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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Levy process "that has only jumps"

I post this today as I'm looking to some caractérisation of a Levy processes "that has only jumps" but I didn't found anything on the web neither on the classic books I know about the ...
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Expected value of exponential Brownian motion

I want to prove that $$E[e^{2B_t}] = e^{2t}$$ where $B_t$ is a Brownian motion. I have been reading up on Mean of exponential Brownian motion but it does not show how the rest of the log-normal ...
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2 votes
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What is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
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1 vote
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Question about how to construct Brownian Motion. [duplicate]

My first question was answered within minutes by Sangchul Lee, but after a week of asking people I am stuck again, and let me describe. I will try to write out the standard definitions carefully. Let $...
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Explicit solution to the following stochastic differential equation

Problem: Consider the function $f:\mathbb{R}^d \to \mathbb{R}$ defined as $f(x) = \|x \| = \sqrt{\sum_{i=1}^d x_i^2}$ and let $(B_t)_{t \geq 0}$ be a $d$-dimensional Brownian motion. Let us ask if ...
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5 votes
1 answer
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Deriving the discretized equation of the Geometric Brownian Motion EDE

I am trying to obtain the discretized equation for the Geometric Brownian Motion EDE, $$ d S_{t}=\mu S_{t} d t+\sigma S_{t} \eta_tdt \tag{1} $$ I am looking for the discretization for the case where $\...
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1 answer
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Covariance matrix and the Cramer-Wold device

In Appendix B of Financial Statistics and Mathematical Finance, after the Cramer-Wold device (Theorem B.1.1) it is said: In particular, the Cramer–Wold technique tells us that $$X_n \xrightarrow{D} N(...
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Expected value of powers of Brownian Motion

At the moment I am following a uni course on Financial mathematics, the current subject is Brownian Motion. A subject I have now encountered a couple of times which I don't really understand is the ...
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1 answer
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Expectation of Brownian function.

$(W_t)t≥0$ be a standard one-dimensional Brownian motion Let $f : R → R$ be a given function and $0 ≤ s ≤ t$. Write down an expression for $E(f(W_t)|W_s = x)$ in terms of $ϕ(x) = Φ′(x)$ Where $Φ(x)$ ...
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Using the Girsanov theorem to construct a solution to an SDE

Suppose that $B$ is a standard Brownian motion and $b$ is a bounded, measurable function. Using the Girsanov theorem, construct a solution to the SDE $$dX_t=b(X_t)dt+dB_t.$$ I really don't know where ...
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3 votes
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Return of Brownian motion to zero

From chapter 4 of Bulinskiy & Shiryaev's Theory of Random Processes (ISBN 5-9221-0335-0): [Exercise 26] Let $W = \{W_t, t \geqslant 0 \}$ be a $m$-dimensional Brownian motion, where $m \geqslant ...
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Sampling distribution of GBM Maximum-Likelihood estimator

Given the geometric Brownian diffusion $$ X_t = \mu X_t dt + \sigma X_t d W_t$$ I learnt that its maximum likelihood estimators are the following as this web article suggests $$\hat \mu = \frac{\delta ...
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How to determine the reflecting horizontal hidden barrier for Ito diffusion

The first article at the link below talks about that the Bi-Directional Grid Constrained (BGC) stochastic processes for a random variable X over time t is one in which the further it departs from the ...
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Fundamental theorem of calculus equivalent for stochastic integrals

Let $B$ be a standard Brownian motion in one dimension and let $H$ be a continuous, adapted, bounded process. Prove that $$\frac{\int_t^{t+h}H_sdB_s}{B_{t+h}-B_t}\to H_t$$ in probability as $h\...
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1 answer
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Distribution of solution to SDE

Let $X_0$ be a standard normal random variable and suppose that $$dX_t=-\frac{1}{2}X_tdt+dB_t.$$ $X_0$ is independent of the Brownian motion. Find the distribution of $X_t$ for $t\geq0$ and find $\...
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2 votes
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Proving uniqueness of weak solution to SDE

This is the SDE: $$dX_t=\operatorname{sign}(X_t+1)dt + dB_t$$ This is a $\mathbb{Q}$-Brownian motion: $$W_t=B_t -\int_{0}^{t}\operatorname{sign}(B_s+1)ds$$ I've already shown that $B_t$ under measure $...
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The maximum of $W_t-t$ with respect to $t$ [duplicate]

I’m asked to calculate $$ P\left(\sup_{t\geq 0} W_t-t-1\geq 0\right)$$ Where $W_t$ is the Brownian motion. I tried to calculate it as the way I calculated the distribution of $\sup_{0\leq s\leq t} W_s$...
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Calculating the transition density function by finding the partial derivative of a conditional probability. [closed]

I've been given a process $Y_t=(1+t)B_t^2,t\ge0$ where $B_t$ is a standard Brownian motion and asked to find the transition density function $f(y,t|x,s)$. I've been instructed that $f(y,t|x,s)$ can be ...
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2 votes
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$exp$ of Half-Normal Distribution

I know that the Half-Normal Distribution has moments of all orders - that is, if $X\sim\mathcal{N}(\mu,\sigma)$, then, $$ E[|X|^p]<\infty $$ However, do we also have $$ E[e^{|X|}]<\infty $$ ? ...
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4 votes
2 answers
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A question about martingale and Brownian motion

It is well known that the Brownian motion $B=(B_t)_{t\ge 0}$ is a martingale with respect to its natural filtration $\mathscr{F}_t$ and the fixed probability measure $\mathbb{P}$, i.e. $$\mathbb{E}(...
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How to prove or calculate $E[\int_{t_{i-1}}^{t_i} e^{-μ({t_i}-s)}\sigma B_s |x_{t_{i-1}}]=0$?

