Abstract

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agent receives labor income which adjusts to financial market shocks in a path dependent way. This path dependency is the novelty of the model and leads to an infinite dimensional stochastic optimal control problem. We solve the problem completely and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional HJB equation, even if state constraints are present. To the best of our knowledge, this is the first infinite dimensional generalization of Merton's optimal portfolio problem for which explicit solutions can be found. The explicit solution allows us to study the properties of optimal strategies and discuss their financial implications.

Keywords

  1. stochastic functional (delay) differential equations
  2. optimal control problems in infinite dimension in state constraints
  3. second order Hamilton--Jacobi--Bellman equations in infinite dimension
  4. verification theorems and optimal feedback controls
  5. life-cycle optimal portfolio with labor income
  6. wages with path dependent dynamics (sticky)

MSC codes

  1. 34K50
  2. 93E20
  3. 49L20
  4. 35R15
  5. 91G10
  6. 91G80

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
J. M. Abowd and D. Card, On the covariance structure of earnings and hours changes, Econometrica, 57 (1989), pp. 411--445.
2.
A. Aksamit and M. Jeanblanc, Enlargement of Filtration with Finance in View, Springer Briefs Quant. Finance, Springer, New York, 2017.
3.
E. Barucci, F. Gozzi, and A. Swiech, Incentive compatibility constraints and dynamic programming in continuous time, J. Math. Econom., 34 (2000), pp. 471--508.
4.
A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd ed., Birkhäuser, Basel, 2007.
5.
L. Benzoni, P. Collin-Dufresne, and R. S. Goldstein, Portfolio choice over the life-cycle when the stock and labor markets are cointegrated, J. Finance, 62 (2007), pp. 2123--2167.
6.
E. Biffis, B. Goldys, C. Prosdocimi, and M. Zanella, A Pricing Formula for Delayed Claims: Appreciating the Past to Value the Future, Working paper, https://arxiv.org/abs/1505.04914, 2015.
7.
Z. Bodie, R. C. Merton, and W. F. Samuelson, Labor supply flexibility and portfolio choice in a life cycle model, J. Econom. Dynam. Control, 16 (1992), pp. 427--449.
8.
S. Bonaccorsi, Stochastic variation of constants formula for infinite dimensional equations, Stoch. Anal. Appl., 17 (1999), pp. 509--528.
9.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
10.
J. Y. Campbell and L. M. Viceira, Strategic Asset Allocation: Portfolio Choice for Long-Term Investors, Oxford University Press, Oxford, UK, 2002.
11.
A. Chojnowska-Michalik, Representation theorem for general stochastic delay equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 26 (1978), pp. 635--642.
12.
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Cambridge University Press, Cambridge, UK, 2014.
13.
M. Di Giacinto, S. Federico, and F. Gozzi, Pension funds with a minimum guarantee: A stochastic control approach, Finance Stoch., 15 (2010), pp. 297--342.
14.
O. Diekmann, S. M. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations-Functional, Complex, and Nonlinear Analysis, Springer, New York, 1995.
15.
W. T. Dunsmuir, B. Goldys, and C. V. Tran, Stochastic Delay Differential Equations as Weak Limits of Autoregressive Moving Average Time Series, Working paper, University of New South Wales, 2016.
16.
P. H. Dybvig and H. Liu, Verification theorems for models of optimal consumption and investment with retirement and constrained borrowing, Math. Oper. Res., 36 (2011), pp. 620--635.
17.
P. H. Dybvig and H. Liu, Lifetime consumption and investment: Retirement and constrained borrowing, J. Econom. Theory, 145 (2010), pp. 885--907.
18.
W. T. Dickens, L. Goette, E. L. Groshen, S. Holden, J. Messina, M. E. Schweitzer, J. Turunen, and M. E. Ward, How wages change: Micro evidence from the International Wage Flexibility Project, J. Econom. Perspectives, 21 (2007), pp. 195--214.
19.
K. J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, New York, 2006.
20.
G. Fabbri and S. Federico, On the infinite-dimensional representation of stochastic controlled systems with delayed control in the diffusion term, Math. Econ. Lett., 2 (2014), https://doi.org/10.1515/mel-2014-0011.
21.
G. Fabbri, F. Gozzi, and A. Swiech, Stochastic Optimal Control in Infinite Dimensions: Dynamic Programming and HJB Equations, Probab. Theory Stoch. Model. 82, Springer, New York, 2017.
