Questions on the calculus of stochastic processes, or processes that have a random component.

learn more… | top users | synonyms

6
votes
1answer
701 views

Expectation of an integral w.r.t. Brownian Motion

I know the following statement: if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
1
vote
1answer
202 views

two question about poisson processes

I'm solving an exercise from a last year exam. Suppose we have an Poisson process $(N_t)$ with parameter $\lambda=\frac{1}{3}$ given with respect to a filtration $(\mathcal{F}_t)$. The first ...
2
votes
1answer
90 views

Fractional Brownian motion as integral, mean zero

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $$X(t)={1\over ...
2
votes
2answers
175 views

How do I derive the Gaussian Mixture distribution of an Ito Integral?

I have a question about the distribution of an Ito Integral. Consider the integral $$ \int_0^1 B_1(r) \mathrm{d}B_2(r), $$ where $B_1$ and $B_2$ are two independent standard Brownian motions. I am ...
1
vote
1answer
106 views

Joint Convergence and Donsker's Theorem

I have a question about joint convergence results derived from an FCLT (i.e., a Functional Central Limit Theorem). To motivate my question, consider the following setup: Let $y_t$ be a random walk ...
1
vote
1answer
100 views

Supermartingale with vanishing drift

Is a continuous supermartingale with vanishing drift already a martingale? In my concrete problem, I have a continuous nonnegative local martingale $ (X_t) $ on $ \left[0, T\right] $ which is bounded ...
2
votes
1answer
678 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
3
votes
0answers
67 views

Find a density function for the endpoint of this stochastic process

$(X_t, Y_t, Z_t)$ is a three-dimensional stochastic process described as follows: $X_t$ is a Brownian Motion. $Y_t = \int_0^t X_s ds$ $Z_t = \inf_{s \in [0, t]} X_s$ I would like to find a density ...
3
votes
1answer
150 views

Find the transition function of this stochastic process

Let $(X_t, Y_t)$ be a two-dimensional Markov stochastic process that runs on time interval $[t_0, t_f]$. Its infintesimal generator is described by the functions $\mu_X, \mu_Y, \sigma_X, \sigma_Y$. I ...
6
votes
0answers
354 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
2
votes
2answers
236 views

Basic stochastic integral

I am new to this stuff. Can some one explain how I could compute the stochastic integral of the form $\int_0^t W_sds$, where $W_t$ is Brownian process? Thanks!
3
votes
0answers
62 views

Is this a valid method for time-integrating a stochastic process?

I have a stochastic process $X_t$, and I have a function $a(x | t)$ that reflects my beliefs about the value of $X_t$ ($a$ is a density function in its first parameter). I am studying the properties ...
2
votes
1answer
75 views

how to derive this form using product formula

Suppose we have the following SDE $$dS(t) = S(t)(\mu(t)dt + \sigma(t)dW(t))=:S(t)dX(t)$$ where $W$ is a Brownian Motion and the processes $\mu,\sigma$ are well defined, such that the expression ...
1
vote
0answers
42 views

How do you convert an infintesimal generator of a Markov process to a transition function?

Suppose a continuous-time continuous-step Markov stochastic process $X_t$ has infinitesimal generator $\mu(x, t)$, $\sigma(x, t)$ ($\mu$, $\sigma$, and $X_0$ are known). How can we use this ...
3
votes
1answer
278 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
3
votes
1answer
278 views

Funny problem about stochastic integrals and Ito' s lemma

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
1
vote
2answers
356 views

Explicit solution of a SDE

I'd like an explicit formula as a function of $W_t$ (standard brownien motion) and $\lambda >0$ for the solution of the following SDE: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
1
vote
1answer
122 views

Upper bound for the $\sup$ of a martingale defined as a stochastic integral of a general continuous martingale

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
5
votes
2answers
255 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
2
votes
2answers
828 views

Expectation of Brownian motion Integral

I want to calculate $\mathbb{E} \left[\left(\int_0^tB_s\text{d}B_s\right)^3\right]$ where $B_t$ is a standard Brownian motion. Using Ito's formula for $f:\mathbb{R}\rightarrow\mathbb{R}$ with ...
1
vote
1answer
294 views

Ito Isometry for conditional expectations

Is Ito's isometry true for conditional expectations too? I mean, is it true that:$$\mathbb{E}\left[\left(\int_0^tX_sdB_s\right)^2\ |\ \mathcal{F}_t^B\right]=\mathbb{E}\left[\int_0^tX^2_sds\ |\ ...
2
votes
2answers
122 views

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
0
votes
2answers
404 views

Conditional Expectation of integral of Wiener process

Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$ where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
3
votes
1answer
130 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
3
votes
2answers
487 views

Show that this process is a martingale

Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$ is a martingale for $t\geq0$.
2
votes
1answer
38 views

Quasimartingale is Quasi-Dirichlet process

a paper I read states, that a Quasimartingale (an process $(X_t)_{t\in [0,T] }$ with $\mathbb E[|X_t|]<\infty$ for all $t\in [0,T]$, which suffices $$\sup_\Delta \sum^{n-1}_{j=0} \left\|\mathbb ...
4
votes
1answer
182 views

