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# 数学代写|随机过程Stochastic Porcesses代考|AMATH562 When does Weak Markov Consistency Imply Strong Markov Consistency?

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## 数学代写|随机过程代写Stochastic Porcesses代考|When does Weak Markov Consistency Imply Strong Markov Consistency?

It is well known that if a process $X$ is a $\mathbb{P}$-Markov chain with respect to a filtration $\mathbb{H}$, and if it is adapted with respect to a filtration $\hat{\mathbb{H}} \subset \mathbb{H}$, then $X$ is a $\mathbb{P}$-Markov chain with respect to $\hat{\mathbb{H}}$. However, the converse is not true in general. Nevertheless, if $X$ is a $\mathbb{P}$-Markov chain with respect to $\hat{\mathbb{H}}$, and $\hat{\mathbb{H}}$ is $\mathbb{P}$-immersed in $\mathbb{H}$ (see Appendix A), then we can deduce from the martingale characterization of Markov chains that $X$ is also a $\mathbb{P}$-Markov chain with respect to $\mathbb{H}$.

Thus, if $\mathbb{F}^{X^i}$ is $\mathbb{P}$-immersed in $\mathbb{F}^X$ then the weak Markov consistency of $X$ with respect to $X^i$ will imply the strong Markov consistency of $X$ with respect to $X^i$. In the following theorem we demonstrate that in fact this property is equivalent to $\mathbb{P}$ immersion between $\mathbb{F}^{X^i}$ and $\mathbb{F}^X$, given that weak Markov consistency holds.

Theorem 3.18 Let $i=1,2$ and assume that $X$ satisfies the weak Markov consistency condition with respect to $X^i$. Then $X$ satisfies the strong Markov consistency condition with respect to $X^i$ if and only if $\mathbb{F}^{X^i}$ is $\mathbb{P}$-immersed in $\mathbb{F}^X$.

## 数学代写|随机过程代写Stochastic Porcesses代考|Strong Markov Consistency of Conditional Markov Chains

Let $X=\left(X^1, \ldots, X^n\right)$ be a multivariate $\left(\mathbb{F}, \mathbb{F}^X\right)$-CMC with values in $\mathcal{X}:=X_{k=1}^n \mathcal{X}_k$, where $\mathcal{X}_k$ is a finite subset of some vector space, $k=1, \ldots, n$. We recall that we consider CMCs on the finite time interval $[0, T]$.

Definition 4.1 1. Let us fix $k \in{1, \ldots, n}$. We say that the process $X$ satisfies the strong Markov consistency property with respect to $\left(X^k, \mathbb{F}\right)$ if, for every $x_1^k, \ldots, x_m^k \in \mathcal{X}k$ and for all $0 \leq t \leq t_1 \leq \cdots \leq t_m \leq T$, it holds that \begin{aligned} & \mathbb{P}\left(X{t_m}^k=x_m^k, \ldots, X_{t_1}^k=x_1^k \mid \mathcal{F}t \vee \mathcal{F}_t^X\right) \ & =\mathbb{P}\left(X{t_m}^k=x_m^k, \ldots, X_{t_1}^k=x_1^k \mid \mathcal{F}_t \vee \sigma\left(X_t^k\right)\right), \end{aligned}
or, equivalently, if $X^k$ is an $\left(\mathbb{F}, \mathbb{F}^X\right)$-CMC.

## 数学代写|随机过程代写Stochastic Porcesses代考|When does Weak Markov Consistency Imply Strong

Markov Consistency? (见附录A)，那么我们可以从马尔可夫链的鞅特征中推导出 $X$ 也是一个 $\mathbb{P}$-马尔可夫链关于 $\mathbb{H}$.

## MATLAB代写

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