Abstract
We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimise their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and along with unspanned income, the market is incomplete. Set in a Brownian framework, the income is driven by a multidimensional diffusion and in particular includes mean-reverting dynamics. The equilibrium is characterised by a system of fully coupled quadratic backward stochastic differential equations, a solution to which is proved to exist under Markovian assumptions. We also show that the equilibrium allocations lead to Pareto-optimal allocations only in exceptional situations.
Similar content being viewed by others
References
Briand, P., Elie, R.: A simple constructive approach to quadratic BSDEs with or without delay. Stoch. Process. Appl. 123, 2921–2939 (2013)
Calvet, L.E.: Incomplete markets and volatility. J. Econ. Theory 98, 295–338 (2001)
Choi, J.H., Larsen, K.: Taylor approximation of incomplete Radner equilibrium models. Finance Stoch. 19, 653–679 (2015)
Christensen, P.O., Larsen, K.: Incomplete continuous-time securities markets with stochastic income volatility. Rev. Asset Pricing Stud. 4, 247–285 (2014)
Christensen, P.O., Larsen, K., Munk, C.: Equilibrium in securities markets with heterogeneous investors and unspanned income risk. J. Econ. Theory 147, 1035–1063 (2012)
Cochrane, J.H.: A mean-variance benchmark for intertemporal portfolio theory. J. Finance 69, 1–49 (2014)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Kardaras, C., Xing, H., Žitković, G.: Incomplete stochastic equilibria with exponential utilities close to Pareto optimality. Working paper (2015). Available online at https://arxiv.org/abs/1505.07224
Kazamaki, N.: Continuous Exponential Martingales and BMO. Lecture Notes in Mathematics, vol. 1579. Springer, Berlin (1994)
Pardoux, É., Peng, S.G.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55–61 (1990)
Telmer, C.I.: Asset-pricing puzzles and incomplete markets. J. Finance 48, 1803–1832 (1993)
Vayanos, D., Vila, J.L.: Equilibrium interest rate and liquidity premium with transaction costs. Econ. Theory 13, 509–539 (1999)
Wang, N.: Precautionary saving and partially observed income. J. Monet. Econ. 51, 1645–1681 (2004)
Wang, N.: Generalizing the permanent-income hypothesis: revisiting Friedman’s conjecture on consumption. J. Monet. Econ. 53, 737–752 (2006)
Weston, K.: Existence of a Radner equilibrium in a model with transaction costs. Math. Financ. Econ. 12, 517–539 (2018)
Xing, H., Žitković, G.: A class of globally solvable Markovian quadratic BSDE systems and applications. Ann. Probab. 46, 491–550 (2018)
Zhang, J.: Backward Stochastic Differential Equations. Springer, Berlin (2017)
Zhao, Y.: Stochastic equilibria in a general class of incomplete Brownian market environments. PhD Thesis, University of Texas at Austin (2012). Available online at https://repositories.lib.utexas.edu/bitstream/handle/2152/ETD-UT-2012-05-5064/ZHAO-DISSERTATION?sequence=1
Žitković, G.: An example of a stochastic equilibrium with incomplete markets. Finance Stoch. 16, 177–206 (2012)
Acknowledgements
The authors are grateful to Kasper Larsen for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author acknowledges the support by the National Science Foundation under Grant No. DMS-1606253 (2016–2018) and No. DMS-1908255 (2019–2022). The second author acknowledges the support by the National Science Foundation under Grant No. DSM-1815017 (2018–2021). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).
Rights and permissions
About this article
Cite this article
Weston, K., Žitković, G. An incomplete equilibrium with a stochastic annuity. Finance Stoch 24, 359–382 (2020). https://doi.org/10.1007/s00780-020-00415-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-020-00415-6