Abstract

Subsistence consumption and investment problems with bankruptcy are classic constrained stochastic optimal control problems in financial economics, where the consumption rate should be greater than a positive number and the investor faces a bankruptcy payment. We derive a novel asymptotic solution to this problem under the fast mean-reverting stochastic volatility model. We rigorously prove that the zeroth-order suboptimal pair of consumption and investment strategies leads to the first-order correction of the objective function in the form of $v^{(0)} +\sqrt{\epsilon}v^{(1)}$. In addition, this zeroth-order suboptimal consumption-investment pair is asymptotically optimal in a class of admissible trading strategy pairs.

Keywords

  1. optimal investment and consumption
  2. subsistence consumption
  3. stochastic volatility
  4. asymptotic optimality
  5. perturbation

MSC codes

  1. 91B28
  2. 35Q93
  3. 35C20

Get full access to this article

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

References

1.
G. Chacko and L.M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete marekts, Rev. Financial Stud., 18 (2005), pp. 1369--1402.
2.
N. Castan̈eda--Leyva and D. Hernández--Hernández, Optimal consumption-investment problems in incomplete markets with stochastic coefficients, SIAM J. Control Optim., 44 (2006), pp. 1322--1344, https://doi.org/10.1137/S0363012904440885.
3.
J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Sø lna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives, Cambridge University Press, Cambridge, 2011.
4.
J.-P. Fouque, C.S. Pun, and H.Y. Wong, Portfolio optimization with ambiguous correlation and stochastic volatilities, SIAM J. Control Optim., 54 (2016), pp. 2309--2338, https://doi.org/10.1137/15M1032533.
5.
J.-P. Fouque and R. Hu, Asymptotic optimal strategy for portfolio optimization in a slowly varying stochastic environment, SIAM J. Control Optim., 55 (2017), pp. 1990--2023, https://doi.org/10.1137/16M1066762.
6.
J.-P. Fouque, R. Sircar, and T. Zariphopoulou, Portfolio optimization and stochastic volatility asymptotics, Math. Finance, 27 (2017), pp. 704--745.
7.
J.-P. Fouque and R. Hu, Optimal portfolio under fast mean-reverting fractional stochastic environment, SIAM J. Finan. Math., 9 (2018), pp. 564--601, https://doi.org/10.1137/17M1134068.
8.
H. Hata, H. Nagai, and S.-J. Sheu, An optimal consumption problem for general factor models, SIAM J. Control Optim., 56 (2018), pp. 3149--3183, https://doi.org/10.1137/17M1135864.
9.
P. Hartman, Ordinary Differential Equations: 2nd ed., Classics Appl. Math. 38, SIAM, Philadelphia, 2002, https://doi.org/10.1137/1.9780898719222.
10.
R. Hu, Asymptotic Optimal Portfolio in Fast Mean-Reverting Stochastic Environments, preprint, https://arxiv.org/abs/1803.07720, 2018.
11.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 2001.
12.
I. Karatzas, J.P. Lehoczky, S.P. Sethi, and S.E. Shreve, Explicit solution of a general consumption/investment problem, Math. Oper. Res., 11 (1986), pp. 261--294.
13.
J.L. Koo, S.R. Ahn, B.L. Koo, H.K. Koo, and Y.H. Shin, Optimal consumption and portfolio selection with quadratic utility and a subsistence consumption constraint, Stoch. Anal. Appl., 34 (2016), pp. 165--177.
14.
R.C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Rev. Econ. Stat., 51 (1969), pp. 247--257.
15.
R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), pp. 373--413.
16.
R.C. Merton, Erratum: “Optimum consumption and portfolio rules in a continuous-time model” $[$J. Econom. Theory, 3 (1971), pp. 373--413$]$, J. Econom. Theory, 6 (1973), pp. 213--214.
17.
H. Nagai, H-J-B equations of optimal consumption-investment and verification theorems, Appl. Math. Optim., 71 (2015), pp. 279--311.
18.
J.-P. Fouque, G. Papanicolaou, R. Sircar, and K. Solna, Singular perturbations in option pricing, SIAM J. Appl. Math., 63 (2003), pp. 1648--1665, https://doi.org/10.1137/S0036139902401550.
19.
S.P. Sethi, M.I. Taksar, and E.L. Presman, Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy, J. Econom. Dynam. Control, 16 (1992), pp. 747--768.
20.
S.P. Sethi, M.I. Taksar, and E.L. Presman, Erratum: “Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy” $[$J. Econom. Dynam. Control, 16 (1992), pp. 747--768], J. Econom. Dynam. Control, 19 (1995), pp. 1297--1298.
21.
S.P. Sethi, Optimal Consumption and Investment with Bankruptcy, Kluwer, Boston, 1997.
22.
G. Shim and Y.H. Shin, Portfolio selection with subsistence consumption constraints and CARA utility, Math. Probl. Eng., (2014), 153793.

Information & Authors

Information

Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 977 - 1005
ISSN (online): 1945-497X

History

Submitted: 25 February 2019
Accepted: 17 September 2019
Published online: 17 December 2019

Keywords

  1. optimal investment and consumption
  2. subsistence consumption
  3. stochastic volatility
  4. asymptotic optimality
  5. perturbation

MSC codes

  1. 91B28
  2. 35Q93
  3. 35C20

Authors

Affiliations

Funding Information

Chinese University of Hong Kong https://doi.org/10.13039/501100004853
Education University of Hong Kong https://doi.org/10.13039/501100010410 : RG3/2018-2019R, SGA2018/19
Research Grants Council, University Grants Committee https://doi.org/10.13039/501100002920 : EdUHK18200114

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.

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.