Assume we have a stochastic process $\{X_n\}_\mathbb{N}$ defined on some underlying probability space that takes values in another measurable space. One of the many definitions that I have seen of the Markov property is as follows:
The process has the Markov property iff for arbitrary $n > s$ and $A$ measurable set
$\mathbb{P}(X_n \in A| \: \sigma(X_1,\dots,X_s)) = \mathbb{P}(X_n \in A| \: \sigma(X_s)) \tag{1}$
Is it possible to define the Markov property as
$$\mathbb{P}(X_n \in A| \: \sigma(X_1,\dots,X_{n-1})) = \mathbb{P}(X_n \in A| \: \sigma(X_{n-1}))$$
and then deduce that $(1)$ holds?
Since $\mathbb{P}(X_n \in A| \: \sigma(X_1,\dots,X_{n-1}))$ = $\mathbb{E}(\mathbb{1}_{X_n \in A}|\: \sigma(X_1,\dots,X_{n-1}))$ I have been trying to use properties of conditional expectation but have not been successful.
Thanks!
Best Answer
Yes, they are equivalent. Let's assume that
$$\mathbb{P}(X_n \in A \mid \sigma(X_1,\ldots,X_{n-1}) = \mathbb{P}(X_n \in A \mid \sigma(X_{n-1})) \tag{1}$$
holds for all measurable sets $A$ and all $n \in \mathbb{N}$. By a standard approximation procedure, this implies $$\mathbb{E}(f(X_n) \mid \sigma(X_1,\ldots,X_{n-1}) ) = \mathbb{E}(f(X_n) \mid \sigma(X_{n-1})) \tag{1'}$$ for any bounded Borel-measurable function $f$.
For fixed $n \in \mathbb{N}$ we prove $$\mathbb{P}(X_n \in A \mid \sigma(X_1,\ldots,X_{n-k})) = \mathbb{P}(X_n \in A \mid \sigma(X_{n-k})), \qquad A \in \mathcal{A}, \tag{2}$$ by induction over $k=1,\ldots,n$.
Base: For $k=1$ this is nothing but $(1)$.
Inductive step: Assume that $(2)$ holds for some $k=1,\ldots,j$; we have to show that $(2)$ holds for $k=j+1$. By the tower property of conditional expectation, we have
$$\mathbb{P}(X_{n} \in A \mid \sigma(X_1,\ldots,X_{n-j-1})) = \mathbb{E} \bigg[ \mathbb{P}(X_k \in A \mid \sigma(X_1,\ldots,X_{n-j})) \mid \sigma(X_1,\ldots,X_{n-j-1}) \bigg].$$
Using our induction hypothesis, we find
$$\mathbb{P}(X_{n} \in A \mid \sigma(X_1,\ldots,X_{n-j-1})) = \mathbb{E} \bigg[ \mathbb{P}(X_n \in A \mid \sigma(X_{n-j})) \mid \sigma(X_1,\ldots,X_{n-j-1}) \bigg].$$
By the factorization lemma, there exists a measurable function $f$ such that
$$\mathbb{P}(X_n \in A \mid \sigma(X_{n-j})) = f(X_{n-j}),$$
and so
$$\mathbb{P}(X_{n} \in A \mid \sigma(X_1,\ldots,X_{n-j-1})) = \mathbb{E}(f(X_{n-j}) \mid \sigma(X_1,\ldots,X_{n-j-1})).$$
It follows from $(1')$ that
$$\mathbb{P}(X_{n} \in A \mid \sigma(X_1,\ldots,X_{n-j-1})) = \mathbb{E}(f(X_{n-j}) \mid \sigma(X_{n-j-1})). \tag{3}$$
If we take on both sides the conditional expectation with respect to $\sigma(X_{n-j-1})$, then we find that
$$\mathbb{P}(X_n \in A \mid \sigma(X_{n-j-1})) = \mathbb{E}(f(X_{n-j}) \mid \sigma(X_{n-j-1})). \tag{4}$$
Combining $(3)$ and $(4)$ we get
\begin{align*} \mathbb{P}(X_{n} \in A \mid \sigma(X_1,\ldots,X_{n-j-1})) &\stackrel{(3)}{=} \mathbb{E}(f(X_{n-j}) \mid \sigma(X_{n-j-1})) \\ &\stackrel{(4)}{=} \mathbb{P}(X_n \in A \mid \sigma(X_{n-j-1})), \end{align*}
i.e. $(2)$ holds for $k=j+1$.