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Abstract: We establish existence of Predictable Forward Performance Processes (PFPPs) in conditionally complete markets, which has been previously shown only in the binomial setting. Our market model can be a discrete-time or a continuous-time model, and the investment horizon can be finite or infinite. We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation, which is the counterpart of the functional equation found in the binomial case. Although this integral equation has been partially studied in the existing literature, we provide a new solution method using the Fourier transform for tempered distributions. We also provide closed-form solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class. We apply our results to two special cases. The first one is the binomial market and is included to relate our work to the existing literature. The second example considers a generalized Black–Scholes model which, to the best of our knowledge, is a new result.
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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 $. 
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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 $.
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[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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