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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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1answer
55 views

Convex functions of a martingale.

Let $X_1(t)=e^{B(t)}$ and $X_2(t)=e^{-B(t)}$ where $B(t)$ is the standard brownian motion and $\{G_t\}$ is the filtration generated by the brownian motion. Determine what kind of martingale $X_1(t)$ ...
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1answer
34 views

Significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$.

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there any significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$. ...
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1answer
44 views

Finite sequence of random step processes such that $\lim_{n\to\infty}E(\int_{0}^{\infty}|f(t)-f_n(t)|^2dt)=0$ for $f(t)=e^{-t^2/4}$

Let $$f(t)=e^{-t^2/4}, \ \ \ t \ge 0$$ I want to show that $f$ is in $M^2$ where $M^2$ denotes the class of stochastic processes $f(t),t\ge0$ such that $$E\left(\int_0^\infty|f(t)|^2dt\right)&...
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1answer
28 views

Independence of Solution of SDE $S^{(x_0, \sigma, \mu)}_t$ of Initial Information $\mathcal{G}_0$

Question Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise: \begin{...
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0answers
52 views

Using Ito's lemma to determine $dY(t)$ when $Y=\sin(t+B_t), \ \ \ t\ge0$

Let $(\Omega, \mathcal F, P)$ be a probability space and $\{B_t\}_{t\ge0}$ a Brownian motion. Furthermore let $\{F_t\}_{t\ge0}$ be the natural filtration of $B$. Let $$Y(t)=\sin(t+B_t), \ \ \ t\ge0$$ ...
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3answers
3k views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
4
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1answer
28 views

Poisson process on Skorokhod's space

For each $n=1,2,\ldots $, let $\ \xi_{n1},\ldots, \xi_{nn}$ be random and independent variables such that $\mathbb{P}(\xi_{ni}=1)=p_n \ \ $ and $\ \ \mathbb{P}(\xi_{ni}=0)=1-p_n$. Let consider the ...
0
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1answer
30 views

Time homogeneity of Ito diffusion

Consider a time homogeneous Ito diffusion satisfying a SDE, \begin{equation}\label{1} dX_t=b(X_t)dt+\sigma(X_t)dB_t, X_s=x \end{equation} $t\geq s$. The unique solution of the SDE is denoted by $...
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2answers
56 views

Find the quadratic variation process of $\int f(s) \, dB_s$

Let $f \in L^2[a,b]$ and let $\displaystyle M(t)=\int_a^tf(s)dB(s)$. Find the quadratic variation process, $[M]_t$ , of $M(t)$. Here the quadratic variation process is the limit in probability of $\...
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0answers
26 views

Basic Questions about Brownian Motion

I was hoping someone could answer a few basic Brownian Motion questions. A Brownian Motion, $B_t$, (let us assume defined on the continuous path space with the Brownian Measure starting at $0$) has ...
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0answers
59 views

How can I numerically compute a stochastic integral?

I am trying simulate a to solve a 2-dimensional stochastic process and $Y_t^1$ is a mean-reverting square root process which I simulated on a time grid using its known conditional distribution. I ...
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1answer
32 views

Derivation of Joint and Conditional density of a Brownian Motion and its Maximum

Given $W(t)$ and $M(t)=\max\limits_{0\leq s\leq t}W(s)$ where $\{W(t),\ t\geq 0\}$ is the standard brownian motion, compute their joint distribution and the conditional distribution of $M(t)$ given $W(...
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0answers
45 views

Every local martingale with respect to the Brownian filtration has its continuous version.

I would appreciate some help on the following. In class, we said that Every local martingale with respect to the Brownian filtration has its continuous version. To prove this, it is apprently enough ...
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1answer
94 views

Girsanov THM and Radon-Nikodym derivative

I've been having a hard time to applicate Girsanov theorem with Radon-Nikodym derivative in the demonstration of German-El Karoui-Rochet formule. I know that $\Pi_0:=S_0\mathbb{Q}^S(S_T\geq K)-K\...
2
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1answer
48 views

Showing if sum converges in $L^2$ (Brownian motion)

Consider a probability space $(\Omega, \mathcal F, P)$ and a Brownian motion $(W(t),t\ge 0)$. Let $T>0$, $t_j^n=jT/n$ and $$\xi_j^n=\frac{1}{3}t_{j+1}^n+\frac{2}{3}t_j^n, j=0,\ldots,n-1$$ ...
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0answers
17 views

Proof of strong Markov continuity of Brownian motion

Let $(B_t)$ a Brownian motion and $\sigma $ a stopping time finite a.s.. I want to prove that $W_t=B_{\sigma +t}-B_\sigma $ is a Brownian motion. The way to prove it is first to prove that for all $0\...
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0answers
12 views

Existence of $L^2$ limit of sequence of martingales

I am currently revising some martingale theory, and was trying out an old past paper question. I haven't come across anything like this before, so was wondering how one would approach it, and also in ...
2
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1answer
73 views

Is the fact that $dW_t\sim (dt)^{1/2}$ come from the $1/2-$holder property of Brownian motion?

(I offer 100 bounty because I really would like to have a constructive answer to this question) I often see that if $W_t$ is a Brownian motion, then $dW_t\sim (dt)^{1/2}$. Can it come from the fact ...
4
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1answer
52 views

Does a strong solution to a SDE imply lipschitz condition?

