Strictly positive multiplicative functionals admit a unique decomposition into exponential trend, martingale, and transient eigenfunction-ratio components

Statement

For any strictly positive multiplicative functional $M$ of a continuous-time strong Markov process $X$, the canonical factorization

$M_t=e^{\rho t}{\hat{M}}_t\frac{\varphi (X_0)}{\varphi (X_t)}$

exists and (after a stochastic-stability selection) is unique up to a scaling of $\varphi$. Here $\rho$ is a real scalar, $\hat{M}$ is a martingale whose logarithm has stationary increments, and $\varphi$ is the strictly positive principal eigenfunction of the associated semigroup of pricing operators. The eigenvalue $\rho$ encodes the long-run growth (or decay) rate, the martingale $\hat{M}$ encodes permanent risk and defines a probability change under which $X$ is stationary, and $\varphi (X_0)/\varphi (X_t)$ encodes the transient state-dependent component.

Evidence summary

Hansen and Scheinkman (2009) prove existence in Corollary 6.1 (assuming an admissible principal eigenpair has been found) and uniqueness in Proposition 7.2 (the stochastically stable principal eigenfunction is the unique one whose distortion induces a stationary Markov process; its eigenvalue is the smallest admissible one). Existence of a principal eigenpair under primitive conditions (drift Lyapunov + irreducibility of a resolvent operator + boundedness) is established in Section 9 using techniques from Nummelin (1984) and Kontoyiannis-Meyn (2003). The companion $L^p$-approximation result (Proposition 7.4) gives the exponential convergence rate at which the rescaled semigroup approaches its limit, providing the quantitative content of the decomposition.

Conditions and scope

• Underlying process is a continuous-time strong Markov semimartingale on $D_0\subset {\mathbb{R}}^n$ with finitely many jumps in any finite interval. Excludes infinite-activity Lévy processes.
• Multiplicative functional $M$ is strictly positive almost surely.
• The distorted Markov process must be Harris recurrent and irreducible (Assumptions 7.3, 7.4) for uniqueness; this is the source of the “stochastic stability selection.”
• The candidate $\hat{M}$ must be a true martingale, not merely a local martingale; sufficient conditions are in Hansen-Scheinkman Appendix C.
• Constructive existence (Section 9) needs a drift Lyapunov condition ($\mathbb{A}V/V\le \underline{\mathbf{a}}$ for some $V\ge 1$) plus boundedness of a constructed candidate $\varphi =\mathbb{G}s$.

Counter-evidence

None known in the parameter regime covered by the assumptions. The paper itself flags two limitations that potentially restrict scope: (1) extension to general (infinite-activity) Lévy processes is open; (2) for genuinely non-Markov environments the decomposition does not directly apply, and Garman-style evolution semigroups must be used instead.

Linked ideas

(none yet — open for follow-up)

Open questions

• Does a meaningful version of the decomposition survive in non-Markov environments via auxiliary state augmentation?
• How does the convergence rate $\eta$ in the $L^p$ approximation depend on model primitives — and can it be computed at moderate horizons rather than just bounded?
• Does the decomposition extend to multiplicative functionals built from infinite-activity Lévy processes?