Figure 1.
Log-returns index of ITLUP (top) and UBI (bottom)
-
Previous Article
Network ANOVA random effects models for node attributes
- JDG Home
- This Issue
-
Next Article
Financial liquidity: An emergent phenomena
July
2020, 7(3): 225-237.
doi: 10.3934/jdg.2020016
Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices
| 1. | Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225, CP 11400, Montevideo, Uruguay |
| 2. | Instituto de Estadística, Facultad de Ciencias Económicas y de Administración, Universidad de la República, Eduardo Acevedo 1139, CP 11200, Montevideo, Uruguay |
One consequence of the fact that a large number of agents with different behaviors operate in financial systems is the emergence of certain statistical properties in some time series. Some of these properties contradict the hypotheses that are established in the traditional models of efficient market and portfolio optimization. Among them is the long-range dependence that is the objective of this work. The approach is proposed by fractional calculus, as a generalization of the classic approach to financial markets through semi-martingales. This paper study the existence of this property in variables dependent on the term structure curves of Uruguayan sovereign debt after the 2002 economic crisis.
Keywords:
Long-range dependence,
fractional calculus,
financial markets,
volatility,
sovereign debt.
Mathematics Subject Classification:
Primary:91G30;Secondary:60G22.
Citation:
Juan Kalemkerian, Andrés Sosa. Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices. Journal of Dynamics and Games,
2020, 7
(3)
: 225-237.
doi: 10.3934/jdg.2020016
![]()
References:
| [1] |
M. Alatriste-Contreras, J. Brida and M. Puchet, Structural change and economic dynamics: Rethinking from the complexity approach, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019).
doi: 10.3934/jdg.2019007. |
| [2] |
E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, (2006). |
| [3] |
W. Arthur, Complexity Economics: A Different Framework for Economic Thought, Complexity and the Economy, Oxford University Press, (2014). |
| [4] |
L. Bacheliere, Théorie de la Spéculation, Phd. Thesis, University of Paris, (1900). |
| [5] |
J. Beran, Statistics for Long Memory Processes, Chapman and Hall, New York, (1994). |
| [6] |
F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, (2006).
doi: 10.1007/978-1-84628-797-8. |
| [7] |
F. Black and M. Scholes,
The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (1973), 637-654.
doi: 10.1086/260062. |
| [8] |
T. Bollerslev,
Generalized autoregressive conditional heterocedasticity, History of Economic Thought Books, McMaster University Archive for Journal of Econometrics, 31 (1986), 307-327.
doi: 10.1016/0304-4076(86)90063-1. |
| [9] |
T. Bollerslev and H. Mikkelsen,
Modeling and pricing long memory in stock market volatility, Journal of Econometric, 73 (1996), 151-184.
doi: 10.1016/0304-4076(95)01736-4. |
| [10] |
J. Bouchaud and M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, (2000). |
| [11] |
F. Breidt, N. Crato and P. de Lima,
The detection and estimation of long memory in stochastic volatility, Journal of Econometrics, 83 (1998), 325-348.
doi: 10.1016/S0304-4076(97)00072-9. |
| [12] |
D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, (2006). |
| [13] |
P. Cheridito, H. Kawaguchi and M. Maejima,
Fractional Ornstein-Uhlenbeck Processes, Electronic Journal of Probability, 8 (2003), 1-14.
doi: 10.1214/EJP.v8-125. |
| [14] |
A. Chronopoulou and F. Viens,
Estimation and pricing under long-memory stochastic volatility, Annals of Finance, 8 (2012), 379-403.
doi: 10.1007/s10436-010-0156-4. |
| [15] |
R. Cont,
Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.
doi: 10.1080/713665670. |
| [16] |
R. Cont, Long range dependence in financial markets, Fractals in Engineering, Springer, (2005), 159–180.
doi: 10.1007/1-84628-048-6_11. |
| [17] |
F. Engle,
Autoregressive conditional heterocedasticity whit estimates of the variance of United Kingdom inflation, History of Economic Thought Books, McMaster University Archive for Econometrica, 50 (1982), 987-1008.
