# American Institute of Mathematical Sciences

October  2018, 5(4): 283-309. doi: 10.3934/jdg.2018018

## Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

 Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received  April 2017 Revised  October 2017 Published  August 2018

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

Citation: Andrew Vlasic. Long-run analysis of the stochastic replicator dynamics in the presence of random jumps. Journal of Dynamics & Games, 2018, 5 (4) : 283-309. doi: 10.3934/jdg.2018018
##### References:
 [1] M. Abundo, On first-passage times problem for one-dimensional jump-diffusion processes, Probability and Mathematical Statistics, 20 (2000), 399-423.   Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge, 2004. Google Scholar [3] N. Balaban, J. Merrin, R. Chait, L. Kowalik and S. Leibler, Bacterial persistence as a phenotypic switch, Science, 305 (2004), 1622-1625.  doi: 10.1126/science.1099390.  Google Scholar [4] R. Bartoszynski, W. J. Bűhler, W. Chan and D. K. Pearl, Population processes under the influence of disasters occurring independently of population size, Journal of Mathematical Biology, 27 (1989), 167-178.  doi: 10.1007/BF00276101.  Google Scholar [5] M. Benaïm, J. Hofbauer and W. Sandholm, Robust permanence and impermanence for the stochastic replicator dynamic, Journal of Biological Dynamics, 2 (2008), 180-195.   Google Scholar [6] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996. Google Scholar [7] R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer-Science+Business Media, B. V., London, 1957. doi: 10.1007/978-94-011-2106-4.  Google Scholar [8] S. Bruan and W. Flűckiger, Increased population of the aphid aphis pomi at a motorway. part 2-the effect of drought and deicing salt, Environmental Pollution Series A, Ecological and Biological, 36 (1984), 261-270.  doi: 10.1016/0143-1471(84)90007-2.  Google Scholar [9] A. Cabrales, Stochastic replicator dynamics, International Economic Review, 41 (2000), 451-481.  doi: 10.1111/1468-2354.00071.  Google Scholar [10] D. Down, S. P. Meyn and R. L. Tweedie, Exponential and uniform ergodicity of markov processes, Annals of Applied Probability, 23 (1995), 1671-1691.  doi: 10.1214/aop/1176987798.  Google Scholar [11] E. Dynkin, Markov Processes, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Google Scholar [12] D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, Journal of Economic Theory, 57 (1992), 420-441.  doi: 10.1016/0022-0531(92)90044-I.  Google Scholar [13] I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972. Google Scholar [14] F. Hanson and H. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187.  doi: 10.1007/s002850050096.  Google Scholar [15] R. Z. Has'minski${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Rockville, Maryland, USA, 2004. Google Scholar [16] J. Hofbauer and L. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamic, Annals of Applied Probability, 19 (2009), 1347-1368.  doi: 10.1214/08-AAP577.  Google Scholar [17] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.  Google Scholar [18] L. Imhof, The long-run behavior of the stochastic replicator dynamics, Annals of Applied Probability, 15 (2005), 1019-1045.  doi: 10.1214/105051604000000837.  Google Scholar [19] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. Google Scholar [20] R. Khasminskii and N. Potsepun, On the replicator dynamics behavior under Stratonovich type random perturbations, Stochastic and Dynamics, 6 (2006), 197-211.  doi: 10.1142/S0219493706001712.  Google Scholar [21] R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Netherlands, 1980. Google Scholar [22] H. Kushner, Stochastic Stability and Control, Academic Press Inc., New York, 1967. Google Scholar [23] H. Masuda, Ergodicity and exponential $β$-mixing bounds for multidimensional diffusions with jumps, Stochastic Processes and their Applications, 25 (2007), 35-56.   Google Scholar [24] M. Menotti-Raymond and S. O'Brien, Dating the genetic bottleneck of the african cheetah, Proc. Natl. Acad. Sci., 90 (1993), 3172-3176.  doi: 10.1073/pnas.90.8.3172.  Google Scholar [25] P. Mertikopoulos and Y. Viossat, Imitation dynamics with payoff shocks, International Journal of Game Theory, 45 (2016), 291-320.  doi: 10.1007/s00182-015-0505-7.  Google Scholar [26] S. P. Meyn and R. L. Tweedie, Stability of markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Annals of Applied Probability, 25 (1993), 518-548.   Google Scholar [27] S. P. Meyn and R. L. Tweedie, A survey of Foster-Lyapunov techniques for general state space Markov processes, Proc. Workshop Stochastic Stability and Stochastic Stabilization. Google Scholar [28] N. Rabalais, R. Turner and W. Wiseman, Gulf of mexico hypoxia, a.k.a, "the dead zone", Ann Rev Ecol Sys, 33 (2002), 235-263.  doi: 10.1146/annurev.ecolsys.33.010802.150513.  Google Scholar [29] W. Sandholm and M. Staudigl, Large deviations and stochastic stability in the small noise double limit, Theoretical Economics, 11 (2016), 279-355.   Google Scholar [30] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. Google Scholar [31] A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, Moscow, 1989. Google Scholar [32] K. Taira, Diffusion processes and partial differential equations, North-Holland Mathematics Studies, 98 (1984), 197-210.  doi: 10.1016/S0304-0208(08)71499-3.  Google Scholar [33] K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer-Verlag, Berlin-Heidelberg, 2004. Google Scholar [34] H. C. Tuckwell, On the first-exit time problem for temporally homogeneous markov processes, Journal of Applied Probability, 13 (1976), 39-48.   Google Scholar [35] A. Vlasic, Stochastic replicator dynamics subject to Markovian switching, Journal of Mathematical Analysis and Applications, 427 (2015), 235-247.  doi: 10.1016/j.jmaa.2015.02.016.  Google Scholar [36] P. Young and D. Foster, Stochastic evolutionary game dynamics, Theoretical Population Biololgy, 38 (1990), 219-232.   Google Scholar

