Tagged Questions
2
votes
1answer
22 views
What is the norm on the functional space used in defining the generator of a homogeneous Markov process?
From Wikipedia:
Given a strongly continuous semigroup $T : \mathbb{R}_+ \to L(B)$ on a Banach space $B$, its infinitesimal generator $A$ of a strongly continuous semigroup $T$ is defined as a ...
0
votes
0answers
32 views
regular/singular decomposition (Cramér-Wold) of weakly stationary stochastic process
Let $(Z(n))_{n\in \mathbb Z}$ be i.i.d. with $Z(1) \sim Uni([-2,2])$ and define
$$X(n)=Z(n)Z(n-1)$$
$X(n)$ is weakly stationary and I want to determine its decomposition in singular (deterministic) ...
0
votes
0answers
26 views
What are $C_b^2 (\mathbb R)$ and $C^{2,1} (\mathbb R × \mathbb R^+ )$?
From a note, for a diffusion process with its transition probability $P(, t|x, s)$,
Theorem 1. (Kolmogorov) Let $f (x) ∈ C_b (\mathbb R)$ and assume that
$$
u(x, s) := ∫ f (y)P (dy, t|x, s) ∈ ...
10
votes
1answer
208 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 ...
3
votes
0answers
54 views
Is there a canonical probability measure on smooth curves?
For continuous curves, we have Brownian motion giving the most natural probability measure. However, the sample paths of Brownian motion are almost surely terribly behaved (not of bounded variation, ...
0
votes
0answers
58 views
A different Markov property definition
In Shreve's Stochastic Calculus in Finance, the Markov property is defined as
Definition 2.3.6. Let $(\Omega,\mathcal F,P)$ be a probability space, let $T$ be a fixed positive number, and let ...
0
votes
0answers
64 views
Exercise (1) on Measure Theory
Let $m$ be a probability measure on $W$, so that $m(W) = 1$.
Let $f: \mathbb{R}^n \times W \rightarrow \mathbb{R}^n$ be locally bounded (in both arguments), and measurable in the second argument.
...
4
votes
1answer
147 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 ...
4
votes
0answers
152 views
Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?
The following is the definition of infinitesimal generator from Oksendal.
Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
1
vote
1answer
91 views
Cylindrical sigma algebra and continuous functions.
Consider the space $\mathbb R^{[0,1]}$ of all functions from $[0,1]$ to $\mathbb R$ and the cylindrical sigma algebra $\mathcal B$ on it. I know how to prove that $C[0,1]\not \in \mathcal B$. My ...
3
votes
1answer
173 views
getting the fundamental solution of Laplace's equation from the heat kernel
Since Laplace's equation is related to the steady state of heat flow problems, I'm guessing that there is a way to get from the heat kernel to the fundamental solution of Laplace's equation by letting ...
1
vote
1answer
130 views
non divergence form vs divergence form operator
Can the non divergence form operator $\mathcal{L}u= u_{xx}+u_{yy} + u_x=\Delta u + u_x$ be put in divergence form? In general, can any constant coefficient non divergence form operator be put into ...
1
vote
0answers
54 views
Equation Involving Bilateral Laplace Transform
Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense:
$$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
1
vote
0answers
67 views
Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal ...
2
votes
0answers
246 views
Can we construct a Hilbert space where the operator $A_u v := -\frac{1}{2} v'' + (vF + v\int_\mathbb{R} Su + u\int_\mathbb{R} Sv )'$ is symmetric?
It seems not to be a easy problem.
I'd like to know if one can define a pertinent Hilbert space where the operator
$$A_p v := -\frac{1}{2} v^{\prime\prime} + (vF + v\int_\mathbb{R} Sp + ...
2
votes
1answer
200 views
How to show that these spaces are Banach spaces
I want to show, that the following spaces are Banach spaces:
$X_1:=\{M=(M_t)_{0\le t \le T} ;\mbox{ M is an adapted RCLL process }\}$ with the norm $\|M\|_{X_1}:=\|\sup_{0\le t\le T}|M_t|\|_{L^2(P)}$ ...
5
votes
1answer
154 views
Reproducing Kernel Hilbert Space is dense?
Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous ...
2
votes
1answer
360 views
Dual of $C[0,1]$, Hilbert space and Riesz representation.
Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim:
...
1
vote
1answer
441 views
Hilbert transform of white noise
What is the Hilbert transform of a white noise $\xi(t)$?
By the Hilbert transform I mean:
http://mathworld.wolfram.com/HilbertTransform.html
Thank you.
3
votes
0answers
80 views
Different definitions of e-property for Markov-Feller chains
Let $X$ be a Polish space. We consider a stochastic kernel $P:X \times \mathcal{B}_X \to [0,1]$ and the Markov semigroup $(P^{\;n})_{n\geq1}$ of iterations of $P$, which satisfy the Chapman–Kolmogorov ...
0
votes
2answers
203 views
Karhunen-Loève / Mercer's theorem. What am I missing?
I'm looking at the eigenfunction expansion of Brownian motion on the interval [0,1]:
$$W_t = \sqrt{2} \sum_{k=1}^\infty Z_k \frac{\sin((k - \frac{1}{2}) \pi t)}{(k - \frac{1}{2}) \pi}.$$
One deduces ...
5
votes
1answer
315 views
Nested sets convergence
Define $\xi\in C^1([-1,1]\times[-1,1])$ such that
$$
\int\limits_{-1}^1 \xi(x,y)\,dy = 1
$$
for all $x\in[-1,1]$ and $\xi\geq 0$. Put $A_0 = [0,1]$ and $$A_{n+1} = \left\{x\in ...
4
votes
1answer
95 views
Closure of an invariant set
Consider a complete metric compact space $X$. For each $x\in X$ we define a probability measure $T(\cdot|x)$ over a Borel sigma-algebra $\mathcal{B}(X)$. We call a set $A\subset X$ invariant if ...
3
votes
2answers
160 views
Is there an analytic solution for the density function of this complex random variable?
The process below yields a distribution of "response times" (RT), and I'd like to know if there is an analytic solution to obtain the density function of this distribution.
An RT is recorded at the ...
