Questions on the calculus of stochastic processes, or processes that have a random component.
2
votes
1answer
64 views
Are these two some kinds of generalized Ornstein–Uhlenbeck processes?
An Ornstein–Uhlenbeck process is
$$
d X_t = (\mu - X_t) dt + d W_t
$$
We try to build a model using some generalized Ornstein–Uhlenbeck processes.
The first one is
$$
d X_t = \exp(-|X_t- \mu|) ...
5
votes
1answer
131 views
Ito's Lemma and Brownian Motion
Show by using Ito's Lemma, for $k \geq 2$ the following result hold.
$$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$
where $W(t) = N(0,t)$ is standard Brownian motion.
I think ...
0
votes
0answers
36 views
Intuition: integration of function with respect to stochastic process
Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function.
With the above conditions, the following equality ...
1
vote
1answer
34 views
Joint distribution of Gaussian process and its derivative
Let $X(t)$ be a Gaussian process with zero mean and covariance function $B(t,s) = 1/(1+(t-s)^2)$. Let $X'(t)$ be its $L^2$-derivative. I am looking for the joint distribution of $X(t)$ and $X'(t)$.
...
1
vote
1answer
51 views
Backward martingale property of quadratic variation
Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
7
votes
1answer
125 views
generating set of predictable sigma algebra
I am solving an exercise in Rogers and Williams and want to ask if my solution is correct. Let me first introduce the notation. The space $b\mathcal{E}$ is the space of processes of the form
...
0
votes
0answers
45 views
Intuitive meaning of Lévy-Khintchine triplet
Let $\varphi$ be the characteristic function of an infinite divisible distribution. It can be expressed in the form $\varphi = e^\psi$ with
$$\psi(\lambda) = i \lambda a - \frac{\sigma^2 ...
5
votes
1answer
46 views
Is this stochastic integral well defined?
Motivation: I want to prove that the existence of a $\sigma$-martingale implies NFLVR (No Free Lunch With Vanishing Risk). This comes from arbitrage theory in mathematical finance and was proved by ...
3
votes
0answers
72 views
locally boundedness of RCLL and LCRL processes
The filrtation in this questions is assumed to fulfill the usual condition. Let $X$ be an adapted RCLL process and we look at $X_-$. It is well known that this process is predictable (hence ...
1
vote
0answers
28 views
Ito's formula for non smooth functions like Tanaka's formula
Does there exist an Ito's formula for function of Brownian Motion which are once differentiable but not twice differentiable like Tanaka's formula?
0
votes
3answers
68 views
Stochastic process with delta correlation in time
I am trying to learn stochastic calculus and when they talk about the Langevin equation they say that the correlation of the gaussian white noise (which i believe is the covariance between two random ...
0
votes
1answer
112 views
Solving Stochastic Differential Equations
Can anyone help me with the following SDE?
Solve the following stochastic differential equation:
$$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$
with $Y_0=0$.
Hint: Try a solution of the form $Z_tH_t$ where $Z_t = ...
3
votes
1answer
103 views
Variance of the Cox Ingersoll Ross model
Consider the Cox-Ingersoll-Ross (CIR) interest rate model: $\displaystyle d r_t = \kappa (\theta - r_t) \, d t + \sigma \sqrt{r_t} \,d W_t$ where $\kappa$, $\theta$, $\sigma$ are positive constants ...
4
votes
1answer
66 views
Computation of basic stochastic integral.
I am trying to compute the covariance of a 1 dimensional Ornstein-Uhlenbeck process $dx_t=-\theta x_t dt+ \sigma dW_t$, $\theta>0$ and I am at the stage,
$$\text{Cov }(x_s,x_t)=\sigma^2 ...
1
vote
1answer
49 views
Brownian Motion and the Functional CLT
Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
2
votes
0answers
43 views
Negative moments of a functional of Wiener process
At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
6
votes
1answer
113 views
Very basic doubt about Itô's lemma
While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following
$$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$
I had some doubt concerning the application of ...
2
votes
1answer
53 views
Some preliminaries for the canonical construction of a Brownian Motion, help needed.
I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
0
votes
0answers
82 views
When are two operators simultaneously diagonalizable?
I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
1
vote
1answer
69 views
Joint density $X^2+Y^2$
Let's say we have a point $(x,y)$ in the unit circle.
