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A variational inequality approach to financial valuation of retirement benefits based on salary


  • Avner Friedman

    () (University of Minnesota, Department of Mathematics, Minneapolis, MN 55455, USA)

  • Weixi Shen

    () (Department of Mathematics, Fudan University, Shanghai 200433, China Manuscript)


We consider a pension plan with the option of early retirement, and paid benefits $\Psi (S,t)$ based on salary S at the time of retirement, but with guaranteed minimum; $S=S(t)$ is a Markov process. Denote by V(S,t) the financial value of the retirement benefits; its formal definition is given in (1.16). Then $\Psi (S,t) = V(S,t)$ at the end period T, while $\Psi (S,t)\leq V(S,t)$ if early retirement is exercised. We prove that V is the unique solution of a variational inequality, and that the set $\{\Psi = V\}$, which corresponds to the optimal time to retire, consists of either one or two continuous curves $S = S_i(t)$, depending on the parameters of the model.

Suggested Citation

  • Avner Friedman & Weixi Shen, 2002. "A variational inequality approach to financial valuation of retirement benefits based on salary," Finance and Stochastics, Springer, vol. 6(3), pages 273-302.
  • Handle: RePEc:spr:finsto:v:6:y:2002:i:3:p:273-302
    Note: received: January 2001; final version received: August 2001

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    Cited by:

    1. Moshe A. Milevsky & Kristen S. Moore & Virginia R. Young, 2006. "Asset Allocation And Annuity-Purchase Strategies To Minimize The Probability Of Financial Ruin," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 647-671.
    2. Moore, Kristen S., 2009. "Optimal surrender strategies for equity-indexed annuity investors," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 1-18, February.

    More about this item


    Retirement benefits; variational inequality; free boundary; stochastic differential equations; optimal time;

    JEL classification:

    • G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors


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