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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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Show $[\int _0 ^t B_{2,s} dB_{1,s}]^2 - \int_0^t (B_{2,s})^2 ds$ is a martingale

I want to show the following: $$[\int _0 ^t B_{2,s} dB_{1,s}]^2 - \int_0^t (B_{2,s})^2 ds$$ is a martingale where $ (B_{2,s},B_{1,s}) $ is a two dim'l Brownian motion. My attempt: By Ito formula, ...
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Stretching the Brownian Motion in [0,1] to get another Brownian Motion in [0,t]

I'm running a simulation of a Standard Brownian Motion by limit of a Symmetric Standard Random Walk $\{S_n ,n\geq 1\}$ and $S_n=\sum_{k=1}^n X_k$, where $P(X_k =-1)=P(X_k =1)=\frac{1}{2}$. And ...
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Random “start afresh” of Brownian Motion [duplicate]

In the book "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve I encountered Problem 6.1 which the authors refer to as "not be hard to show": Let $\{B_t, \cal{F}_t;t\geq 0\}$ be a ...
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32 views

For a Brownian motion $(B_t)_{t \geq 0}$, do we have $E[(B_\tau - B_\sigma)^2]=E[B_\tau^2 - B_\sigma^2]$ for stopping times $\sigma \leq \tau$?

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0}, P)$ be a filtered probability space and let $(B_t)_{t \geq 0}$ be a Brownian motion on that space. The question is if the following is true: For ...
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12 views

If $W$ a brownian motion, what is $\mathbb P\{W(t_2)\in [a_2,b_2]\mid W(t_1)=x_1\}$

Let $W(t)$ a standard Brownian motion. In the book "Introduction to stochastic differential equation" of Evans, page 41 they try to find $$\mathbb P\{a_1\leq W(t_1)\leq a_2, a_2\leq W(t_2)\leq b_2\}.$$...
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22 views

Introduction to SDE : why does the noise is the derivative of Brownian motion?

In "An introduction to stochastic differential equations" of Evans, he define a SDE as follow : Consider a ODE $$\begin{cases}\dot x(t)=b(x(t))\\ x(0)=x_0\end{cases} $$ We call $x(t)$ the state of ...
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30 views

How can I show that the stochastic integral of a jump process w.r.t. Brownian motion is a local martingale by using this special localizing sequence?

Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<\infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then \begin{equation} M_t=\int_0^...
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Supremum of a general Gaussian Process

I have a stochastic integral of the form \begin{align*} X(t) = \int_0^t h(v) W(v) dv \end{align*} where $W(v)$ is the standard Brownian motion and $h(v)$ is a positive, integrable function. While $X(t)...
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Expected value of a random process which is a function of a Wiener process

I do not quite know how to attack and proceed with the following problem: Let W(t) be a Wiener process. In that case, calculate the expected value of random process Y(t)=W^2(t)cos^4(W(t))exp(−3W(t)/2)...
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1answer
17 views

Measurable state space

I have come across the following question in my lecture notes. I don't feel that comfortable with filtrations and showing measureability. We consider the integral $X_t=\int_0^tB_sds$, where $Bt:t\...
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Integrals and Wiener process in Unit root test

I was reading Unit Root Tests in Econometrics which involves integral of Wiener process or stochastic integral. There I found two equivalences that I couldn't prove, so may could you help me. I'm a ...
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Relation between time-changed solutions to SDEs

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a $\mathcal F$-...
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Malliavin Calculus: when a seminorm of Brownian motion belongs to $\mathbb D^\infty$?

I am a new entry in this community, so I apologize for any possible mistake. I'm studying the Malliavin Calculus, and I am stuck with the following problem: For every $0 < \gamma < 1$ and $...
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28 views

Two different answers when integrating with respect to Brownian motion

Consider the integral with respect to Brownian motion $$\int_{0}^{t}s \ dB_{s} \ . $$ A textbook I am reading uses integration by parts to rewrite the above integral as $$tB_{t}-\int_{0}^{t}B_{s} \ ...
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38 views

Calculating expected value of a complex wiener process (geometric, cosine, quadratic multiplications)

$$W_t$$ is defined as a wiener process. How could the expected value of the equation below be calculated on condition that t=1? $$Y_t = W^2_t * \cos^4(W_t) e^{-\frac32 W^2_t}.$$ At first, I have ...
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1answer
39 views

Expectation of a stopping time w.r.t Brownian motion

For a real-valued standard Brownian motion $B= (B_t)_{t\geq 0}$ we define the stopping times $ \tau_{a} := \inf \left\{ t> 0: B_t \leq a \right\},~a <0$, $ \tau_{b} := \inf \left\{ t&...
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Adding a linear drift to a diffusion

