In this talk, I will introduce a Brownian motion to the Lotka-Volterra system within its economical interpretation, the Goodwin model:
dx_t & = & x_t (\Phi(y_tt) − \alpha + v^2(y_t ))dt + x_t v(y_t )dW_t \\
dy_t & = & y_t (\kappa(x_t ) − \gamma + v^2(y_t))dt + y_t v(y_t)dW_t\\
We motivate this by heterogeneity of workers productivity growth rate in the capitalist economy. We expect it to provide a richer set of possible trajectories than the deterministic counterpart, the latter allowing a rather simple form for business cycles and growth rate evolution. The addition of stochasticity, in the treatment of this system implies different methods also. This model is thus an opportunity to manipulate stochastic Lyapunov techniques. First, we prove the existence of a solution to the system, even with $\phi$ or $\kappa$ exploding for finite values. Then we provide estimates using a Lyapunov function for the deterministic system. Finally, we study the trajectories of the system.
In the deterministic system, we obtain closed orbits around a non-hyperbolic equilibrium point. Here, we obtain successive recurrent domains around a non stable point. This provides an opportunity to develop additional stochastic stability results. We prove under a specific assumption that trajectories indeed loop around the centre in finite time almost surely. However, other regimes of variations are possible.
This is part of a Research Project with the Fields Institute for Research in Mathematical Sciences.