Show that $M_t$ is a Standard Brownian Motion

Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$ where $(B_t)_{t\geq0}$ is a Standard Brownian Motion. Show that $M$ is also a Standard Brownian Motion and compute ...
2
votes
0answers
68 views

How to calculate the following expectation

I have a problem to find the expectation of the following expression, $$E\left[W_T e^{\int_0^T(W_s)ds}\right].$$ Here, $W_T$ is a Brownian motion. Any suggestions as to how to proceed with it? Many ...
6
votes
0answers
316 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
1
vote
0answers
415 views

Show that this semimartingale is a local martingale

Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
3
votes
1answer
67 views

Checking a solution for a SDE

I want to show that the process $Y(t) = e^t \int_0^t e^{-s}dW(s)$ satisfies the following SDE: $dX(t) = X(t)dt + dW(t), \ \ t\geq 0 , \quad X(0) = 0$ I think the right approach is to use Ito's ...
1
vote
0answers
80 views

Fractional Brownian motion, selfsimilar

Let $0<H<1$. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and ...
2
votes
2answers
258 views

Martingale representation theorem

Trying to figure out how to solve problems on the 'form': Find a real number $z$ and a square integrable, adapted process $\psi(s,w)$ such that $$G(w) = z + \int \psi(s,w)\,dB_s(w)$$ for som ...
1
vote
1answer
31 views

a homework question about Levy air

I have a question in my homework: Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define $$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$$ Show that $$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
3
votes
1answer
193 views

Expectation of stopping time

Let $X_t$be the solution to the SDE: $dX_t=-X_tdt+dB_t$, $X_0=0$ Then $X_t$ is the Ornstein–Uhlenbeck process $X_t=e^{-t}\int_0^te^sdB_s$. I want to calculate $\mathbb{E}[e^\tau X_\tau]$ when ...
1
vote
2answers
245 views

Ito Process $\Longrightarrow$ continuous semimartingale

I know that the Ito integral is defined in general for continuous semimartingales. But it can also be defined only for Ito processes. My question is if every process $X_t$ satisying a SDE of the form ...
6
votes
1answer
263 views

Expected Value of Brownian motion using ito isometry

Find $$ E\ \left[\left(\int_{0}^T e^{s+W_s}dW_s \right)^2\right], $$ where $(W_s)$ is a Brownian motion. I tried to use Ito isometry to solve this question, but still not yet to find the right ...
2
votes
1answer
371 views

Table of Ito Integrals

Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of? Did a search but didn't come up with anything and was wondering if anyone knew of ...
0
votes
0answers
126 views

Can we apply Ito's formula?

Suppose that we are given the following processes: $B=(B_t)_{t\geq0}\ $ a standard Brownian motion starting at zero, $I=I_t=\int_0^t|B_s|^2ds,\ S=S_t=\sup_{0\leq s\leq t} B_s$ for $t\geq0$ and a ...
2
votes
0answers
138 views

Integral representation of fractional Brownian motion

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
votes
1answer
190 views

A question about Itô's representation for $\cos(B_T)$

According to Itô’s representation, any $\xi \in L_2(\Omega, F_T , P)$ has a unique representation: $ \xi = E(\xi) + \int_0^T H_s dBs$ where $(H_s)$ is an adapted process belonging to $L_2$. $B$ is a ...
2
votes
1answer
110 views

Long Range Dependence, Fractional Brownian Motion

A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $$\lim\limits_{n\to\infty}{\rho(n) \over ...
3
votes
0answers
101 views

Stochastic differential equation solution suggestion

Any suggestion on solving the stochastic differential equation \begin{align} dW(t) = d\widetilde{W}(t) + \left(\frac{\kappa - W(t)}{\tau-t} - \frac{1}{\kappa - W(t)}\right)dt \end{align} where ...
6
votes
3answers
732 views

Expected value of average of Brownian motion

For a standard one-dimensional Brownian motion $W(t)$, calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$ Note: I am not able to figure out how to approach this problem. All ...
5
votes
0answers
98 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
4
votes
1answer
194 views

Limiting distribution of Ornstein-Uhlenbeck process

Let $X_t = e^{-\lambda t} \left(X_0 + \int _0^t e^{\lambda u} dW_u\right)$ where $(W_u)_{u \geqslant 0}$ is a Wiener process, $X_0$ random variable of law $\nu$ and independent of $\int _0^t ...
0
votes
1answer
37 views

Question on weak convergence of random variables

Let $X_n, Y_n, X$ be real random variables such that $X_n \to X$ weakly and $\mathbb{P}_{Y_n} = N(0, 1/n)$ for all positive integers $n$. I am trying to prove that $X_n + Y_n \to X$ weakly as well. ...
0
votes
1answer
117 views

Weak convergence discrete space

Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most countably many integer values. Prove that that $X_n \to X$ weakly if and only if $\lim_{n \to \infty} P (X_n = j) = P(X = ...
0
votes
1answer
319 views

Show that this continuous local martingale is a martingale

We are given the following SDE: $$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$ and $$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$ We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
17
votes
2answers
826 views

Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...