doi: 10.2307/1912773. |
| [18] |
E. Ghysels, A. Harvey and E. Renault,
Stochastic volatility, Handbook of Statistics, 14 (1996), 119-191.
doi: 10.1016/S0169-7161(96)14007-4. |
| [19] |
A. Gloter and M. Hoffmann,
Stochastic volatility and fractional Brownian motion, Stochastic Processes and their Applications, 113 (2004), 143-172.
doi: 10.1016/j.spa.2004.03.008. |
| [20] |
C. Granger and R. Joyeux,
An introduction to long memory time series models and fractional differencing, Journal of Time Series Analysis, 1 (1980), 15-29.
doi: 10.1111/j.1467-9892.1980.tb00297.x. |
| [21] |
W. Granger, S. Spear and Z. Ding, Statistics and Finance: An Interface. Stylized Facts on the Temporal and Distributional Properties of Absolute Returns: An Update, Imperial College Press, London, (2000), 97–120. |
| [22] |
T. Graves, R. Gramacy, N. Watkins and C. Franzke, A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980, Entropy, 19 (2017).
doi: 10.3390/e19090437. |
| [23] |
A. Harvey, Long memory in stochastic volatility, Forecasting Volatility in Financial Markets, (1998), 307–320.
doi: 10.1016/B978-075066942-9.50018-2. |
| [24] |
J. Hasslett and E. Raftery,
Space-time modelling with long memory dependence: Assessing Irelands win power resource, Journal of Applied Statistics, 38 (1989), 1-50.
doi: 10.2307/2347679. |
| [25] |
S. Heston,
A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, Society for Financial Studies, 6 (1993), 327-343.
doi: 10.1093/rfs/6.2.327. |
| [26] |
H. Hurst,
The long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 1116 (1951), 770-808.
|
| [27] |
B. Jacobsen,
Long term dependence in stock returns, Journal of Empirical Finance, 3 (1996), 393-417.
doi: 10.1016/S0927-5398(96)00009-6. |
| [28] |
J. Kalemkerian and J. R. León,
Fractional iterated Ornstein-Uhlenbeck processes, Latin American Journal of Probability and Mathematical Statistics, 16 (2019), 1105-1128.
doi: 10.30757/ALEA.v16-41. |
| [29] |
J. Kalemkerian, Parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes, preprint, arXiv: 2004.10369 (2020). |
| [30] |
A. Kolmogorov, Wienersche Spiralen und Einige Andere Interessante Kurven im Hilbertschen Raum., C. R. Acad. Sci. URSS, (1940), 115–118. |
| [31] |
S. London and F. Tohmé, Economic evolution and uncertainty: Transitions and structural changes, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019).
doi: 10.3934/jdg.2019011. |
| [32] |
L. Lima and Z. Xiao,
Is there long memory in financial time series?, Applied Financial Economics, 20 (2013), 487-500.
doi: 10.1080/09603100903459733. |
| [33] |
T. Lux,
Long-term stochastic dependence in financial prices: Evidence from the German stock market, Applied Economics Letters, 3 (1996), 701-706.
doi: 10.1080/135048596355691. |
| [34] |
B. Mandelbrot, Une Classe Processus Stochastiques Homothétiques a soi: Application a la loi Climatologique H. E. Hurst, C.R. Acad. Sci. Paris, (1965), 3274–3277. |
| [35] |
B. Mandelbrot and J. Wallis,
Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water Resour. Res., 5 (1969), 967-988.
|
| [36] |
R. Merton,
On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 22 (1974), 449-470.
doi: 10.1142/9789814759588_0003. |
| [37] |
M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, 2nd edition, Springer, (2005). |
| [38] |
H. Niu and J. Wang,
Volatility clustering and long memory of financial time series and financial price model, Digital Signal Processing, 23 (2013), 489-498.