I've read (without proof :( ) that the joint density of $z$, where $z^2=x^2+y^2$, is:
$$f_{X,Y}(x,y) = ...
1
vote
1answer
98 views
Distribution of integral with respect to Brownian motion
Let $B$ be a Brownian motion and define the complex sequence $(X(n))_{n\in \mathbb Z}$ as
$$ X(n) := \int^\pi_{-\pi} e^{inx} dB(\pi + x)$$
What is the distribution of $X(n), n\in \mathbb Z$?
0
votes
0answers
58 views
diffusion processes and Ito diffusion processes
If I am correct, a diffusion process is defined as a Markov process with a.s. continuous sample paths.
A Ito diffusion process is defined via a SDE. From Wikipedia:
A (time-homogeneous) Itō ...
2
votes
1answer
83 views
Haar basis on $L^2(0,1)$ - proof?
I have the following problem.
We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$,
$$f_{j,n}(t)=\left\{ ...
0
votes
2answers
40 views
Identity for exponential of Brownian motion using scaling relation
Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$.
I stumbled over the following identity:
$$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))]
\\=\mathbb ...
4
votes
1answer
126 views
Stopping time and Brownian motion (specific example)
Let $B$ be a Brownian motion. I want to show that
$$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$
is not a stopping time w.r.t. the standard filtration.
How can one intuitively see that this ...
1
vote
1answer
53 views
What is $1_{\{\tau_n>0\}}X^{\tau_n}$ process saying?
As title says, what is $1_{\{\tau_n>0\}}X^{\tau_n}$ process? I do have understanding of what stochastic processes are, but not sure what is this specific process saying.
-1
votes
1answer
69 views
what is F-previsible process? And what would be F?
What is F-previsible process? I tried to search in the Internet but I couldn't find it... Also what is F here?
context: http://en.m.wikipedia.org/wiki/Martingale_representation_theorem#section_2
0
votes
0answers
23 views
Mathematics courses during summer break in London. [closed]
I'm not sure if I should ask this question here:
I'm going to study an MSc which requires high level of mathematical knowledge specially stochastic calculus (mathematical finance).
Is anyone aware of ...
0
votes
0answers
32 views
Relation between diffusion coefficient and diffusion process?
In a SDE
$$
\mathrm{d} X_t = \mu(X_t,t)\, \mathrm{d} t + \sigma(X_t,t)\, \mathrm{d} B_t ,
$$
$\mu$ is called drift coefficient and $\sigma$ is called diffusion coefficient.
An Ito diffusion ...
0
votes
0answers
81 views
Ways to solve SDEs besides Ito's formula
I learn that Ito formula is often used to solve SDEs.
I was wondering if there are other ways to solve SDEs?
I searched in Oksendal's SDE (section 5.1), and Shreve's Stochastic Calculus in Finance, ...
1
vote
1answer
62 views
What is the rationale of solving SDE by Ito's formula?
When solving a SDE by Ito's formula, we have to find a function $f(t, X_t)$ of index $t$ and the process $X$ to be solved for.
I was wondering what is the criterion of choosing $f$? Is it to
make ...
0
votes
0answers
21 views
On the stochastic dominance between two multivariate Gaussians
Are there known sufficient conditions for one multivariate Gaussian distribution to stochastically dominate another, under different covariance matrices?
0
votes
1answer
47 views
Expectation of a stochastic exponential
In class a while ago we used the following simplification:
$$ \mathbb E \left[ \exp\left(\langle \boldsymbol a,\mathbf W_t\rangle \right) \right] \quad =\quad \exp\left(\frac12 |\boldsymbol a|^2 ...
2
votes
2answers
43 views
Questions regarding filtration - more information
So for stochastic process $X_k$, We can define probability space, and filtration $\mathcal{F}_k$.
As far as I know, as $\mathcal{F}$ is sigma algebra, filtration represents sequences of events that ...
0
votes
0answers
137 views
Are these processes martingales?
Determine and prove if the following processes $ Y(t) $ are martingales. Assume that $ X(t) $ is the standard Brownian Motion
$$ Y(t) = e^{\sigma X(t)-0.5\sigma^2t} $$
$$ Y(t) = e^{0.5t}\Bigg(1 - ...
0
votes
0answers
135 views
Analysis of Brownian Motion
The following tasks consider transformation an analysis of Brownian Motion.
