Optimal investment strategy for an investor with partial information under exchange rate risk based on Malliavin calculus

Hongwei Liu; Tianjing Kan; Hongwei Liu; Tianjing Kan

doi:10.3934/math.20251339

AIMS Mathematics

2025, Volume 10, Issue 12: 30528-30543. doi: 10.3934/math.20251339

Previous Article Next Article

Research article

Optimal investment strategy for an investor with partial information under exchange rate risk based on Malliavin calculus

Hongwei Liu ^{1,2,3
,
,},
Tianjing Kan ¹

1.
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China
2.
Center for Applied Mathematics of Guangxi(GUET), Guilin 541004, China
3.
Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China

Received: 20 October 2025 Revised: 03 December 2025 Accepted: 15 December 2025 Published: 25 December 2025
MSC : 91G80, 93E20

This paper investigates the optimal investment decision for an investor with partial information under the criterion of maximizing the expected utility of terminal wealth. The domestic and foreign stock prices, as well as the exchange rate, are modeled as jump-diffusion processes with stochastic coefficients. By employing Malliavin calculus, we derive a sufficient and necessary condition for the optimal investment strategy in cross-border transactions. In some special cases, a closed-form expression is obtained. Finally, a numerical example is provided to illustrate the impacts of parameters $ \rho_1 $, $ \rho_2 $, and $ \sigma_R $ on the optimal investment strategy.
- optimal investment,
- partial information,
- exchange rate,
- jump-diffusion model,
- Malliavin calculus
Citation: Hongwei Liu, Tianjing Kan. Optimal investment strategy for an investor with partial information under exchange rate risk based on Malliavin calculus[J]. AIMS Mathematics, 2025, 10(12): 30528-30543. doi: 10.3934/math.20251339

Related Papers:

Abstract

This paper investigates the optimal investment decision for an investor with partial information under the criterion of maximizing the expected utility of terminal wealth. The domestic and foreign stock prices, as well as the exchange rate, are modeled as jump-diffusion processes with stochastic coefficients. By employing Malliavin calculus, we derive a sufficient and necessary condition for the optimal investment strategy in cross-border transactions. In some special cases, a closed-form expression is obtained. Finally, a numerical example is provided to illustrate the impacts of parameters $ \rho_1 $, $ \rho_2 $, and $ \sigma_R $ on the optimal investment strategy.