doi: 10.1016/j.dsp.2012.11.004. |
| [39] |
W. Palma, Long Memory Time Series-Theory and Methods, John Wiley, Hoboken, NJ, (2007).
doi: 10.1002/9780470131466. |
| [40] |
W. Petty, Political Arithmetick, History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, (1690). |
| [41] |
M. Taqqu, M. Teverovsky and W. Willinger,
Estimators for long range dependence: An empirical study, Fractals, 3 (1995), 785-788.
doi: 10.1142/S0218348X95000692. |
| [42] |
P. Theodossiou,
Financial data and the skewed generalized T distribution, Management Science, 44 (1998), 1650-1661.
|
| [43] |
G. Uhlenbeck and L. Ornstein,
On the theory of Brownian Motion, Physical Review, 36 (1930), 823-841.
doi: 10.1103/PhysRev.36.823. |
| [44] |
R. Weron,
Estimating long-range dependence: Finite sample properties and confidence intervals, Physica A: Statistical Mechanics and its Applications, 312 (2002), 285-299.
doi: 10.1016/S0378-4371(02)00961-5. |
show all references
References:
| [1] |
M. Alatriste-Contreras, J. Brida and M. Puchet, Structural change and economic dynamics: Rethinking from the complexity approach, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019).
doi: 10.3934/jdg.2019007. |
| [2] |
E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, (2006). |
| [3] |
W. Arthur, Complexity Economics: A Different Framework for Economic Thought, Complexity and the Economy, Oxford University Press, (2014). |
| [4] |
L. Bacheliere, Théorie de la Spéculation, Phd. Thesis, University of Paris, (1900). |
| [5] |
J. Beran, Statistics for Long Memory Processes, Chapman and Hall, New York, (1994). |
| [6] |
F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, (2006).
doi: 10.1007/978-1-84628-797-8. |
| [7] |
F. Black and M. Scholes,
The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (1973), 637-654.
doi: 10.1086/260062. |
| [8] |
T. Bollerslev,
Generalized autoregressive conditional heterocedasticity, History of Economic Thought Books, McMaster University Archive for Journal of Econometrics, 31 (1986), 307-327.
doi: 10.1016/0304-4076(86)90063-1. |
| [9] |
T. Bollerslev and H. Mikkelsen,
Modeling and pricing long memory in stock market volatility, Journal of Econometric, 73 (1996), 151-184.
doi: 10.1016/0304-4076(95)01736-4. |
| [10] |
J. Bouchaud and M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge University Press, (2000). |
| [11] |
F. Breidt, N. Crato and P. de Lima,
The detection and estimation of long memory in stochastic volatility, Journal of Econometrics, 83 (1998), 325-348.
doi: 10.1016/S0304-4076(97)00072-9. |
| [12] |
D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2nd edition, Springer Verlag, (2006). |
| [13] |
P. Cheridito, H. Kawaguchi and M. Maejima,
Fractional Ornstein-Uhlenbeck Processes, Electronic Journal of Probability, 8 (2003), 1-14.
doi: 10.1214/EJP.v8-125. |
| [14] |
A. Chronopoulou and F. Viens,
Estimation and pricing under long-memory stochastic volatility, Annals of Finance, 8 (2012), 379-403.
doi: 10.1007/s10436-010-0156-4. |
| [15] |
R. Cont,
Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.
doi: 10.1080/713665670. |
| [16] |
R. Cont, Long range dependence in financial markets, Fractals in Engineering, Springer, (2005), 159–180.
doi: 10.1007/1-84628-048-6_11. |
| [17] |
F. Engle,
Autoregressive conditional heterocedasticity whit estimates of the variance of United Kingdom inflation, History of Economic Thought Books, McMaster University Archive for Econometrica, 50 (1982), 987-1008.
doi: 10.2307/1912773. |
| [18] |
E. Ghysels, A. Harvey and E. Renault,
Stochastic volatility, Handbook of Statistics, 14 (1996), 119-191.
doi: 10.1016/S0169-7161(96)14007-4. |
| [19] |
A. Gloter and M. Hoffmann,
Stochastic volatility and fractional Brownian motion, Stochastic Processes and their Applications, 113 (2004), 143-172.
