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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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14 views

Why $\mathbb P(B_t\geq L)=\mathbb P(B_t\geq \ell, \tau\leq t)$?

Let $(B_t)$ a Brownian motion. I want to prove that for all $L\geq 0$, $$\mathbb P(\sup_{0\leq s\leq t}B_s\geq L)=2\mathbb P(B_t\geq L).$$ The proof start by : let $\tau=\inf\{t\geq 0\mid B_t= L\}$. ...
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0answers
24 views

Write a summation to approximate the integral

Let $W(t)$ be a Brownian motion and let $I$ be the integral $I$ = $\int_0^T$ $W(t)dt$ $1)$ Given the partition $π$ : $0 = $$t_0$$ < $$t_1$$ < $$t_2$$ < . . . < T$, write a summation to ...
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2answers
52 views

Simple process in Itô calculus

For the definition of Itô integral, one uses simple stochastic processes. I have found two definitions for simple stochastic process, given a filtration $(\mathcal{F}_t)_{t\geq0}$, an interval $[0,T]$ ...
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1answer
26 views

Integrating $e^{at}$ with respect to Brownian Motion

I am having trouble figuring out how to integrate $$ \int_0^t e^{as} dW_s $$ where $a \in \mathbb{R}$ and $W_t$ is a Brownian motion. I (think) that when you have an integral w.r.t. Brownian motion ...
3
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1answer
42 views

Strong Markov property, Brownian motion

I have a question about the strong Markov property of Brownian motions. Let $(\{X_t\}_{t \ge 0}, P_x)$ be a $d$-dimensional Brownian motion starting from $x \in \mathbb{R}^d$. Let $\tau=\inf\{t>0 ...
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0answers
27 views

Solving a Stratonovich SDE

I am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\...
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1answer
43 views
+50

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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1answer
28 views

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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1answer
15 views

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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1answer
13 views

Proving standard Brownian motion [closed]

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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1answer
14 views

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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0answers
24 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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1answer
16 views

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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0answers
29 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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1answer
16 views

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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1answer
14 views

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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1answer
30 views

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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1answer
27 views

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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0answers
19 views

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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0answers
11 views

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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1answer
42 views

Stochastic Integration $\int_0^T \exp[W(t)-t/2]\,\mathrm d W(t)$ [closed]

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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0answers
17 views

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:...
5
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1answer
52 views

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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0answers
20 views

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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0answers
11 views

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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0answers
39 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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1answer
56 views

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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1answer
59 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
40 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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0answers
18 views

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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0answers
15 views

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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1answer
30 views

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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0answers
25 views

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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0answers
22 views

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
19 views

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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1answer
31 views

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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0answers
25 views

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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0answers
21 views

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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0answers
29 views

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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0answers
25 views

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
24 views

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
32 views

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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2answers
52 views

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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0answers
13 views

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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0answers
9 views

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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0answers
87 views

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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0answers
12 views

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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1answer
23 views

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 ...