigma_n$ uniformly elliptic. Then Krylov's estimation ...
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1answer
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Justification of interchanging Expectation and Limit in Ito Integral Approximation

My reason for asking this question is because I can't seem to justify extending the results from the Ito Integral of elementary functions to the continuous form after taking the limit. For example, if ...
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How to show $\int_{0}^{t} s \mathop{dW_{s}} = tW_{t} - \int_{0}^{t} W_{s} \mathop{ds}$?

I'm new to stochastic integration, and I've been stuck on this exercise. I want to show $$\int_{0}^{t} s \mathop{dW_{s}} = tW_{t} - \int_{0}^{t} W_{s} \mathop{ds}$$ holds, but I don't really know ...
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How can I simulate increments of a two dimensional brownian motion?

I am attempting to simulate an sde system of the following form $$ dX_t=\sqrt{\vert aX+bY\vert}dW^1_t \\ dY_t=\sqrt{\vert cX+dY \vert}dW^2_t $$ where $W=(W^1,W^2)$ is a standard two dimensional ...
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Passage time and historiacl maximum of geometric Brownian Motion

I need to prove two ineqaulities about a geometric Brownian Motion: for $v>0$, $c>0$, define $$ R_c:=\inf \{t>0:e^{vW(t)-\frac12v^2t}=c\} $$ as the first passage time of a geometric BM ...
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Strong markov property of a transformation of the Brownian motion

Let $(B_t)$ be a standard Brownian motion and consider $(X_t)$ defined as: $$X_t=e^{-t}B_{e^{2t}}$$ I've proved that this process is markov, however I can't prove that is strong markov. I know that ...
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Meaning of dummy calibration procedure of GBM

Ciao, I was doing a dummy check on calibration of GBM. Let me explain. We will consider the usual process: $$ dX_t = \mu X_t dt + \sigma X_t dW_t $$ with $X_0$ initial data and $t\in [0, T]$ Suppose ...
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Brownian motion question for $\mathbb{E}\left[W_s W_t^2 \right]$ and $\mathbb{E}\left[W_s^2 W_t^2 \right]$

Let $\{W_t:t \ge 0\}$ be a Brownian motion. Find for all $0 \le s < t$: $\mathbb{E}\left[W_s W_t \right]$ $\mathbb{E}\left[W_s W_t^2 \right]$ $\mathbb{E}\left[W_s^2 W_t^2 \right]$ $\...
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1answer
62 views

How to bound increments of Ito integral?

Given a Brownian motion $\{W_t\}_{t\in[0;T]}$ and a continuous, adapted and square-integrable (bounded if you want) process $\{\sigma_t\}_{t\in[0;T]}$ and $\varepsilon > 0$, I want to prove that ...
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1answer
27 views

Cumulative distribution function of $W(T)/\sqrt{T}$, where $W$ is a standard Wiener process.

$W(T)$ is the standard Wiener process and $\phi(\cdot)$ is the CDF of $\mathcal N(0,1)$. Can anyone explain how I can go from (7.9) to the last step; i.e. how does $W(T)$$/$$\sqrt(T)$ disappear? I am ...
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2answers
74 views

Shifted two-sided Brownian Motion

Let $(B_t)_{t\in\mathbb{R}}$ be a two-sided Brownian motion, defined as $B(t) = \begin{cases} B_1(t),\quad t >0 \\ 0, \quad t = 0 \\ B_2(-t), \quad t < 0 \end{cases}$. For some $a>0$ let $T:=\...
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2answers
32 views

Probability of Brownian Motion hitting -2 before 1?

Why is the probability of Brownian Motion hitting -2 before 1 is equal to 1/3? This is an interview question asked for Quant roles. I found a similar question was previously asked: Brownian motion ...
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38 views

Revuz and Yor's Book “Continuous Martingales ans Brownian Motion” - Chapter 1 - Exercise 1.11 [closed]

Context : This post is the first of a post taken from exercises in Revuz and Yor's Book "Continuous Martingales ans Brownian Motion". The reason for doing so is that the exercises of this book are ...
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1answer
33 views

expectation of the sign of product of bivariate normal variables

Let $X$ and $Y$ be two normal random variables with mean zero, variance 1 and correlation $\rho$. Let $Z=sign(XY)$ with $sign(x)=1$ if $x>0$ and $sign(x)=0$ otherwise. Now, I calculate $\mathbb{E}[...
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1answer
27 views

What is the distribution of minimum of Brownian motion on arbitrary interval?

We know that $P(\min_{0 \leq s\leq t} B_t \leq x)=2P(B_t\leq x)$. This can be found in any standard stochastic calculus textbook. However I am curious about instead of the interval $[0,t]$ if we ...