Stochastic Optimal Control Theory Applied in Finance

Ndondo Mboma Apollinaire; Pandi Ngumba Amanda

doi:doi:10.11648/j.mcs.20220704.11

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Stochastic Optimal Control Theory Applied in Finance

Ndondo Mboma Apollinaire, Pandi Ngumba Amanda

Published in Mathematics and Computer Science (Volume 7, Issue 4)

Received: 6 June 2022 Accepted: 4 July 2022 Published: 2 August 2022

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Abstract

In the field of investment, depending on their structures and in order to make the best decisions that are optimal, some companies are subject to some restrictions on their assets. And generally speaking, these constraints concern assets evolving in uncertainty. This paper focuses on studying a financial continuous-time Merton optimal investment problem in the case where there is a reallocation constraint with regard to the risky asset. Under this constraint, a certain rate is fixed such that the stock asset cannot be liquidated sooner than the rate. It is a stochastic control pure investment case for a large investor who faces a discounted infinite time horizon with utility function of only wealth, subject to a risk aversion coefficient. Our main goal is to characterise an optimal trading strategy for investors expecting high returns for low risks. We propose the dynamic programming method whose value function satisfies a nonlinear partial differential equation. Under homotheticity of the value function, a reduction of dimension is used in order to reduce the original two spatial dimensions problem to one dimension in a bounded domain. Numerical approximations are used to study the dynamic programming by finite difference discretisation and the convergence between the finite and the infinite time horizon problem is presented.

Published in	Mathematics and Computer Science (Volume 7, Issue 4)
DOI	10.11648/j.mcs.20220704.11
Page(s)	59-67
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Merton Problem, Stochastic Optimal Control, Dynamic Programming, Reallocation Constraint

References

[1]	R. C. Merton (1969) Lifetime portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics, pages 247-257.
[2]	M. J. Magill, and G. M. Constantinides (1976) Portfolio selection with transactions costs. Journal of Economic Theory, 13 (2): 245-263.
[3]	M. Dai, L. Jiang, P. Li, and F. Yi (2009) Finite horizon optimal investment and consumption with transaction costs. SIAM Journal on Control and Optimization, 48 (2): 1134-1154.
[4]	M. Dai, and Y. Zhong (2010) Penalty methods for continuous-time portfolio selection with proportional transaction costs. Journal of Computational Finance, pages 1-31.
[5]	T. Arun (2012) The merton problem with a drawdown constraint on consumption. arXiv preprint arXiv: 1210.5205.
[6]	G.S.Bakshi, andZ.Chen(1996)Thespiritofcapitalism and stock-market prices. The American Economic Review, pages 133-157.
[7]	S. P. Sethi, and G. L. Thompson (2000) What is optimal control theory? Springer.
[8]	A. M. Ndondo, S. Kasereka, S. F. Bisuta, K. Kyamakya, E. G. Doungmo, R. B. M. Ngoie (2021) Analysis, modelling and optimal control of covid-19 out break with three forms of infection in democratic republic of the congo, Results in Physics, 24 104096.
[9]	A. M. Ndondo, R. O. Walo, M. Yengo Vala-ki sisa (2016). Optimal control of a model of gambiense sleeping sickness in humans and cattle, American Journal of Applied Mathematics, 4 (5): 204-216.
[10]	F. Egriboyun, and H. M. Soner (2010) Optimal investment strategies with a reallocation constraint. Mathematical methods of operations research, 71 (3): 551-585.
[11]	M. H. Davis, and A. R. Norman (1990) Portfolio selection with transaction costs. Mathematics of operations research, 15 (4): 676-713.
[12]	N. Touzi (2004) Stochastic control problems, viscosity solutions and application to finance. In Publications of the Scuola Normale Superiore of Pisa. Citeseer.
[13]	J. Holth (2011) Merton’s portfolio problem, constant fraction investment strategy and frequency of portfolio rebalancing. Master’s thesis.
[14]	H. M. Soner (2004) Stochastic optimal control in finance. Scuola Normale Superiore.
[15]	N. O. Umeorah (2017) Pricing barrier and lookback options using finite difference numerical methods. Master’s thesis, North-West University (South Africa), Potchefstroom Campus.

