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Continuous-time contracting problems with one-sided limited commitment under Knightian uncertainty

  • School of Mathematics and Physics,Anhui Polytechnic University,Wuhu 241000,China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.02.010
  • Received Date: 09 June 2018
  • Accepted Date: 25 October 2018
  • Rev Recd Date: 25 October 2018
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    The optimal contract design problem with one-sided limited commitment under Knightian uncertainty was studied.First，based on the agent’s endowment process and the consumption returned by the principal under Knightian uncertainty，the contract model was established with the agent’s one-side limited commitment which characterizes the maximization of the principal expected profit under the agent’s expected utility (the agent’s continuation value) being not lower than the agent’s outside option value (keeping the participation constraint).By using the dynamic programming principle under Peng’s sublinear expectation theory， the Hamilton-Jacobi-Bellman (HJB) equation of the principal’s value function on her maximal expected utility was derived.Next，using the sublinear theory， the weak and strong dual theorems and the verification theorem of optimal strategy were obtained.Finally，for an example of consumption，the numerical simulation for results and the corresponding economic analysis were provided.

    Abstract

    The optimal contract design problem with one-sided limited commitment under Knightian uncertainty was studied.First，based on the agent’s endowment process and the consumption returned by the principal under Knightian uncertainty，the contract model was established with the agent’s one-side limited commitment which characterizes the maximization of the principal expected profit under the agent’s expected utility (the agent’s continuation value) being not lower than the agent’s outside option value (keeping the participation constraint).By using the dynamic programming principle under Peng’s sublinear expectation theory， the Hamilton-Jacobi-Bellman (HJB) equation of the principal’s value function on her maximal expected utility was derived.Next，using the sublinear theory， the weak and strong dual theorems and the verification theorem of optimal strategy were obtained.Finally，for an example of consumption，the numerical simulation for results and the corresponding economic analysis were provided.
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    • References(21)
  • [1]
    COASE R H.The nature of the firm[J].Economica，1937，4(9): 386-405.
    [2]
    COCHRANE J H.A simple test of full consumption insurance[J].Journal of Political Economy，1991，99(2): 957-976.
    [3]
    TOWNSEND R M.Risk and insurance in rural Philippines[J].Econometrica，1997，65(1): 171-184.
    [4]
    ALVAREZ F，JERMANN U J.Efficiency，equilibrium，and asset pricing with risk of default[J].Econometrica，2000，68(4): 775-797.
    [5]
    LIGON E，THOMAS J P，WORRALL T.Informal insurance arrangements with limited commitment: Theory and evidence from village economies[J].Review of Economic Studies，2002，69(1): 209-244.
    [6]
    THOMAS J，WORRALL T.Self-enforcing wage contracts[J].Review of Economic Studies，1988，55(4): 541-554.
    [7]
    SANNIKOV Y.A continuous-time version of the principal-agent problem[J].Review of Economic Studies，2010，75(3): 957-984.
    [8]
    MIAO J，RIVERA A.Robust contracts in continuous time[J].Econometrica，2016，84(4): 1405-1440.
    [9]
    GROCHULSKIY B，ZHANG Y.Optimal risk sharing and borrowing constraints in a continuous-time model with limited commitment[J].Journal of Economic Theory，2011，146(3): 2356-2388.
    [10]
    WILLIAMS N.Persistent private information[J].Econometrica，2011，79(4): 1233-1275.
    [11]
    ZHANG Y.Characterization of a risk sharing contract with one-sided commitment[J].Journal of Economic Dynamics & Control，2013，37(4): 794-809.
    [12]
    MIAO J，ZHANG Y.A duality approach to continuous-time contracting problems with limited commitment [J].Journal of Economic Theory，2015，159: 929-988.
    [13]
    CHOQUET G.Theory of capacities[J].Ann Inst Fourier，1953，5: 131-295.
    [14]
    ELLSBERG D.Risk，ambiguity，and the savage axiom[J].Quaterly Journal of Economics，1961，7: 643-669.
    [15]
