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An incomplete equilibrium with a stochastic annuity

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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.

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Acknowledgements

The authors are grateful to Kasper Larsen for helpful discussions.

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Correspondence to Kim Weston.

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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).

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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

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  • DOI: https://doi.org/10.1007/s00780-020-00415-6

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