Consider a multidimensional diffusion in $\mathbb{R}^d$ whose generator is given by: $$Gf(x)=\langle b(x), Df(x)\rangle+\frac{1}{2}\operatorname{Trace}(Q(x)D^2f(x)),$$ where $b(x)\in\mathbb{R}^d$ and ...
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22 views

Wiener space, probability measure, local martingale

I hope you can help me the following problem. Assume a Wiener space, that means a probability space $(\Omega,\mathbb F,\mathbb P)$, where $\Omega = C([0,\infty))$, $X$ is the coordinate process, $\...
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21 views

Distribution of the Itō integral process with respect to Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $X$ be a $\mathcal F$-progressive process on $(\Omega,\mathcal A,\...
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“Straightforward” estimate on intersection-probabilities of Brownian Motion

I am currently working on a paper and there appears this so called "straightforward estimate": $$\mathbf{P}\{B[0,1] \cap B[3,n] = \emptyset\} \leq \frac{c}{\ln(n)} \quad \text{where}\ c<\infty\ \...
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Difference between weak ( or martingale ) and strong solutions to SDEs

Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} $$ ...
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53 views

Calculating $\mathbb{E}(|B_t|^{-2})$ for 3-dim. Brownian motion

Ok so I'm given a standard 3-dimensional Brownian motion $B(t) = (B_{1}(t),B_{2}(t),B_{3}(t))$, function $f(x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}$ and the process $A(t) = f(B(t))$ and $t \in [1;\infty)...
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13 views

What does a 'pathwise' solution mean

I have not done very much stochastic calculus but know the basics. (Ito formula, Ito lemma, Stochastic Integral construction, the properties of B.M, Some properties of the stochastic integral, I have ...
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13 views

What is cylindrical Brownian Motion / Wiener Process

I have been given some reading on the Krylov-Bogoliubov Method for constructing invariant measures. An SDE in Hilbert space H is introduced as $$d(X)=b(X)dt + \sigma(X)dW $$ Where W is the ...
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35 views

Brownian Motion and Variance

If $(W_t) _t$ is a Brownian motion regarding to a filtration $(F_t) _t$ and the process $Z(t)$ is defined by $$ Z_t= \int_0^t W_s ds$$ What is $\operatorname{Var}(Z_t)$?
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Showing Brownian motion is a martingale

I am now self-studying stochastic calculus textbook. I have a question regarding the martingale property of Brownian motion. The book says: $$\mathbb E[B(t)-B(s)\mid F_s]=\mathbb E[B(t)-B(s)]$$...
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1answer
31 views

Showing independence of increments of a stochastic process

The textbook on stochastic calculus I am now reading says that if $X\colon [0,\infty)\times\Omega\rightarrow\mathbb R$ is a stochastic process such that $X(t)-X(s)\sim N(0,t-s)$ for all $t \geq s \...
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16 views

Fubini-Tonelli Brownian Motion

Can't understand the order of computing the next integral where B is real Brownian motion with $0\leq s \leq t$ by using Fubini-Tonelli: $$cov(X_s,X_t)=E\biggl( \int_0^sB_rdr\int_0^tB_udu \biggr)=E\...
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Is there a process that is not square-integrable but still gives rise to a martingale when integrated w.r.t. Brownian motion?

When an adapted process $X$ satisfies $\int_0^TX_t^2dt<\infty$ a.s. but not $E\int_0^TX_t^2dt<\infty$, the stochastic integral $\int_0^tX_sdB_s$, $0\le t\le T$, is only guaranteed to be a local ...
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54 views

Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$. Let ...
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Proof of expected present value of constant rebate

I am (so far unsuccessfully) looking for a specific proof for my research. The following is taken from Y.-K. Kwok. 'Mathematical models of financial derivatives' (Springer, 2008) on page 193 (original ...
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46 views

Conditional expected value of Ito integral

Let $0<\sigma_s<M$ be a bounded continuous stochastic process independent from a Brownian motion $W$. Let $\mathcal{F}_{\sigma}$ be the sigma-algebra generated by $\left\{\sigma_s|0\leq s\leq 1\...
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27 views

“Time”-Laplace transforms of Brownian motion transition and first hitting time laws

I was having a look at Girsanov's theorem and some of its applications. One of them is the possibility of affirming the following theorem. Given a probability space $(\Omega, \mathcal{F},P)$ equipped ...
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What is concretely the variance of a stochastic process ? For example, what mean $Var(B_t)=t$?

What is concretely the variance of a stochastic process ? Let for example $(B_t)$ the brownian motion in $\mathbb R$. What mean $Var(B_t)=t$ ? I know that $\mathbb E[B_t]=0$, so in the brownian motion ...
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Find $E[X(t)^2]$ given the partial differential equation… where $B(t), t\geq 0$ is a Brownian motion.