doi: 10.1016/j.spa.2004.03.008. |
| [20] |
C. Granger and R. Joyeux,
An introduction to long memory time series models and fractional differencing, Journal of Time Series Analysis, 1 (1980), 15-29.
doi: 10.1111/j.1467-9892.1980.tb00297.x. |
| [21] |
W. Granger, S. Spear and Z. Ding, Statistics and Finance: An Interface. Stylized Facts on the Temporal and Distributional Properties of Absolute Returns: An Update, Imperial College Press, London, (2000), 97–120. |
| [22] |
T. Graves, R. Gramacy, N. Watkins and C. Franzke, A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980, Entropy, 19 (2017).
doi: 10.3390/e19090437. |
| [23] |
A. Harvey, Long memory in stochastic volatility, Forecasting Volatility in Financial Markets, (1998), 307–320.
doi: 10.1016/B978-075066942-9.50018-2. |
| [24] |
J. Hasslett and E. Raftery,
Space-time modelling with long memory dependence: Assessing Irelands win power resource, Journal of Applied Statistics, 38 (1989), 1-50.
doi: 10.2307/2347679. |
| [25] |
S. Heston,
A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, Society for Financial Studies, 6 (1993), 327-343.
doi: 10.1093/rfs/6.2.327. |
| [26] |
H. Hurst,
The long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 1116 (1951), 770-808.
|
| [27] |
B. Jacobsen,
Long term dependence in stock returns, Journal of Empirical Finance, 3 (1996), 393-417.
doi: 10.1016/S0927-5398(96)00009-6. |
| [28] |
J. Kalemkerian and J. R. León,
Fractional iterated Ornstein-Uhlenbeck processes, Latin American Journal of Probability and Mathematical Statistics, 16 (2019), 1105-1128.
doi: 10.30757/ALEA.v16-41. |
| [29] |
J. Kalemkerian, Parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes, preprint, arXiv: 2004.10369 (2020). |
| [30] |
A. Kolmogorov, Wienersche Spiralen und Einige Andere Interessante Kurven im Hilbertschen Raum., C. R. Acad. Sci. URSS, (1940), 115–118. |
| [31] |
S. London and F. Tohmé, Economic evolution and uncertainty: Transitions and structural changes, Journal of Dynamics and Games, American Institute of Mathematical Sciences, 6 (2019).
doi: 10.3934/jdg.2019011. |
| [32] |
L. Lima and Z. Xiao,
Is there long memory in financial time series?, Applied Financial Economics, 20 (2013), 487-500.
doi: 10.1080/09603100903459733. |
| [33] |
T. Lux,
Long-term stochastic dependence in financial prices: Evidence from the German stock market, Applied Economics Letters, 3 (1996), 701-706.
doi: 10.1080/135048596355691. |
| [34] |
B. Mandelbrot, Une Classe Processus Stochastiques Homothétiques a soi: Application a la loi Climatologique H. E. Hurst, C.R. Acad. Sci. Paris, (1965), 3274–3277. |
| [35] |
B. Mandelbrot and J. Wallis,
Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water Resour. Res., 5 (1969), 967-988.
|
| [36] |
R. Merton,
On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 22 (1974), 449-470.
doi: 10.1142/9789814759588_0003. |
| [37] |
M. Musiela and M. Rutkowski, Martingale Methods in Financial Modeling, 2nd edition, Springer, (2005). |
| [38] |
H. Niu and J. Wang,
Volatility clustering and long memory of financial time series and financial price model, Digital Signal Processing, 23 (2013), 489-498.
doi: 10.1016/j.dsp.2012.11.004. |
| [39] |
W. Palma, Long Memory Time Series-Theory and Methods, John Wiley, Hoboken, NJ, (2007).
doi: 10.1002/9780470131466. |
| [40] |
W. Petty, Political Arithmetick, History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, (1690). |
| [41] |
M. Taqqu, M. Teverovsky and W. Willinger,
Estimators for long range dependence: An empirical study, Fractals, 3 (1995), 785-788.
