2
votes
1answer
36 views

Markov processes on function spaces

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$? It seems Dirichlet Form is a ...
2
votes
1answer
61 views

Why this function is continuous?

Let $(\Omega,\Sigma,\mu)$ be a sample space and let $L^2= \lbrace f:\Omega \rightarrow R / \int f^2d\mu <\infty \rbrace$ be a Hilbert space. Let $L_n=L^2\times L^2 \times .... \times L^2$ ($n$ ...
4
votes
1answer
53 views

Understanding of Brownian Motion

My background is functional analysis rather than probability, but I would like to understand what is a Brownian motion. Below I'm giving my current understanding, can anyone verify whether I'm ...
1
vote
0answers
26 views

Convergence of cadlag functions to a continuous one

I want to prove a convergence result as simple as possible. Using a straight forward approach I can prove the result, but I'm 100% sure that there must be a much simpler (and shorter) argument using ...
1
vote
0answers
21 views

Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
2
votes
1answer
64 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a ...
3
votes
1answer
60 views

A question about extensions of Markov semigroups

I've cross-posted this to MO, if a reply appears on that post I'll update this one. Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on ...
4
votes
1answer
121 views

Kolmogorov continuity theorem for Banach space valued random processes

I am interested in the Kolmogorov continuity theorem. I would like to know if this theorem holds for Banach space valued random processes (probably separable Banach space). I cannot find a paper or a ...
3
votes
0answers
131 views

infinitesimal generator of reflecting Brownian motion

Suppose $f\in C_0^{\infty}([0,\infty))$ and $f'(0)=0$. I'm having trouble proving that $$\frac{1}{t}E_x[f(|W_t|)-f(x)]\to\frac{1}{2}f''(x)$$ uniformly on $[0,\infty)$ as $t\downarrow0$. Showing the ...
1
vote
0answers
50 views

Total set of functions in $L^2(\Omega)$

Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
2
votes
1answer
41 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
35 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) ∈ ...
11
votes
1answer
406 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
66 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
99 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 ...
5
votes
2answers
257 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
254 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
342 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
246 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
262 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
58 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
95 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
269 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
227 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 $\|M\|_{X_1}:=\|\sup_{0\le t\le T}|M_t|\|_{L^2(P)}$ ...
7
votes
1answer
209 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 ...
3
votes
1answer
599 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: ...
2
votes
1answer
685 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
92 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 ...
2
votes
2answers
270 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
355 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
116 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
175 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 ...