$B_s$ is brownian motion. Because $\int_{t_{i-1}}^{t_i} e^{-μ({t_i} -s)}\sigma dB_s $ has a Brownian component, it is normally distributed with the mean zero according to Taylor & Karin 1998 's ...
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5 votes
1 answer
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Brownian motion with a stopping time

Let $x \geq 0,c<0,$ and a Brownian motion $(W_u)_{u}.$ Let $T:=\inf\{u \geq 0, B_u +cu\geq x\}.$ It follows that $Y:=\sup_{u \geq 0}(B_u+cu) \in ]0,\infty[.$ We want to verify that $\{Y \geq x\} \...
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Maximal inequality for Bessel process

Let $W = \{W_t, t \geq 0\}$ be an $m$-dimensional Brownian motion. The process $( \|W_t\|, \mathcal{F}_t )_{t ≥ 0}$ is then a Bessel process, where $\|\cdot\|$ is the Euclidean norm in $R^m$. Question....
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1 vote
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Brownian motion, submartingale

I'm trying to prove whether a given process is a submartingale. Let $W = \{W_t, t ≥ 0\}$ - m-dimensional Brownian motion. Prove that $( ||W_t||, F_t )_{t ≥ 0}$ is a submartingale (with a.s. continuous ...
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6 votes
1 answer
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Exercise 3.13 Paolo Baldi stochastic calculus: bounds on Brownian motion

I have been brushing up on stochastic analysis before the start of my PhD, and I encountered this exercise on the book "Stochastic Calculus with applications" by Paolo Baldi. The text is as ...
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1 vote
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Attempt to show that local martingale is a true martingale

Consider the process $X_t=e^{\frac{1}{2}t}\cos(B_t)$, where $B$ is a Brownian motion in $\mathbb{R}$. Using Ito's formula (unless I'm mistaken) implies that $$dX_t=-e^{\frac{1}{2}t}\sin(B_t)dB_t,$$ ...
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1 vote
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Passage time of a Brownian motion

Let $X=\{X_t\}_{t\geq 0}$ be a weak solution of $$ dX_t = \mu(t,X)\,dt+\sigma(t,X)\,dW_t $$ where $\mu$ and $\sigma$ are measurable and W is a Brownian motion. In this paper it is said that $$ \inf\{t&...
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Simulate the integral of the a brownian bridge process

Alright, so I am trying to simulate a value and its not working out the way I expect. I think I narrowed down my an issue to me simulating the integral of brownian bridge process. I know that 1000 ...
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2 votes
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A simple procedure to simulate multifractional Brownian motion paths

In a paper by Peltier and Vehel the multifractional Brownian motion (mBm) was defined for the first time, and they also give a procedure to simulate mBm sample paths. Briefly, mBm generalizes the ...
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-1 votes
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What are common ways to realistically simulate the stock market using historical market data? [closed]

I am currently using the FinRL library to try to automate Trading using Reinforcement Learning. However, I wanted to understand how FinRL simulates the stock market using historical data. I read here ...
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2 answers
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Expectation of product of brownian motion and stochastic integral

Let $f:[0,\infty)\to\mathbb{R}$ be a deterministic continuous function and $B$ a Brownian motion with $B_0=0$. I need to prove that $$\mathbb{E}\left(B_t\int_0^tf(s)dB_s\right)=\int_0^tf(s)ds.$$ I ...
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2 votes
1 answer
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Finding a weak solution to an SDE

Consider the SDE $$dX_t=\text{sign}(X_t)dB_t$$ with $X_0=0$ and where $$\text{sign}(x)=\begin{cases}-1&\text{if }x\leq0\\1&\text{if }x>0\end{cases}.$$ I am asked to find a weak solution to ...
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Showing that probability of BM being in part of a boundary is harmonic

Let $D$ be a domain in $\mathbb{R}^d$ and let $A$ be a measurable subset of its boundary $\partial D$. For $x\in D$, define $$\phi(x)=\mathbb{P}(X_T\in A)$$ where $(X_t)_{t\geq0}$ is a Brownian motion ...
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1 vote
1 answer
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Distribution of stochastic integral to stopping time

Let $B$ be a standard Brownian motion, and let $H$ be a continuous adapted process with $\int_0^\infty H_s^2ds=\infty$. For $\sigma>0$, let $T_\sigma=\inf\{t\geq0:\int_0^tH_s^2ds>\sigma^2\}$. ...
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An eigenvalue formula related to the transition semigroup of Brownian motion on the sphere

