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Predictable forward performance processes in complete markets

Angoshtari Bahman

Angoshtari Bahman. Predictable forward performance processes in complete markets[J]. JOURNAL OF MECHANICAL ENGINEERING. doi: 10.3934/puqr.2023007
Citation: Angoshtari Bahman. Predictable forward performance processes in complete markets[J]. JOURNAL OF MECHANICAL ENGINEERING. doi: 10.3934/puqr.2023007

Predictable forward performance processes in complete markets

doi: 10.3934/puqr.2023007
Funds: I would like to thank Thaleia Zariphopoulou and Xunyu Zhou for motivating my interest in this topic and for their invaluable feedback on my work. I am grateful for helpful comments and suggestions from Samuel Cohen, Sigrid Källblad, Gechun Liang, Moris Strub, and the referees. I acknowledge support through a start-up grant at the University of Miami. Part of this research was performed while I was visiting the Institute for Mathematical and Statistical Innovation (IMSI), which is supported by the National Science Foundation (Grant No. DMS-1929348).
  • Algorithm 1 Investment policy according to a PFPP
    Require: initial wealth $ x_0 $ and initial inverse marginal $ I_0=U_0' $
      $ {\bf{g}}\gets[\; ] $, $ X^*_0\gets x_0 $
      for $ n=0,1,\dots $ do
        Step 1 Observe $ {\boldsymbol{\Theta}}_{n+1} $. Set $ {\bf{g}}\gets {\bf{g}}\oplus {\boldsymbol{\Theta}}_{n+1} $ and $ \nu\gets $ the distribution of $ \rho_{n+1}|_{ {\bf{G}}_{n+1}= {\bf{g}}} $.
        Step 2 Find $ I_{n+1}\in {\cal{I}} $ satisfying $ \int_{ \mathbb{R}_+}I_{n+1}(\rho y) {\rm{d}} \nu(\rho)<+\infty $ and $ \int_{ \mathbb{R}_+}\rho I_{n+1}(\rho y) {\rm{d}}\nu(\rho) = $ $ I_n(y) $ for all $ y > 0 $. This is Problem 3.5.
        Step 3 Starting with wealth $ X^*_n $, invest over time period $ [n,n+1] $ to replicate the payoff $ X^*_{n+1}:= I_{n+1}\big(\rho_{n+1}I_n^{-1}(X^*_n)\big) $ at $ n+1 $. This is possible since the market is complete and $ \mathbb{E}[Z_{n+1}X^*_{n+1}| {\mathscr{F}}_n]=Z_nX^*_n $.
    下载: 导出CSV
    Algorithm 2 Investment policy according to a PFPP with CMIM functions
    Require: $ 0< \gamma_1\le \gamma_2 $ satisfying (4.15). Initial wealth $ x_0>0 $.
    Require: Initial risk-aversion measure $ m_0 $ satisfying $ \operatorname{supp}(m_0)\subset( \gamma_1, \gamma_2) $.
      $ {\bf{g}}\gets[\; ] $, $ X^*_0\gets x_0 $, $ I_0(y)\gets\int_{ \gamma_1}^{ \gamma_2} y^{-1/ \gamma} \mathrm{d} m_0( \gamma) $.
      for $ n=0,1,\dots $ do
       Step 1Observe $ {\boldsymbol{\Theta}}_{n+1} $. Set $ {\bf{g}}\gets {\bf{g}}\oplus {\boldsymbol{\Theta}}_{n+1} $ and $ \nu\gets $ the distribution of $ \rho_{n+1}|_{ {G}_{n+1}= {\bf{g}}} $.
       Step 2$ I_{n+1}(y)\gets\int_{ \gamma_1}^{ \gamma_2} y^{-1/ \gamma} \mathrm{d} m_{n+1}( \gamma) $ in which $ m_{n+1} $ is a measure equivalent to $ m_n $ with the Radon–Nikodym derivative $ \frac{ \mathrm{d} m_{n+1}}{ \mathrm{d} m_n}( \gamma)=\left( \int_{ \mathbb{R}_+} \rho^{1-\frac{1}{ \gamma}} \mathrm{d}\nu(\rho)\right)^{-1} $ for $ \gamma\in( \gamma_1, \gamma_2) $.
       Step 3Starting with wealth $ X^*_n $, invest over time period $ [n,n+1] $ to replicate the payoff $ X^*_{n+1}:= I_{n+1}\big(\rho_{n+1}I_n^{-1}(X^*_n)\big) $ at $ n+1 $. This is possible since the market is complete and $ \mathbb{E}[Z_{n+1}X^*_{n+1}| {\mathscr{F}}_n]=Z_nX^*_n $.
    下载: 导出CSV
  • [1] Angoshtari, B., Zariphopoulou, T. and Zhou, X. Y., Predictable forward performance processes: The binomial case, SIAM Journal on Control and Optimization, 2020, 58(1): 327−347. doi: 10.1137/18M1188653
    [2] Hörmander, L., The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd edition, Classics in Mathematics, Springer, Berlin, Heidelberg, 1990.
    [3] He, X. D., Strub, M. S. and Zariphopoulou, T., Forward rank-dependent performance criteria: Time-consistent investment under probability distortion, Mathematical Finance, 2021, 31(2): 683−721. doi: 10.1111/mafi.12298
    [4] Källblad, S., Black’s inverse investment problem and forward criteria with consumption, SIAM Journal on Financial Mathematics, 2020, 11(2): 494−525. doi: 10.1137/17M1143812
    [5] Liang, G., Strub, M. S. and Wang, Y., Predictable forward performance processes: Infrequent evaluation and robo-advising applications, arXiv: 2110.08900, 2021.
    [6] Mostovyi, O., Sîrbu, M. and Zariphopoulou, T., On the analyticity of the value function in optimal investment and stochastically dominant markets, arXiv: 2002.01084, 2020.
    [7] Musiela, M. and Zariphopoulou, T., Portfolio choice under dynamic investment performance criteria, Quantitative Finance, 2009, 9(2): 161−170. doi: 10.1080/14697680802624997
    [8] Musiela, M. and Zariphopoulou, T., Portfolio choice under space-time monotone performance criteria, SIAM Journal on Financial Mathematics, 2010, 1(1): 326−365. doi: 10.1137/080745250
    [9] Musiela, M. and Zariphopoulou, T., Initial investment choice and optimal future allocations under time-monotone performance criteria, International Journal of Theoretical and Applied Finance, 2011, 14(1): 61−81. doi: 10.1142/S0219024911006267
    [10] Rockafellar, R. T., Convex Analysis, Princeton Landmarks in Mathematics and Physics, Princeton University Press, 1970.
    [11] Strub, M. S. and Zhou, X. Y., Evolution of the Arrow–Pratt measure of risk-tolerance for predictable forward utility processes, Finance and Stochastics, 2021, 25(2): 331−358. doi: 10.1007/s00780-020-00444-1
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出版历程
  • 收稿日期:  2022-06-07
  • 录用日期:  2022-09-21
  • 网络出版日期:  2022-11-11

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