Find $E[X(t)^2]$ given, $dX(t) = (\frac{4(1+t)^2 - X(t)^2)}{8X(t)})dt + \frac{1}{2}X(t)dB(t)$, where $B(t), t\geq 0$ is a Brownian motion. I think I must apply $It\hat{o}$'s formula to $X(t)^2$ but I ...
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2answers
47 views

Proving $\int_a^\infty \frac{e^{-t}}{t}dt\geq -\log a - 1$

I am trying to prove the following inequaility: $$\int_a^\infty \dfrac{e^{-t}}{t}dt\geq -\log a - 1$$ for all $a$ positive and real. Now, I already try breaking the domain at 1 and using integration ...
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1answer
46 views

Calculating the conditional expectation of a Brownian Motion

Let $B$ be a BM, $t\in (0,1)$. Calculate $E(B_t|B_1)$. There is a hint: If we have a sequence of i.i.d. variables $(X_i)_{i\in \mathbb{N}}$ and the first moment of $X_1$ exists, then for $S_n:=\sum_{...
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Standardised Brownian Bridge integral

Let $Z(\lambda)$ denote a standardised Brownian Bridge $Z(\lambda)=(W(\lambda)-\lambda W(1))/\sqrt{\lambda(1-\lambda)}$. As $\sup_{\lambda \in [0,1]}Z(\lambda)\rightarrow \infty$, $\int_{0}^{1}Z(\...
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Conditioned Maximum of Brownian Motion (alternative metthod)

I have the following question related with this problem I was trying to solve this problem with the following reasoning: I use the join distribution of $M_t=x$ and $X_t=y$ $f_{M_t,X_t}(x,y)=\frac{2}{...
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Expected value of running minimum of BM

I need to calculate the following: $E[1_{\{Max_{0 \leq r \leq t} B(r) \ge 2\}}(-1 - min_{T_{2} \leq r \leq t}B(r))^{+}]$, where + means maximum of $0$ and the value in parenthesis. My attempt: The ...
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1answer
27 views

Does each PDE's have a strong link to a Particular Stochastic Process

Brownian Motion has a deep link to the heat PDE. Studying the dynamics of a particle moving (as if undergoing Brownian Motion) one can derive the heat equation. Also looking at the generator of the ...
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Understanding Itô's lemma

I am new to stochastic calculus and I am learning it during off-hours, so I am full of doubts that maybe someone more expert may dispel. Let's say I get the usual geometric Brownian process, where $...
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The density function of Bt on {Ta ∧ Tb > t}

enter image description here sources: Durrett Theory and Examples pag 394 From the graph I assumed that I had to prove that $\underset{n\to \infty }{lim}\mathbb{P}_x(T_a<T_b,B_t\in (\mathcal{P}...
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Product of two processes Differential

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
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Intuition behind reflexive principle of the brownian motion

Let $B$ a brownian motion and $S_t=\sup_{0\leq s\leq t}B_s$ for all $t\geq 0$. Let $a\geq 0$ and $b\in\mathbb R$, $b\leq a$. Then$$\mathbb P\{S_t>a, B_t<b\}=\mathbb P\{B_t>2a-b\}.$$ It looks ...
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28 views

first exit time from the interval (a,b)(Brownian motion)

Let ${\left({B}_{t}\right)}_{t\ge 0}$ un $ BM^{1} $( Brownian motion),$\; a<0<b $, denote by $ \displaystyle\tau =\mathrm{inf}\left\{t\ge 0:{B}_{t}\notin (a,b)\right\} $ the first exit time from ...
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first exit time from the interval (a,b)

Let ${\left({B}_{t}\right)}_{t\ge 0}$ un $ BM^{1} $( Brownian motion) and denote by ${m}_{t}=\underset{s\leq t }{\mathrm{inf}}{B}_{s}, \;$ ${M}_{t}=\underset{s\leq t }{\mathrm{sup}}{B}_{s}\,$, let $ a&...
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0answers
16 views

Position of 2D Brownian motion exiting quarter plane

Let $X_t = (X_t^1,X_t^2)$ a planar brownian motion without drift with independent components startet at $X_0 = (1,1)$ and $\tau := \inf \lbrace t\ge 0: X_t \notin (0,\infty)^2 \rbrace$ the first time ...
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28 views

Analytical solution of Langevin equation.

I am looking for analytical solution of Langevin equation where I will have a an external force ($a \cos \theta $), thermal force and drag force where I can able to find MSD.
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1answer
28 views

Exercise in Wiener processes

Exercise is from the book "Wstęp do rachunku prawdopodobieństwa" Jakubowski, Sztencel I need to show that family of random variables given by $$ Y_t = tW_{1/t},\,t>0 $$ And $Y_0=0$ where $W_t$ is ...