show all references

##### References:
 [1] M. Abundo, On first-passage times problem for one-dimensional jump-diffusion processes, Probability and Mathematical Statistics, 20 (2000), 399-423.   Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge, 2004. Google Scholar [3] N. Balaban, J. Merrin, R. Chait, L. Kowalik and S. Leibler, Bacterial persistence as a phenotypic switch, Science, 305 (2004), 1622-1625.  doi: 10.1126/science.1099390.  Google Scholar [4] R. Bartoszynski, W. J. Bűhler, W. Chan and D. K. Pearl, Population processes under the influence of disasters occurring independently of population size, Journal of Mathematical Biology, 27 (1989), 167-178.  doi: 10.1007/BF00276101.  Google Scholar [5] M. Benaïm, J. Hofbauer and W. Sandholm, Robust permanence and impermanence for the stochastic replicator dynamic, Journal of Biological Dynamics, 2 (2008), 180-195.   Google Scholar [6] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996. Google Scholar [7] R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer-Science+Business Media, B. V., London, 1957. doi: 10.1007/978-94-011-2106-4.  Google Scholar [8] S. Bruan and W. Flűckiger, Increased population of the aphid aphis pomi at a motorway. part 2-the effect of drought and deicing salt, Environmental Pollution Series A, Ecological and Biological, 36 (1984), 261-270.  doi: 10.1016/0143-1471(84)90007-2.  Google Scholar [9] A. Cabrales, Stochastic replicator dynamics, International Economic Review, 41 (2000), 451-481.  doi: 10.1111/1468-2354.00071.  Google Scholar [10] D. Down, S. P. Meyn and R. L. Tweedie, Exponential and uniform ergodicity of markov processes, Annals of Applied Probability, 23 (1995), 1671-1691.  doi: 10.1214/aop/1176987798.  Google Scholar [11] E. Dynkin, Markov Processes, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Google Scholar [12] D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, Journal of Economic Theory, 57 (1992), 420-441.  doi: 10.1016/0022-0531(92)90044-I.  Google Scholar [13] I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972. Google Scholar [14] F. Hanson and H. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187.  doi: 10.1007/s002850050096.  Google Scholar [15] R. Z. Has'minski${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Rockville, Maryland, USA, 2004. Google Scholar [16] J. Hofbauer and L. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamic, Annals of Applied Probability, 19 (2009), 1347-1368.  doi: 10.1214/08-AAP577.  Google Scholar [17] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.  Google Scholar [18] L. Imhof, The long-run behavior of the stochastic replicator dynamics, Annals of Applied Probability, 15 (2005), 1019-1045.  doi: 10.1214/105051604000000837.  Google Scholar [19] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. Google Scholar [20] R. Khasminskii and N. Potsepun, On the replicator dynamics behavior under Stratonovich type random perturbations, Stochastic and Dynamics, 6 (2006), 197-211.  doi: 10.1142/S0219493706001712.  Google Scholar [21] R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Netherlands, 1980. Google Scholar [22] H. Kushner, Stochastic Stability and Control, Academic Press Inc., New York, 1967. Google Scholar [23] H. Masuda, Ergodicity and exponential $β$-mixing bounds for multidimensional diffusions with jumps, Stochastic Processes and their Applications, 25 (2007), 35-56.   