For the proces $ Y(t) = -\theta \mu t + \sigma X(t) $ design an algebraic substitution to $ X(t) $ that removes the drift ...
0
votes
0answers
118 views
Geometric Brownian Motion
Consider asset price $S$ that evolves according to Geomtric Brownian Motion with constant $\mu$ and $\sigma$
$$dS = \mu Sdt + \sigma SdX$$
Show by the application of Itô's Lemma to function $\log S$ ...
0
votes
0answers
69 views
Definition of Ito process
About SDE, Wikipedia says:
A typical equation is of the form $$
\mathrm{d} X_t = \mu(X_t,t)\, \mathrm{d} t + \sigma(X_t,t)\, \mathrm{d} B_t , $$ where $B$ denotes a Wiener process. ... ...
0
votes
1answer
25 views
what does $X_{s-}$ mean in the integration by parts formula for the Ito integral?
The integration by parts formula for the Itō integral is
If $X$ and $Y$ are semimartingales then
$$
X_tY_t = X_0Y_0+\int_0^t X_{s-}\,dY_s + \int_0^t Y_{s-}\,dX_s + [X,Y]_t
$$
where $[X, ...
4
votes
0answers
88 views
Is Queueing Theory dead? [closed]
I was studying queueing theory for my class and noticed that we are now able to either solve with certainity most queiening problems or simulate them.
is queueing a dead research area?
I read this ...
0
votes
0answers
60 views
What are some open research problems in Stochastic Processes?
I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes.
Any examples or recent papers or similar would be appreciated.
The motivation for ...
0
votes
0answers
24 views
Coefficients in spectral representation of a stochastic process
I want to find the spectral representation of the weakly stationary process $X(t)$ with $\mathbb E[X(t)] \equiv 0$, i.e. the spectral process $Z(t)$ such that
$$X(t) = \int_{\mathbb R} e^{i\lambda ...
1
vote
1answer
60 views
Spectral representation of specific stochastic process
Let $\gamma_1,\gamma_2,\ldots$ be uncorrelated random variables with $\mathbb E[\gamma_k]=0, \mathbb E[\gamma_k^2]=c_k$ and $\sum_{k\geq 1} c_k < \infty$. Define
$$X(t) = \sum_{k=0}^\infty ...
10
votes
1answer
200 views
Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?
Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels.
Feynman-Kac formula is also a pde corresponding to a stochastic process ...
0
votes
0answers
36 views
The identity of two parameters derived via conditioning arguments
Suppose I have a random variable $X_1\in\mathbb{R}$ and a random vector $X_2\in\mathbb{R}^d$. Furthermore, there are two measurable functions $f_1$ and $f_2$, and two deterministic vectors $\theta_1, ...
2
votes
2answers
65 views
Distribution of $\int_0^t e^s dB(s)$
Consider $$\left(\int_0^t e^s dB(s)\right)_{t\in [0,1]}$$ where $B$ is a Brownian motion.
What is the distribution of this process?
Since $f(s) = e^s$ is continuously differentiable, $B$ is process ...
0
votes
2answers
47 views
Incorrect use of the scaling relation for Brownian motion?
I want to calculate
$$CoV\left(B_1,\int_0^1 B_t dt\right) = \int_0^1 CoV\left(B_t,B_1\right) dt= \int_0^1 \min(t,1)dt = 1/2$$
On the other hand, one could also use the scaling relation for Brownian ...
0
votes
0answers
41 views
Converting Fokker-Planck equation to some form
Fokker-Planck equation (one-dimension) is:
$$\tag{0} \frac{\partial}{\partial t}f(x,t) = -\frac{\partial}{\partial x}\left[\mu(x,t)f(x,t)\right] +
\frac{\partial^2}{\partial x^2}\left[
...
1
vote
1answer
91 views
Implication of Lévy-Khintchine theorem/representation
I have trouble understanding the use/implication of the Lévy-Khintchine theorem. One possible way to state it is the following:
The characteristic function $\varphi$ is infinitely divisible if and ...
1
vote
1answer
160 views
Different versions of Girsanov theorems?
I am reading two different versions of Girsanov theorem regarding change of measure to preserve Brownian motion.
Wikipedia has the following Girsanov theorem:
If $X$ is a continuous process and ...
