Stability in a Discrete-Time Dynamic Model with Delay for a Stock Market

The time evolution of prices and savings in a stock market is modeled by a discrete-delay nonlinear dynamic system. The proposed model has a unique and unstable steady-state, so its time evolution is determined by the nonlinear effects acting out of the equilibrium. We perform the analysis of the linear approximation through the study of the eigenvalues of the Jacobian matrix in order to characterize the local stability properties and the local bifurcations in the parameter space. If the delay is equal to zero, Lyapunov exponents are calculated. For certain values of the parameters, we prove that the system has a chaotic behaviour. The discrete nonlinear model is associated with a discrete stochastic model. For the liniarization of this model, we establish the conditions for which the mean and quadratic mean values of the state variables are asymptotically stable. Some numerical examples are finally given to justify the theoretical results.