Consider $dX_t=b(X_t,t)dt+\sigma(X_t,t)dB_t$. I know that, $|b(x,t)-b(y,t)|+|\sigma(x,t)-\sigma(y,t)|\leq D|x-y|$ for some constant D implies the existence and uniqueness of a strong solution. ...
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0answers
29 views

Implementation of Brownian local time process

I am trying to replicate results of Grigoriu's Solution Of Boundary Value Problem By Monte Carlo Simulation. Essentially, we are looking for the local solution of a PDE and for this, we take samples ...
0
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1answer
769 views

Find the covariance of a brownian motion.

Given a standard Brownian motion $\{W_t\}_{t\geq0}$, find the value of $\operatorname{Cov}\left(W_t,W_s\right)$. Is there a way to simplify $$\operatorname{Cov}\left(W_t,W_s\right)?$$
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0answers
28 views

Density of Maximum brownian motion

Why is it that for $M(t):=\max\limits_{0\leq s\leq t}B(s)$ $$P(M(t)\geq a) = 2P(B(t)\geq a)=\frac{2}{\sqrt{2\pi}}\int\limits_{a/\sqrt{\sigma^2t}}^{\infty}e^{-y^2/2}dy, \ a \geq 0$$ Doesn't result in: $...
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0answers
23 views

Bound on Second Moment of Solution of Stochastic Differential Equation

Question Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise: \begin{...
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0answers
23 views

How to map the coefficients of an SDE to its solution process in a measurable way?

Preliminaries and Standard Technical Framework Let $T \in (0, \infty)$ be fixed. Let $d \in \mathbb{N}_{\geq 1}$ be fixed. Let $$(\Omega, \mathcal{G}, (\mathcal{G}_t)_{t \in [0,T]}, \mathbb{P})$$ ...
2
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1answer
25 views

Brownian Motion infinitesimal generator $\int Lf(x) \mu(dx) = 0$

Take $B(t)$ as a standard Brownian Motion. Consider a real-valued Markov process with infinitesimal generator L. Recall that µ is an invariant measure if $$\int E[f(B(t))] \mu(dx) = \int f(x) \...
4
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1answer
245 views

Show that $O_t$ is a Gaussian Process

Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process. I think $\mathsf{E}[O_t]=e^{-\alpha t} ...
2
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1answer
54 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
34 views

Distribution of a function of Brownian motion

Q) Let $W = \int_{0}^{t}B_sds$, $B$ is a Brownian motion. Find $EW$ and $EW^2$. My attempt: $B_s \sim N(0,s) $ $$W = \int_{0}^{t}\frac{1}{\sqrt{2\pi s}}e^{-\frac{x^2}{2s}}ds , EW = \int_{-\infty}^...
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1answer
74 views

How to easily see the time integral of a Brownian motion is normally distributed?

It's well known that $X_t:=\int_0^tB_\tau d\tau$ where $\{B_\tau\}$ is a 1D standard Brownian motion is distributed as $N(0, t^3/3)$. Is there any "immediate" way to see this fact? The easiest one I ...
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0answers
33 views

Are Ito integral processes only unique up to indistinguishability?

Technically speaking, are Ito integrals of stochastic processes $S$ with respect to Brownian motion $B$ $$ \int_{0}^{t} S_s dB_s $$ concrete stochastic processes in $t$ or only equivalence classes ...
4
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2answers
78 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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0answers
29 views

Are Ito integrals equivalence classes or conrete random variables?

Technically speaking, are Ito integrals of stochastic processes $S$ with respect to Brownian motion $B$ $$ \int_{0}^{T} S_s dB_s $$ random variables or equivalence classes of almost surely equal ...
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0answers
19 views

In which sense we have unicity of solution in stochastic differential equation?

Let $$dX_t=f(X_t)dt+g(X_t)dB_t,$$ and SDE with $f$ and $g$ nice enough to have existence and unicity of the solution. I'm not sure what mean unicity here. For example, consider the equation $dX_t=dB_t\...
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1answer
19 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 $\tau=\inf\{t\geq 0\mid B_t= L\}$. ...
4
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1answer
71 views

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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2answers
4k views

Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
4
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1answer
65 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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1answer
38 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 ...
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1answer
291 views

How to plot a SDE $dx(t)=Ax(t)dt+Bx(t)dW(t) $ using Matlab, and how will the graph look like? [closed]

Could you please provide a full Matlab programme (which can be copied and pasted directly) that can plot the solution to the stochastic differential equation (SDE) $dx(t)=Ax(t)dt+Bx(t)dW(t)$, (where $...
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1answer
29 views

Find values of $a$ and $\lambda$ for which $Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale

Find values of $a$ and $\lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,b\in\mathbb{R}$ can be positive as well as negative, since $...
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1answer
22 views

Brownian motion with zero volatility [closed]

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
31 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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0answers
30 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 ...
0
votes
1answer
21 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{\...
0
votes
1answer
20 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 \...
1
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0answers
27 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 ...
1
vote
1answer
45 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 ...
1
vote
2answers
961 views

Proof that $p$-th total variation of a brownian motion is $0$ while $p>2$

The p-th total variation is defined as $$|f|_{p,TV}=\sup_{\Pi_n}\lim_{||\Pi_n||\to n}\sum^{n-1}_{i=0}|f(x_{i+1}-f(x_{i})|^p$$ And I know how to calculate the first total variation of the standard ...
1
vote
1answer
21 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 ...
0
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1answer
17 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 ...