References

[1]	R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3 (1976), 125–144. https://doi.org/10.1016/0304-405X(76)90022-2 doi: 10.1016/0304-405X(76)90022-2
[2]	J. C. Cox, S. A. Ross, The valuation of options for alternative stochastic processes, J. Financ. Econ., 3 (1976), 145–166. https://doi.org/10.1016/0304-405X(76)90023-4 doi: 10.1016/0304-405X(76)90023-4
[3]	P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations, 1978,195–263.
[4]	G. D. Nunno, T. Meyer-Brandis, B. Øksendal, F. Proske, Optimal portfolio for an insider in a market driven by Lévy processes, Quant. Financ, 6 (2006), 83–94. https://doi.org/10.1080/14697680500467905 doi: 10.1080/14697680500467905
[5]	X. Peng, F. Chen, Deterministic investment strategy in a DC pension plan with inflation risk under mean-variance criterion, Probab. Eng. Inform. Sci., 36 (2022), 201–216. https://doi.org/10.1017/S026996482000025X doi: 10.1017/S026996482000025X
[6]	F. Chen, B. Li, X. Peng, Portfolio selection and risk control for an insurer with uncertain time horizon and partial information in an anticipating environment, Methodol. Comput. Appl. Probab., 24 (2022), 635–659. https://doi.org/10.1007/s11009-022-09941-6 doi: 10.1007/s11009-022-09941-6
[7]	A. T. T. Kieu, B. Øksendal, Y. Y. Okur, A Malliavin calculus approach to general stochastic differential games with partial information, In: Malliavin calculus and stochastic analysis: a festschrift in honor of David Nualart, Boston: Springer, 2013,489–510. https://doi.org/10.1007/978-1-4614-5906-4_22
[8]	O. M. Pamen, F. Proske, H. Binti Salleh, Stochastic differential games in insider markets via Malliavin calculus, J. Optim. Theory Appl., 160 (2014), 302–343. https://doi.org/10.1007/s10957-013-0310-z doi: 10.1007/s10957-013-0310-z
[9]	K. Kiyota, S. Urata, Exchange rate, exchange rate volatility and foreign direct investment, World Econ., 27 (2004), 1501–1536. https://doi.org/10.1111/j.1467-9701.2004.00664.x doi: 10.1111/j.1467-9701.2004.00664.x
[10]	C. Guo, X. Zhuo, C. Constantinescu, O. Pamen, Optimal reinsurance-investment strategy under risks of interest rate, exchange rate and inflation, Methodol. Comput. Appl. Probab., 20 (2018), 1477–1502. https://doi.org/10.1007/s11009-018-9630-7 doi: 10.1007/s11009-018-9630-7
[11]	C. Fei, W. Fei, Y. Rui, L. Yan, International investment with exchange rate risk, Asia-Pac. J. Account. Econ., 28 (2021), 225–241. https://doi.org/10.1080/16081625.2019.1569539 doi: 10.1080/16081625.2019.1569539
[12]	W. Wang, Q. Li, Q. Li, S. Xu, Robust optimal investment strategies with exchange rate risk and default risk, Mathematics, 11 (2023), 1550. https://doi.org/10.3390/math11061550 doi: 10.3390/math11061550
[13]	R. J. Elliott, J. B. Moore, L. Aggoun, Hidden Markov models: estimation and control, New York: Springer, 1995. https://doi.org/10.1007/978-0-387-84854-9
[14]	W. Pang, Y. Ni, X. Li, K. Cedric Yiu, Continuous-time mean-variance portfolio selection with partial information, Journal of Mathematical Finance, 4 (2014), 353–365. https://doi.org/10.4236/jmf.2014.45033 doi: 10.4236/jmf.2014.45033
[15]	P. Lakner, Optimal trading strategy for an investor: the case of partial information, Stoch. Proc. Appl., 76 (1998), 77–97. https://doi.org/10.1016/S0304-4149(98)00032-5 doi: 10.1016/S0304-4149(98)00032-5
[16]	I. Pikovsky, I. Karatzas, Anticipative portfolio optimization, Adv. Appl. Probab., 28 (1996), 1095–1122. https://doi.org/10.2307/1428166 doi: 10.2307/1428166
[17]	I. D. Baltas, N. E. Frangos, A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Appl. Stoch. Model. Bus., 28 (2012), 506–528. https://doi.org/10.1002/asmb.925 doi: 10.1002/asmb.925
[18]	Z. Liang, M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insur. Math. Econ., 65 (2015), 66–76. https://doi.org/10.1016/j.insmatheco.2015.08.008 doi: 10.1016/j.insmatheco.2015.08.008
[19]	G. Di Nunno, B. Øksendal, Optimal portfolio, partial information and Malliavin calculus, Stochastics, 81 (2009), 303–322. https://doi.org/10.1080/17442500902917979 doi: 10.1080/17442500902917979
[20]	X. Peng, Y. Hu, Optimal proportional reinsurance and investment under partial information, Insur. Math. Econ., 53 (2013), 416–428. https://doi.org/10.1016/j.insmatheco.2013.07.004 doi: 10.1016/j.insmatheco.2013.07.004
[21]	X. Peng, F. Chen, W. Wang, Optimal investment and risk control for an insurer with partial information in an anticipating environment, Scand. Actuar. J., 2018 (2018), 933–952. https://doi.org/10.1080/03461238.2018.1475300 doi: 10.1080/03461238.2018.1475300
[22]	G. D. Nunno, B. Øksendal, F. Proske, Malliavin calculus for Lévy processes with applications to finance, Heidelberg: Springer, 2008. https://doi.org/10.1007/978-3-540-78572-9

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)