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Ndondo Mboma Apollinaire, Pandi Ngumba Amanda. (2022). Stochastic Optimal Control Theory Applied in Finance. Mathematics and Computer Science, 7(4), 59-67. https://doi.org/10.11648/j.mcs.20220704.11

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ACS Style

Ndondo Mboma Apollinaire; Pandi Ngumba Amanda. Stochastic Optimal Control Theory Applied in Finance. Math. Comput. Sci. 2022, 7(4), 59-67. doi: 10.11648/j.mcs.20220704.11

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AMA Style

Ndondo Mboma Apollinaire, Pandi Ngumba Amanda. Stochastic Optimal Control Theory Applied in Finance. Math Comput Sci. 2022;7(4):59-67. doi: 10.11648/j.mcs.20220704.11

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@article{10.11648/j.mcs.20220704.11,
  author = {Ndondo Mboma Apollinaire and Pandi Ngumba Amanda},
  title = {Stochastic Optimal Control Theory Applied in Finance},
  journal = {Mathematics and Computer Science},
  volume = {7},
  number = {4},
  pages = {59-67},
  doi = {10.11648/j.mcs.20220704.11},
  url = {https://doi.org/10.11648/j.mcs.20220704.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20220704.11},
  abstract = {In the field of investment, depending on their structures and in order to make the best decisions that are optimal, some companies are subject to some restrictions on their assets. And generally speaking, these constraints concern assets evolving in uncertainty. This paper focuses on studying a financial continuous-time Merton optimal investment problem in the case where there is a reallocation constraint with regard to the risky asset. Under this constraint, a certain rate is fixed such that the stock asset cannot be liquidated sooner than the rate. It is a stochastic control pure investment case for a large investor who faces a discounted infinite time horizon with utility function of only wealth, subject to a risk aversion coefficient. Our main goal is to characterise an optimal trading strategy for investors expecting high returns for low risks. We propose the dynamic programming method whose value function satisfies a nonlinear partial differential equation. Under homotheticity of the value function, a reduction of dimension is used in order to reduce the original two spatial dimensions problem to one dimension in a bounded domain. Numerical approximations are used to study the dynamic programming by finite difference discretisation and the convergence between the finite and the infinite time horizon problem is presented.},
 year = {2022}
}

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TY  - JOUR
T1  - Stochastic Optimal Control Theory Applied in Finance
AU  - Ndondo Mboma Apollinaire
AU  - Pandi Ngumba Amanda
Y1  - 2022/08/02
PY  - 2022
N1  - https://doi.org/10.11648/j.mcs.20220704.11
DO  - 10.11648/j.mcs.20220704.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 59
EP  - 67
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20220704.11
AB  - In the field of investment, depending on their structures and in order to make the best decisions that are optimal, some companies are subject to some restrictions on their assets. And generally speaking, these constraints concern assets evolving in uncertainty. This paper focuses on studying a financial continuous-time Merton optimal investment problem in the case where there is a reallocation constraint with regard to the risky asset. Under this constraint, a certain rate is fixed such that the stock asset cannot be liquidated sooner than the rate. It is a stochastic control pure investment case for a large investor who faces a discounted infinite time horizon with utility function of only wealth, subject to a risk aversion coefficient. Our main goal is to characterise an optimal trading strategy for investors expecting high returns for low risks. We propose the dynamic programming method whose value function satisfies a nonlinear partial differential equation. Under homotheticity of the value function, a reduction of dimension is used in order to reduce the original two spatial dimensions problem to one dimension in a bounded domain. Numerical approximations are used to study the dynamic programming by finite difference discretisation and the convergence between the finite and the infinite time horizon problem is presented.
VL  - 7
IS  - 4
ER  -