22.
G. Freni, F. Gozzi, and N. Salvadori, Existence of optimal strategies in linear multisector models, Econom. Theory, 29 (2006), pp. 25--48.
23.
F. Gozzi and C. Marinelli, Stochastic optimal control of delay equations arising in advertising models, in Stochastic PDE's and Applications VII, Lect. Notes Pure Appl. Math. 245, Springer, New York, 2006, pp. 133--148.
24.
F. Guvenen, Learning your earning: Are labor income shocks really very persistent?, Amer. Econom. Rev., 97 (2007), pp. 687--712.
25.
F. Guvenen, An empirical investigation of labor income processes, Rev. Econom. Dynam., 12 (2009), pp. 58--79.
26.
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
27.
R. G. Hubbard, J. Skinner, and S. P. Zeldes, Precautionary saving and social insurance, J. Political Economy, 103 (1995), pp. 360--399.
28.
M. Jeanblanc, M. Yor, and M. Chesney, Mathematical Methods for Financial Markets, Springer, New York, 2009.
29.
I. Karatzsas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1991.
30.
I. Karatzsas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998.
31.
S. Khan, Evidence of nominal wage stickiness from microdata, Amer. Econom. Rev., 87 (1997), pp. 993--1008.
32.
M. Kocan and P. Soravia, A viscosity approach to infinite-dimensional Hamilton--Jacobi equations arising in optimal control with state constraints, SIAM J. Control Optim., 36 (1998), pp. 1348--1375.
33.
H. Le Bihan, J. Montornes, and T. Heckel, Sticky wages: Evidence from quarterly microeconomic data, Amer. Econom. J. Macroeconomics, 4 (2012), pp. 1--32.
34.
R. Lorenz, Weak Approximation of Stochastic Delay Differential Equations with Bounded Memory by Discrete Time Series, Ph.D. dissertation, Humboldt University, 2006.
35.
T. E. MaCurdy, The use of time series processes to model the error structure of earnings in a longitudinal data analysis, J. Econometrics, 18 (1982), pp. 83--114.
36.
C. Meghir and L. Pistaferri, Income variance dynamics and heterogeneity, Econometrica, 72 (2004), pp. 1--32.
37.
R. Merton, Continuous-Time Finance, Basil Blackwell, Oxford, UK, 1990.
38.
R. A. Moffitt and P. Gottschalk, Trends in the transitory variance of earnings in the United States, Econom. J., 112 (2002), pp. C68--C73.
39.
S.-E. A. Mohammed, Stochastic differential systems with memory: Theory, examples and applications, in Stochastic Analysis and Related Topics VI. Progress in Probability, L. Decreusefond, B. Øksendal, J. Gjerde, and A. S. Üstünel, eds., Progr. Probab. 42. Birkhäuser, Boston, MA, 1996, pp. 1--77.
40.
H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, New York, 2009.
41.
P. E. Protter, Stochastic Integration and Differential Equations, Springer, New York, 2005.
42.
M. Reiß, Nonparametric Estimation for Stochastic Delay Differential Equations, Ph.D. dissertation, Humboldt University, 2002.
43.
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, New York, 1998.
44.
M. Rosestolato, Path-dependent SDEs in Hilbert spaces, in Proceedings of the International Symposium on BSDEs, 2017, pp. 261--300.
45.
P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints, Differential Integral Equations, 12 (1999), pp. 275--293.
46.
K. Storesletten, C. I. Telmer, and A. Yaron, Cyclical dynamics in idiosyncratic labor market risk, J. Political Economy, 112 (2004), pp. 695--717.
47.
R. B. Vinter, A Representation of Solution to Stochastic Delay Equations, Computing and Control Department, Imperial College, 1975.
48.
J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999.
49.
Y. Zhu and G. Zhou, Technical analysis: As asset allocation perspective on the use of moving averages, J. Financial Econom., 92 (2009), pp. 519--544.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1906 - 1938
ISSN (online): 1095-7138

History

Submitted: 6 May 2019
Accepted: 23 April 2020
Published online: 8 July 2020

Keywords

  1. stochastic functional (delay) differential equations
  2. optimal control problems in infinite dimension in state constraints
  3. second order Hamilton--Jacobi--Bellman equations in infinite dimension
  4. verification theorems and optimal feedback controls
  5. life-cycle optimal portfolio with labor income
  6. wages with path dependent dynamics (sticky)

MSC codes

  1. 34K50
  2. 93E20
  3. 49L20
  4. 35R15
  5. 91G10
  6. 91G80

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.