Long-Term Risk: An Operator Approach

• 成果类型: Article
• 署名作者: Hansen, Lars Peter; Scheinkman, Jose A.
• 署名单位: University of Chicago; Princeton University
• 刊物名称: ECONOMETRICA
• ISSN/ISSBN: 0012-9682
• DOI: 10.3982/ECTA6761
• 发表日期: 2009
• 页码: 177-234
• 关键词: Stochastic differential utility; CONTINUOUS-TIME PROCESSES; MARKOV-PROCESSES; SPECTRAL THEORY; consumption; STABILITY; RESOLUTION; criteria; chains; CHOICE

摘要

We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous-time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. The members of this family are indexed by the elapsed time between payoff and valuation dates, and they are necessarily related via a mathematical structure called a semigroup. We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale, and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state. We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk-return trade-off.