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# 数学代写|随机过程Stochastic Porcesses代考|MA53200 Strong Markov Consistency of Feller–Markov Families

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## 数学代写|随机过程代写Stochastic Porcesses代考|Strong Markov Consistency of Feller–Markov Families

We can conduct a more comprehensive study of strong Markov consistency in the case of time-homogeneous Feller-Markov families. In particular, we can prove in this case that, for each $j \in{1, \ldots, d}$, the strong $j$ th coordinate and the weak $j$ th coordinate of an $\mathcal{M} \mathcal{M} \mathcal{F H}$ that is strongly Markovian consistent with respect to $X^j$ are $\mathbb{R}$-Feller-Markov families. In the time-inhomogeneous case an analogous property was assumed rather than proved (cf. Propositions $2.21$ and 2.22).
Definition $2.23$ Let
$$\mathcal{M} \mathcal{M} \mathcal{F H}=\left{\left(\Omega, \mathcal{F}, \mathbb{F},\left(X_t\right)_{t \geq 0}, \mathbb{P}_x, P\right): x \in \mathbb{R}^n\right}$$

be a time-homogeneous Markov family. For all $t \geq 0$ we define an operator $T_t$ on $C_0\left(\mathbb{R}^n\right)$ by
$$T_t u(x)=\mathbb{E}x u\left(X_t\right)=\int{\mathbb{R}^n} u(y) P(x, t, d y), \quad x \in \mathbb{R}^n .$$
If $\left(T_t, t \geq 0\right)$ is a $C_0\left(\mathbb{R}^n\right)$-Feller semigroup then $\mathcal{M M} \mathcal{F H}$ is called a $\mathbb{R}^n$-FellerMarkov family. If, in addition, $C_c^{\infty}\left(\mathbb{R}^n\right) \subseteq D(A)$, where $A$ is the generator of $\left(T_t, t \geq\right.$ 0 ), then the family $\mathcal{M} \mathcal{M} \mathcal{F H}$ is called a nice $\mathbb{R}^n$-Feller-Markov family.

## 数学代写|随机过程代写Stochastic Porcesses代考|Definition and Characterization of Strong Markov Consistency

We begin with the specification of Definition $2.15$ to the present case of Markov chains.

Definition 3.1 1. Let us fix $i \in{1, \ldots, n}$. We say that the Markov chain $X=$ $\left(X^1, \ldots, X^n\right)$ satisfies the strong Markov consistency condition with respect to the coordinate $X^i$ if, for every $B \subset \mathcal{X}i$ and all $t, s \geq 0$, $$\mathbb{P}\left(X{t+s}^i \in B \mid \mathcal{F}t^X\right)=\mathbb{P}\left(X{t+s}^i \in B \mid X_t^i\right), \quad \mathbb{P} \text {-a.s., }$$
or, equivalently,
$$\mathbb{P}\left(X_{t+s}^i \in B \mid X_t\right)=\mathbb{P}\left(X_{t+s}^i \in B \mid X_t^i\right), \quad \mathbb{P} \text {-a.s., }$$
so that $X^i$ is a Markov chain in the filtration of $X$, i.e. in the filtration $\mathbb{F}^X$.

1. If $X$ satisfies the strong Markov consistency condition with respect to $X^i$ for each $i \in{1, \ldots, n}$ then we say that $X$ satisfies the strong Markov consistency condition.

## 数学代写|随机过程代写Stochastic Porcesses代考|Strong Markov Consistency of Feller-Markov Families

\left 缺少或无法识别的分隔符

$$T_t u(x)=\mathbb{E} x u\left(X_t\right)=\int \mathbb{R}^n u(y) P(x, t, d y), \quad x \in \mathbb{R}^n$$

## 数学代写随机过程代写Stochastic Porcesses代考|Definition and Characterization of Strong Markov Consistency

1. 如果 $X$ 唡是强妳可夫一致生条件 $X^i$ 每个 $i \in 1, \ldots, n$ 然扁我1说 $X$ 满足强尔可夫一致生 条件。

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