Markov-Doob integrability

markov_function_theorem doob_transform

The Markov function theorem and Doob's transform can be combined to show certain "intergrability".

Let \(f: S \to T\) be a function. Let \(w: S \to [0, \infty)\) be a weight function, and \(h: T \to (0, \infty)\) its "generating function":

\[ h(y) = \sum_{x \in f^{-1} (y)} w(x) \qquad (-2) \]

Define stochastic kernel \(K: T \times S \to [0, 1]\) by

\[ K(y, x) = {w(x) \over h(y)} 1_{y = f(x)}. \qquad (-1) \]

These notations will play the same role as they appear in the Markov function theorem. We show that an equation implies both Doob transform and the intertwining relation in the Markov function theorem.

Lemma. Let \(X\) be Markov with state space \(S\) and transition kernel \(P\), and let \(Y_n := f(X_n)\). Then if there exists kernel \(A: T \times T \to [0, \infty)\) such that

\[ A(y, f(x')) w(x') = \sum_{x \in f^{-1}(y)} w(x) P(x, x') \qquad (0) \]

for all \(x' \in S\) and \(y \in T\), then

  1. \(h\) is \(A\)-harmonic. \[ \sum_{y'} A(y, y') h(y') = h(y). \]
  2. \(Y|_{X_0 \sim K(f(X_0), \cdot)}\) is a Markov process with transition kernel \[ Q(y, y') = h(y)^{-1} A(y, y') h(y'), \] which is the Doob \(h\)-transform of \(A\).
  3. \(\prob(X_n = x | Y_n = f(x), Y_{1 : n - 1}) = {w(x) \over h(f(x))}.\)

Proof. By plugging in \(K\) and \(Q\) and using (0), it is straightforward to verify the intertwining relation

\[ K P = Q K \qquad (1) \]

Therefore we can apply the Markov function theorem to show Items 2 and 3. For item 1, simply sum both sides over \(x'\) and use (-1) and the fact that \(P\) is a transition kernel. \(\square\)

The Markov-Doob integrability can be carried over to continuous-time case involving generators as well.