    SCHMEIDLER D.Subjective probability and expected utility without additivity[J].Econometrica，1989，57: 571-587.
    [16]
    CHEN Z，EPSTEIN L.Ambiguity，risk and asset returns in continuous time[J].Econometrica，2002，70: 1403-1443.
    [17]
    韩立岩，周娟.Knight不确定环境下基于模糊测度的期权定价模型[J].系统工程理论与实践，2007，27(12): 123-132.
    HAN Liyan，ZHOU Juan.Option pricing with fuzzy measures under Knightian uncertainty[J].Systems Engineering: Theory & Practice，2007，27(12): 123-132.
    [18]
    费为银，李淑娟.Knight不确定下带通胀的最优消费和投资模型研究[J].工程数学学报，2012，29(6): 799-806.
    FEI Weiyin,LI Shujuan.Study on optimal consumption and portfolio with inflation under Knightian uncertainty[J].Chinese Journal of Engineering Mathematics，2012，29(6): 799-806.
    [19]
    费为银，高贵云，梁勇.奈特不确定下带通胀的跨国直接投资问题[J].数学杂志，2016,36(3): 598-608.
    FEI Weiyin，GAO Guiyun，LIANG Yong.On study of foreign direct investment with inflation under ambiguity[J].Journal of Mathematics，2016,36(3): 598-608.
    [20]
    彭实戈.非线性期望的理论、方法及意义[J].中国科学: 数学，2017，47(10): 1223-1254.
    PENG Shige.Theory，methods and meaning of nonlinear expectation theory[J].Sci Sin Math，2017，47(10): 1223-1254.
    [21]
    FEI W Y，FEI C.Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion[EB/OL].[2018-06-01]. https://arxiv.org/abs/1309.0209.)
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    [1]
    COASE R H.The nature of the firm[J].Economica，1937，4(9): 386-405.
    [2]
    COCHRANE J H.A simple test of full consumption insurance[J].Journal of Political Economy，1991，99(2): 957-976.
    [3]
    TOWNSEND R M.Risk and insurance in rural Philippines[J].Econometrica，1997，65(1): 171-184.
    [4]
    ALVAREZ F，JERMANN U J.Efficiency，equilibrium，and asset pricing with risk of default[J].Econometrica，2000，68(4): 775-797.
    [5]
    LIGON E，THOMAS J P，WORRALL T.Informal insurance arrangements with limited commitment: Theory and evidence from village economies[J].Review of Economic Studies，2002，69(1): 209-244.
    [6]
    THOMAS J，WORRALL T.Self-enforcing wage contracts[J].Review of Economic Studies，1988，55(4): 541-554.
    [7]
    SANNIKOV Y.A continuous-time version of the principal-agent problem[J].Review of Economic Studies，2010，75(3): 957-984.
    [8]
    MIAO J，RIVERA A.Robust contracts in continuous time[J].Econometrica，2016，84(4): 1405-1440.
    [9]
    GROCHULSKIY B，ZHANG Y.Optimal risk sharing and borrowing constraints in a continuous-time model with limited commitment[J].Journal of Economic Theory，2011，146(3): 2356-2388.
    [10]
    WILLIAMS N.Persistent private information[J].Econometrica，2011，79(4): 1233-1275.
    [11]
    ZHANG Y.Characterization of a risk sharing contract with one-sided commitment[J].Journal of Economic Dynamics & Control，2013，37(4): 794-809.
    [12]
    MIAO J，ZHANG Y.A duality approach to continuous-time contracting problems with limited commitment [J].Journal of Economic Theory，2015，159: 929-988.
    [13]
    CHOQUET G.Theory of capacities[J].Ann Inst Fourier，1953，5: 131-295.
    [14]
    ELLSBERG D.Risk，ambiguity，and the savage axiom[J].Quaterly Journal of Economics，1961，7: 643-669.
    [15]
    SCHMEIDLER D.Subjective probability and expected utility without additivity[J].Econometrica，1989，57: 571-587.
    [16]
    CHEN Z，EPSTEIN L.Ambiguity，risk and asset returns in continuous time[J].Econometrica，2002，70: 1403-1443.
    [17]
    韩立岩，周娟.Knight不确定环境下基于模糊测度的期权定价模型[J].系统工程理论与实践，2007，27(12): 123-132.
    HAN Liyan，ZHOU Juan.Option pricing with fuzzy measures under Knightian uncertainty[J].Systems Engineering: Theory & Practice，2007，27(12): 123-132.
    [18]
    费为银，李淑娟.Knight不确定下带通胀的最优消费和投资模型研究[J].工程数学学报，2012，29(6): 799-806.
    FEI Weiyin,LI Shujuan.Study on optimal consumption and portfolio with inflation under Knightian uncertainty[J].Chinese Journal of Engineering Mathematics，2012，29(6): 799-806.
    [19]
    费为银，高贵云，梁勇.奈特不确定下带通胀的跨国直接投资问题[J].数学杂志，2016,36(3): 598-608.
    FEI Weiyin，GAO Guiyun，LIANG Yong.On study of foreign direct investment with inflation under ambiguity[J].Journal of Mathematics，2016,36(3): 598-608.
    [20]
    彭实戈.非线性期望的理论、方法及意义[J].中国科学: 数学，2017，47(10): 1223-1254.
    PENG Shige.Theory，methods and meaning of nonlinear expectation theory[J].Sci Sin Math，2017，47(10): 1223-1254.
    [21]
    FEI W Y，FEI C.Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion[EB/OL].[2018-06-01]. https://arxiv.org/abs/1309.0209.)
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