NOTE: This question is related to the econometric method explored in the following two questions:

• Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector
• Example of the change of measure proposed in Hansen (2012)

QUESTION: Suppose that $X_t$ is an $n$-state Markov chain with transition probability matrix $\mathbb P$ and realized values given by $n$-dimensional coordinate vectors. Suppose that $\{W_{t+1} \}$ is an iid sequence of multivariate normally distributed random vectors. How would I represent an equation of the form $$ E[\exp(D'X_t + X_t' F W_{t+1}) e(X_{t+1}) \mid X_t = x] = \exp(\eta) e(x) $$ as an eigenvector problem for a matrix $\mathbb M$? How can I represent $\mathbb M$ in terms of the primitives of the problem?

NOTE: This question is related to the econometric method explored in the following two questions:

• Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector
• Example of the change of measure proposed in Hansen (2012)

QUESTION: Suppose that $X_t$ is an $n$-state Markov chain with transition probability matrix $\mathbb P$ and realized values given by $n$-dimensional coordinate vectors. Suppose that $\{W_{t+1} \}$ is an iid sequence of multivariate normally distributed random vectors. How would I represent an equation of the form $$ E[\exp(D'X_t + X_t' F W_{t+1}) e(X_{t+1}) \mid X_t = x] = \exp(\eta) e(x) $$ as an eigenvector problem for a matrix $\mathbb M$? How can I represent $\mathbb M$ in terms of the primitives of the problem?