This paper extends the long-term factorization of the stochastic discount factor introduced and studied by in discrete-time ergodic environments and by Hansen andScheinkman (2009) andHansen (2012) in Markovian environments to general semimartingale environments. The transitory component discounts at the stochastic rate of return on the long bond and is factorized into discounting at the long-term yield and a positive semimartingale that extends the principal eigenfunction of Hansen and Scheinkman * The authors thank Lars Peter Hansen (the co-editor) and the anonymous referees for their insightful comments and suggestions that helped improve the paper, and Jaroslav Borovicka, Peter Carr, Timothy Christensen (discussant) and Jose Scheinkman for stimulating discussions. This paper is based on research supported by the grant CMMI-1536503 from the National Science Foundation.† likuanqin2012@u.northwestern.edu ‡ linetsky@iems.northwestern.edu 1 (2009) to the semimartingale setting. The permanent component is a martingale that accomplishes a change of probabilities to the long forward measure, the limit of T -forward measures. The change of probabilities from the data generating to the long forward measure absorbs the long-term risk-return trade-off and interprets the latter as the long-term risk-neutral measure.

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the longmaturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping. * The authors thank Peter Carr for bringing Ross Recovery Theorem to their attention and for numerous stimulating discussions,

A. PROOFS FOR SECTION 3 WE FIRST RECALL SOME RESULTS about semimartingale topology originally introduced by Émery (1979) (see Czichowsky and Schweizer (2006), Kardaras (2013), andCuchiero andTeichmann (2015) for recent applications in mathematical finance). The semimartingale topology is stronger than the topology of uniform convergence in probability on compacts (ucp). In the latter case, the supremum in Eq. (2.1) is only taken over integrands in the form η t = 1 [0 s] (t) for every s > 0:The following inequality due to Burkholder is useful for proving convergence in the semimartingale topology in Theorem 3.1 (see Meyer (1972, Theorem 47, p. 50) for discrete martingales and Cuchiero and Teichmann (2015) for continuous martingales, where a proof is provided inside the proof of their Lemma 4.7).LEMMA A.1: For every martingale X and every predictable process η bounded by 1, |η t | ≤ 1, it holds that aP supfor all a ≥ 0 and t > 0.We will also use the following result (see Kardaras (2013, Proposition 2.10)).We will also make use of the following lemma.LEMMA A.3: Let (X n t ) t≥0 be a sequence of martingales such that X n t

We show that the martingale component in the long-term factorization of the stochastic discount factor due to Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) is highly volatile, produces a downward-sloping term structure of bond Sharpe ratios, and implies that the long bond is far from growth optimality. In contrast, the long forward probabilities forecast an upward sloping term structure of bond Sharpe ratios that starts from zero for short-term bonds and implies that the long bond is growth optimal. Thus, transition independence and degeneracy of the martingale component are implausible assumptions in the bond market.