doi: 10.1142/S0218348X95000692. |
| [42] |
P. Theodossiou,
Financial data and the skewed generalized T distribution, Management Science, 44 (1998), 1650-1661.
|
| [43] |
G. Uhlenbeck and L. Ornstein,
On the theory of Brownian Motion, Physical Review, 36 (1930), 823-841.
doi: 10.1103/PhysRev.36.823. |
| [44] |
R. Weron,
Estimating long-range dependence: Finite sample properties and confidence intervals, Physica A: Statistical Mechanics and its Applications, 312 (2002), 285-299.
doi: 10.1016/S0378-4371(02)00961-5. |
Figure 2.
Autocorrelation Function (left) and Partial Autocorrelation Function (right) in ITLUP returns. Top to bottom: log returns index, absolute returns, square returns
Figure 3.
Autocorrelation Function (left) and Partial Autocorrelation Function (right) in UBI returns. Top to bottom: log-returns index, absolute returns, square returns
Figure 4.
Time-lagged relations between absolute and squared returns in the indices
Table 1.
ITLUP index
| Method Estimation | Parameter (square) | Parameter (absolute) |
| Simple |
0.7308 | 0.7489 |
| Corrected R over S | 0.8407 | 0.8633 |
| Empirical Hurst exponent | 0.7703 | 0.7876 |
| Corrected empirical Hurst | 0.7158 | 0.7342 |
| Method Estimation | Parameter (square) | Parameter (absolute) |
| Simple |
0.7308 | 0.7489 |
| Corrected R over S | 0.8407 | 0.8633 |
| Empirical Hurst exponent | 0.7703 | 0.7876 |
| Corrected empirical Hurst | 0.7158 | 0.7342 |
Table 2.
UBI index
| Method Estimation | Parameter (square) | Parameter (absolute) |
| Simple |
0.6987 | 0.7679 |
| Corrected R over S | 0.7887 | 0.8892 |
| Empirical Hurst exponent | 0.7481 | 0.8304 |
| Corrected empirical Hurst | 0.6953 | 0.7776 |
| Method Estimation | Parameter (square) | Parameter (absolute) |
| Simple |
0.6987 | 0.7679 |
| Corrected R over S | 0.7887 | 0.8892 |
| Empirical Hurst exponent | 0.7481 | 0.8304 |
| Corrected empirical Hurst | 0.6953 | 0.7776 |
| [1] |
Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems and Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217 |
| [2] |
Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 |
| [3] |
Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235 |
| [4] |
Yiju Chen, Xiaohu Wang, Kenan Wu. Wong-Zakai approximations of stochastic lattice systems driven by long-range interactions and multiplicative white noises. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022113 |
| [5] |
Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006 |
| [6] |
Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021120 |
| [7] |
Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, 2018 doi: 10.3934/bdia.2017018 |
| [8] |
Qihong Chen. Recovery of local volatility for financial assets with mean-reverting price processes. Mathematical Control and Related Fields, 2018, 8 (3&4) : 625-635. doi: 10.3934/mcrf.2018026 |
| [9] |
Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022068 |
| [10] |
Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357 |
| [11] |
Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 539-560. doi: 10.3934/dcdss.2020030 |
| [12] |
Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417 |
| [13] |
Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, 2022, 14 (1) : 57-89. doi: 10.3934/jgm.2021012 |
| [14] |
Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557 |
| [15] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks and Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003 |
| [16] |
Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272 |
| [17] |
Marina Núñez, Carles Rafels. A survey on assignment markets. Journal of Dynamics and Games, 2015, 2 (3&4) : 227-256. doi: 10.3934/jdg.2015003 |
| [18] |
Alfredo Daniel Garcia, Martin Andrés Szybisz. Financial liquidity: An emergent phenomena. Journal of Dynamics and Games, 2020, 7 (3) : 209-224. doi: 10.3934/jdg.2020015 |
| [19] |
Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312 |
| [20] |
Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3685-3701. doi: 10.3934/dcdss.2020466 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]