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# 数学代写|随机过程代写Stochastic Porcess代考|Stochastically Continuous Processes

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## 数学代写|随机过程代写Stochastic Porcess代考|Stochastically Continuous Processes

Assume now that $\mathscr{X}$ is a Hausdorff topological space. Let $\mathfrak{B}$ be the $\sigma$-algebra of Borel sets in $\mathscr{X}$, i. e. we choose the $\sigma$-algebra generated by the open sets in $\mathscr{X}$ as the $\sigma$-algebra $\mathfrak{B}$. A homogeneous Markov process $\left{\mathscr{F}, \mathscr{N}, \mathrm{P}x\right}$ in the phase space ${\mathscr{X}, \mathfrak{B}}$ is called stochastically continuous if for each $x \in \mathscr{X}$ and any neighborhood $U_x$ of point $x$ the relation $\lim {t \rightarrow 0} P\left(t, x, U_x\right)=1$ is satisfied (here $P(t, x, B)$ is the transition probability of the process). The transition probability which satisfies this condition is also called stochastically continuous.
Denote by $\mathscr{C}{\mathscr{X}}$ the set of continuous bounded functions $f(x)$ defined on $\mathscr{X}$ with the norm $|f|=\sup _x|f(x)|$; the space $\mathscr{C}{\mathscr{X}}$ is a Banach space. We verify certain properties of stochastically continuous processes.
I. For all $f \in \mathscr{C}x$ $$\lim {t \downarrow 0} \mathbf{T}t f(x)=f(x)$$ for each and every $x \in \mathscr{X}$. Indeed, choosing a neighborhood $U$ of point $x$ such that $|f(y)-f(x)|<\varepsilon$ for $y \in U$ we have \begin{aligned} & \varlimsup{t \downarrow 0}\left|\int P(t, x, d y) f(y)-f(x)\right| \ & \quad=\varlimsup_{t \downarrow 0}\left|\int_U P(t, x, d y)[f(y)-f(x)]+\int_{\mathscr{X}-U} P(t, x, d y) f(y)+P(t, x, \mathscr{X} \backslash U) f(x)\right| \leqslant \varepsilon, \end{aligned}
since $P(t, x, \mathscr{X} \backslash U) \rightarrow 0$.
II. For all $f \in \mathscr{C}x$ the function $\mathbf{T}_t f(x)$ is continuous from the right in $t$ for each $x$. This follows from the relation $$\lim {h \downarrow 0} \mathbf{T}{t+h} f(x)=\lim {h \downarrow 0} \int P(t, x, d y) \mathbf{T}h f(y)=\int P(t, x, d y) \lim {h \downarrow 0} \mathbf{T}h f(y)=\mathbf{T}_t f(x) ;$$ the Lebesgue bounded convergence theorem and property I were utilized in this relation. III. The resolvent $\mathbf{R}\lambda$ satisfies for all $f \in \mathscr{C}{\mathscr{X}}$ the relation $$\lim {\lambda \rightarrow+\infty} \lambda \mathbf{R}_\lambda f(x)=f(x) \text { for each } \quad x \in \mathscr{X}$$

## 数学代写|随机过程代写Stochastic Porcess代考|Feller Processes in Locally Compact Spaces

The set of functions $f \in \mathscr{C}{\mathscr{X}}$ for which (1) is satisfied forms a sub-space of $\mathscr{C}{\mathscr{X}}$. We denote this subspace by $\mathscr{L}$. It follows from (8) in Section 2 that $\mathbf{R}{\hat{\lambda}}\left(\mathscr{C}_x\right) \subset \mathscr{L}$. Assume that $\mathscr{L}$ is different from $\mathscr{C}{\mathscr{X}}$. Then there exists a linear functional $l(f)$ on $\mathscr{C}x$ such that $l(f)=0$ for all $f \in \mathscr{L}$ and $|l|=1$. However, every linear functional on $\mathscr{C}{\mathscr{X}}$ is of the form
$$l(f)=\int f(y) l(d y)$$
where $l(B)$ is a finite countably-additive function on the $\sigma$-algebra $\mathfrak{B}$ of Borel sets $B$ in $\mathscr{X}$. Since $\lambda \mathbf{R}\lambda f \in \mathscr{L}$ for all $f \in \mathscr{C}{\mathscr{X}}$, it follows that
$$0=\int \lambda \mathbf{R}\lambda f(y) l(d y)$$ Since $\lambda \mathbf{R}\lambda f$ is bounded and convergent to $f$ as $\lambda \rightarrow \infty$, it follows by the Lebesgue bounded convergence theorem that
$$\int f(y) l(d y)=l(f)=0$$
for all $f \in \mathscr{C}{\mathscr{X}}$, which contradicts the condition $|l|=1$. Hence $\mathscr{L}=\mathscr{C}{\mathscr{X}}$ and relation (1) is proved.
We now describe the generating operators of Feller processes on a compact space.

# 随机过程代写

## Matlab代写

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