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Author Information

Ndondo Mboma Apollinaire

Mathematics and Computer Science Department, Faculty of Sciences, University of Lubumbashi, Lubumbashi, DRC
Pandi Ngumba Amanda

Faculty of Technological Sciences, New Horizons University, Lubumbashi, DRC

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APA Style

Ndondo Mboma Apollinaire, Pandi Ngumba Amanda. (2022). Stochastic Optimal Control Theory Applied in Finance. Mathematics and Computer Science, 7(4), 59-67. https://doi.org/10.11648/j.mcs.20220704.11

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ACS Style

Ndondo Mboma Apollinaire; Pandi Ngumba Amanda. Stochastic Optimal Control Theory Applied in Finance. Math. Comput. Sci. 2022, 7(4), 59-67. doi: 10.11648/j.mcs.20220704.11

Copy | Download

AMA Style

Ndondo Mboma Apollinaire, Pandi Ngumba Amanda. Stochastic Optimal Control Theory Applied in Finance. Math Comput Sci. 2022;7(4):59-67. doi: 10.11648/j.mcs.20220704.11

Copy | Download

@article{10.11648/j.mcs.20220704.11,
  author = {Ndondo Mboma Apollinaire and Pandi Ngumba Amanda},
  title = {Stochastic Optimal Control Theory Applied in Finance},
  journal = {Mathematics and Computer Science},
  volume = {7},
  number = {4},
  pages = {59-67},
  doi = {10.11648/j.mcs.20220704.11},
  url = {https://doi.org/10.11648/j.mcs.20220704.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20220704.11},
  abstract = {In the field of investment, depending on their structures and in order to make the best decisions that are optimal, some companies are subject to some restrictions on their assets. And generally speaking, these constraints concern assets evolving in uncertainty. This paper focuses on studying a financial continuous-time Merton optimal investment problem in the case where there is a reallocation constraint with regard to the risky asset. Under this constraint, a certain rate is fixed such that the stock asset cannot be liquidated sooner than the rate. It is a stochastic control pure investment case for a large investor who faces a discounted infinite time horizon with utility function of only wealth, subject to a risk aversion coefficient. Our main goal is to characterise an optimal trading strategy for investors expecting high returns for low risks. We propose the dynamic programming method whose value function satisfies a nonlinear partial differential equation. Under homotheticity of the value function, a reduction of dimension is used in order to reduce the original two spatial dimensions problem to one dimension in a bounded domain. Numerical approximations are used to study the dynamic programming by finite difference discretisation and the convergence between the finite and the infinite time horizon problem is presented.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Stochastic Optimal Control Theory Applied in Finance
AU  - Ndondo Mboma Apollinaire
AU  - Pandi Ngumba Amanda
Y1  - 2022/08/02
PY  - 2022
N1  - https://doi.org/10.11648/j.mcs.20220704.11
DO  - 10.11648/j.mcs.20220704.11
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 59
EP  - 67
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20220704.11
AB  - In the field of investment, depending on their structures and in order to make the best decisions that are optimal, some companies are subject to some restrictions on their assets. And generally speaking, these constraints concern assets evolving in uncertainty. This paper focuses on studying a financial continuous-time Merton optimal investment problem in the case where there is a reallocation constraint with regard to the risky asset. Under this constraint, a certain rate is fixed such that the stock asset cannot be liquidated sooner than the rate. It is a stochastic control pure investment case for a large investor who faces a discounted infinite time horizon with utility function of only wealth, subject to a risk aversion coefficient. Our main goal is to characterise an optimal trading strategy for investors expecting high returns for low risks. We propose the dynamic programming method whose value function satisfies a nonlinear partial differential equation. Under homotheticity of the value function, a reduction of dimension is used in order to reduce the original two spatial dimensions problem to one dimension in a bounded domain. Numerical approximations are used to study the dynamic programming by finite difference discretisation and the convergence between the finite and the infinite time horizon problem is presented.
VL  - 7
IS  - 4
ER  -

Copy | Download