TY - CHAP
T1 - On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations
AU - Baccas, Ricardo
AU - Kelly, Cónall
AU - Rodkina, Alexandra
N1 - Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
PY - 2019
Y1 - 2019
N2 - We consider the stochastically perturbed cubic difference equation with variable coefficients xn+1=xn(1-hnxn2)+ρn+1ξn+1,n∈N,x0∈R. Here (ξn)n∈N is a sequence of independent random variables, and (ρn)n∈N and (hn)n∈N are sequences of nonnegative real numbers. We can stop the sequence (hn)n∈N after some random time N so it becomes a constant sequence, where the common value is an FN -measurable random variable. We derive conditions on the sequences (hn)n∈N, (ρn)n∈N and (ξn)n∈N, which guarantee that lim n→∞xn exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value x0∈ R.
AB - We consider the stochastically perturbed cubic difference equation with variable coefficients xn+1=xn(1-hnxn2)+ρn+1ξn+1,n∈N,x0∈R. Here (ξn)n∈N is a sequence of independent random variables, and (ρn)n∈N and (hn)n∈N are sequences of nonnegative real numbers. We can stop the sequence (hn)n∈N after some random time N so it becomes a constant sequence, where the common value is an FN -measurable random variable. We derive conditions on the sequences (hn)n∈N, (ρn)n∈N and (ξn)n∈N, which guarantee that lim n→∞xn exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value x0∈ R.
KW - Global almost sure asymptotic stability
KW - Nonlinear stochastic difference equation
KW - Nonuniform timestepping
UR - https://www.scopus.com/pages/publications/85069151877
U2 - 10.1007/978-3-030-20016-9_6
DO - 10.1007/978-3-030-20016-9_6
M3 - Chapter
AN - SCOPUS:85069151877
SN - 9783030200152
T3 - Springer Proceedings in Mathematics and Statistics
SP - 171
EP - 197
BT - Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 23, 2017
A2 - Elaydi, Saber
A2 - Pötzsche, Christian
A2 - Sasu, Adina Luminiţa
PB - Springer New York LLC
T2 - 23rd International Conference on Difference Equations and Applications, ICDEA 2017
Y2 - 24 July 2017 through 28 July 2017
ER -