Google Scholar [24] M. Menotti-Raymond and S. O'Brien, Dating the genetic bottleneck of the african cheetah, Proc. Natl. Acad. Sci., 90 (1993), 3172-3176.  doi: 10.1073/pnas.90.8.3172.  Google Scholar [25] P. Mertikopoulos and Y. Viossat, Imitation dynamics with payoff shocks, International Journal of Game Theory, 45 (2016), 291-320.  doi: 10.1007/s00182-015-0505-7.  Google Scholar [26] S. P. Meyn and R. L. Tweedie, Stability of markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Annals of Applied Probability, 25 (1993), 518-548.   Google Scholar [27] S. P. Meyn and R. L. Tweedie, A survey of Foster-Lyapunov techniques for general state space Markov processes, Proc. Workshop Stochastic Stability and Stochastic Stabilization. Google Scholar [28] N. Rabalais, R. Turner and W. Wiseman, Gulf of mexico hypoxia, a.k.a, "the dead zone", Ann Rev Ecol Sys, 33 (2002), 235-263.  doi: 10.1146/annurev.ecolsys.33.010802.150513.  Google Scholar [29] W. Sandholm and M. Staudigl, Large deviations and stochastic stability in the small noise double limit, Theoretical Economics, 11 (2016), 279-355.   Google Scholar [30] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. Google Scholar [31] A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, Moscow, 1989. Google Scholar [32] K. Taira, Diffusion processes and partial differential equations, North-Holland Mathematics Studies, 98 (1984), 197-210.  doi: 10.1016/S0304-0208(08)71499-3.  Google Scholar [33] K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer-Verlag, Berlin-Heidelberg, 2004. Google Scholar [34] H. C. Tuckwell, On the first-exit time problem for temporally homogeneous markov processes, Journal of Applied Probability, 13 (1976), 39-48.   Google Scholar [35] A. Vlasic, Stochastic replicator dynamics subject to Markovian switching, Journal of Mathematical Analysis and Applications, 427 (2015), 235-247.  doi: 10.1016/j.jmaa.2015.02.016.  Google Scholar [36] P. Young and D. Foster, Stochastic evolutionary game dynamics, Theoretical Population Biololgy, 38 (1990), 219-232.   Google Scholar
 [1] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [2] Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010 [3] Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140 [4] Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076 [5] Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049 [6] Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47 [7] Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091 [8] Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080 [9] Alex Potapov, Ulrike E. Schlägel, Mark A. Lewis. Evolutionarily stable diffusive dispersal. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3319-3340. doi: 10.3934/dcdsb.2014.19.3319 [10] Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1 [11] Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049 [12] Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 [13] Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198 [14] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [15] Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113 [16] Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models†. Journal of Dynamics & Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012 [17] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [18] Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 [19] Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697 [20] Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332

Impact Factor:

## Metrics

• HTML views (556)
• Cited by (0)

• on AIMS