A Stochastic Optimal Control of DC-Pension Fund Optimization Driven By Fractional Brownian motion With Hurst Parameter>α=1/2
Keywords:
DC-Pension Scheme, fractional Brownian motion, non-random optimal control, Stochastic Optimal Control.
Abstract
We considered the goal of the pension fund manager as to minimize the utility loss and maximizeutility at exit from the scheme, the noises involved in the dynamics of the process are fractional Brownian motions with short-range dependence. We use the DPP method to derive the HJB equation for the value function, and obtain the optimal strategy explicitly driven by fractional Brownian motion with Hurst parameter 
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References
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Society: Mathematical, Physical and Engineering Sciences, 2004, 460(2041):
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[27] Yan Li “ An Optimal Portfolio Problem Presented by fractional Brownian motion and
its applications. Wuham University Journal of Natural Sciences, 2022, vol. 27,
No.1,053-056. DOI: https://doi.org/10.1051/wujns/2022271053
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respect to (dt)aAppl. Math. Lett., 18,739-748 (2005)
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fractional Brownian motion with Hurst parameter. Stat. & Prob. Lett., 8,1013-1020
(2011).
[30] Jianwei G, Stochastic Optimal Control of DC Pension Fund under the fractional
Brownian motion. Appl. Math. Inf. Sci. 7, No 2, 571-578(2013).
[2] Blake D, Cairns AJ, Dowd K. Pensionmetrics II: stochastic pension plan design
during the distribution phase. Insur Math Econ.2003; 33(1):29–47
[3] Blake D, Wright D, Zhang Y. Optimal funding and investment strategies in defined
contribution pension plans under Epstein-Zin utility. Pension Institute Discussion Paper; 2008;No. PI-0808.
[4] Blake D, Wright D, Zhang Y. Target-driven investing: optimal investment strategies
in defined contribution pension plans under loss aversion. Pension Institute
Discussion. 2011; Paper No. PI-1112.
[5] Battocchio P, Menoncin F. Optimal pension management in a stochastic framework. Insur Math Econ. 2004; 34(1):79–95
[6] Boulier J-F, Huang S, Taillard G. Optimal management under stochastic interest
rates: the case of a protected defined contribution pension fund. Insur Math Econ.2001; 28(2):173–189
[7] Cairns A. Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time. Astin Bull.2000; 30(1):19–55
[8] He L, Liang Z. Optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework. Insur Math Econ. 2013; 53:
643–649
[9] Zhao H, Rong X, Zhao Y. Optimal excess-of-loss reinsurance and investment problem for an insurer withjump–diffusion risk process under the Heston model model. Insur Math Econ. 2013; 53(3):504–514
[10] Zheng X, Taksar M. A stochastic volatility model and optimal portfolio selection. Quant Financ. 2013; 13(10):1547–1558
[11] Guan G, Liang Z. Optimal management of DC pension plan in a stochastic interest rate and Stochastic volatility framework. Insur Math Econ. 2014; 57:58–66.
[12] Hao C, Xue-Yan L. Optimal consumption and portfolio decision with convertible bond in affine interest rate and Heston’s SV framework. Math Probl Eng 2016:1–12
[13] Kapur, S. and Orszag, J.M. : A portfolio Approach to Investment and Annuitization during Retirement. Proceedings of the Third International Congress on Insurance: Mathematics and Economics,London, 1999
[14] He L, Liang Z. Optimal assets allocation and benefit outgo policies of DC pension plan with compulsory conversion claims. Insur Math Econ. 2015; 61:227–234
[15] He L, Liang Z. Optimal asset allocation and benefit outgo policies of the DC pension plan during the decumulation phase. DEStech Trans Econ Manag (icem). 2016; https://doi.org/10.12783/dtem/icem2016/3952
[16] R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics, 51, 247-257 (1969).
[17] G. Deelstra, M. Grasselli and P.F. Koehl, Optimal design of the guarantee for
defined contribution funds. J. of Econ. Dyna. and Contr., 28, 2239-2260 (2004).
[18] J.F. Boulier, S. Huang and G.Taillard, Optimal management under stochastic interest
rates: The case of a protected defined contribution pension fund. Insur.: Math.and
Econo., 28, 173-189 (2001).
[19] G. Deelstra, M.Grasselli and P.F.Koehl, Optimal investment strategies in the presence of a minimum guarantee. Insur.: Math.and Econo., 33, 189-207 (2003).
[20] J. Cox and C.F.Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. of Econ. Theory, 49, 33-83 (1989).
[21] B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM review 10(1968), No. 4, 422-437
[22] C.Bender, An Itô formal for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter, Stochastic Process. Appl. 104(2003), No.1, 81-106, MR2003M: 60137
[23] Taqqu, M.S.(2003). Fractional Brownian motion and long-range dependence
(P.Doukhan, G. Oppenheim and M.S. Taqqu, eds) 5-38, Birkhäuser, Boston, MA. MR1956042
[24] Biagini, F., Hu, Y. , Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for
Brownian motion and Applications. Springer, London. MR2387368.
[25] Taqqu, M.S. “Benoȋt Mandelbrot and Fractional Brownian Motion” Statistical Science, 2013, vol28, No1, 131-134. https://doi:10.1214/12-STS389.
[26] Wallner N, Øksendal B, Sulem A, et al. An introduction to white noise theory and
Malliavan Calculus for fractional Brownian motion [J]. proceedings of the Royal
Society: Mathematical, Physical and Engineering Sciences, 2004, 460(2041):
347-372.
[27] Yan Li “ An Optimal Portfolio Problem Presented by fractional Brownian motion and
its applications. Wuham University Journal of Natural Sciences, 2022, vol. 27,
No.1,053-056. DOI: https://doi.org/10.1051/wujns/2022271053
[28] G. Jumarie, On the representation of fractional Brownian motion as an integral with
respect to (dt)aAppl. Math. Lett., 18,739-748 (2005)
[29] A.D.Mamadou and O. Youssef, A linear stochastic differential equation driven by a
fractional Brownian motion with Hurst parameter. Stat. & Prob. Lett., 8,1013-1020
(2011).
[30] Jianwei G, Stochastic Optimal Control of DC Pension Fund under the fractional
Brownian motion. Appl. Math. Inf. Sci. 7, No 2, 571-578(2013).
Published
2023-11-04
How to Cite
Ijioma, O. (2023). A Stochastic Optimal Control of DC-Pension Fund Optimization Driven By Fractional Brownian motion With Hurst Parameter>α=1/2. GPH - International Journal of Mathematics, 6(09), 01-13. https://doi.org/10.5281/zenodo.10070948
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Copyright (c) 2023 Onwukwe Ijioma
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