Let $(P_t)_{t\ge0}$ the transition semigroup of a standard Brownian motion on the $n$-dimensional unit sphere $S^n$ and let $f_i\colon S^n\to\Bbb R $ denote the function with $f_i(x)=x_i$ for all $x=(...
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Distribution of a rotationally-invariant $\alpha$-stable process

Let $B_{t^{2/\alpha}}$ be a rotionally invariant Brownian motion and let $Y$ be a nonnegative $\alpha/2$ stable process. I would like to check that $X_t:= \sqrt{Y}B_{t^{2/\alpha}}$ has the ...
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How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

$B_s$ is brownian motion. Because $\int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ has a Brownian component, it is normally distributed according to Taylor & Karin 1998 's Introduction to Stochastic ...
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2 votes
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Proving $E\bigg[\bigg(\int_a^b X_s dW_s \bigg)^2 \ \bigg\vert \ \mathcal{F}_a \bigg] =\int_a^b (X_s)^2 ds $

Suppose $(\mathcal{F}_t)_{t \geq0}$ is a filtration on a probability space and $W=(W_t)_{t \geq0}$ is a Brownian Motion with respect to this filtration. Let $(X_t)_{t \geq0}$ denote some ...
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Expectation of a brownian motion given two states [duplicate]

I have a following question about standard brownian motion. For $0 < s < t < u$. How could I derive $E(B_t|B_s,B_u)$? Here is my try: First, $$E(B_t|B_s,B_u) = E(B_t\cdot B_s|B_u) / P(B_s|B_u)...
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3 votes
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Supremum of Brownian motion increments

Let $W=(W(t))_{t\geq 0}$ be a Brownian motion. Consider the random variable $$Y(t):=\sup_{1\leq s\leq t}[W(s)-W(s-1)],$$ for some fixed instant $t\geq 0$. I am interested in this $Y(t).$ Could anyone ...
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0 votes
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Expect return of a standard Brownian motion

I'm working on the problem for a university project, I struggling for several days, I would post the same question with different coefficients from the problem I currently working on, so please show ...
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27 views

Variance of Stock Price - St - given by GBM

I am working on a problem, where I'm interested in computing the variance of the stock price in the next two years. Using the GBM notation for the stock price, I can write St as $ S_t = S_0e^{(\mu - \...
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2 votes
1 answer
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The stochastic differential of $\cos (B_t^{(1)}B_t^{(2)})$

Let ($B_1$, $B_2$) be a bi-dimensional correlated Brownian motions Calculate the stochastic differential equation of the process $\cos(B_{1,t}B_{2,t})$. Attempt: Let $X_t$ be the stochastic process ...
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3 votes
1 answer
42 views

Hitting time for 2-dimensional Brownian motion

Fix $a>0$ and let $(A_t^-)_{t\geq0}$ and $(A_t^+)_{t\geq 0}$ be two independent one-dimensional Brownian motions, starting from $-a$ and $a$, respectively. Set $$T=\inf\{t\geq0:A_t^-=A_t^+\}.$$ I ...
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2 votes
1 answer
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Surprisingly simple expected time for the "range" of a Brownian motion to extend beyond $a$ - is there a martingale method?

I will call the range $R(t)$ of a standard Brownian motion the difference between its maximum $M(t) = \max_{0\leq s \leq t} B(t)$ and its minimum $m(t) = \min_{0\leq s \leq t} B(t)$. That is, I am ...
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2 votes
1 answer
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Question from Exercise 3.3.17 in Karatzas and Shreve. Levy’s characterization theorem

I'm trying to do the following exercise from the mentioned book. As for $(M^{(1)}, M^{(2)})$ that's obvious from Levy's characterization. However, I'm now sure how to deal with $(M^{(1)},M^{(3)})$. I ...
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3 votes
1 answer
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Is Brownian motion a semimartingale?

I have read an article about semimartingales on Wikipedia and it says that: "A Brownian motion is a semimartingale". However, it is hard for me to find any proof of this statement, and I ...
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4 votes
1 answer
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Showing that a function of two brownian motions is a martingale.

Let $B$ be a standard Brownian motion, let $f$ be a smooth function taking values in $[a,b]$ where $0<a<b<\infty$ and assume that the derivative $f^\prime$ is bounded. For $t\in[0,1]$ and $x\...
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2 votes
1 answer
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A local martingale subtract half its quadratic variation tends to negative infinity

Let $M$ be a continuous local martingale with $M_0=0$ and $[M]_\infty=\infty$ almost surely. I am required to show that $M_t-\frac{1}{2}[M]_t\to-\infty$ almost surely as $t\to\infty$. Unfortunately I ...
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0 votes
0 answers
8 views

Showing that the product of a constant vector and a vector of independent Brownian motions is a Brownian motion

I'm trying to show that $\textbf{c}^T\textbf{Z}_t$ is a Brownian motion, where $\textbf{c}$ is a real constant vector and $\textbf{Z}_t$ a vector of independent Brownian motions. I can show that $c_1(...
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