Adjustment dynamics in a regular stochastic network

Stochastic parameters are introduced in the network game model with production and knowledge externalities. The original model was formulated by V. D. Matveenko and A.V. Korolev and was a generalization of a simple two-period Romer model transferred to networks. Each economic agent has an initial supply of goods, which in the first period it can use for consumption and for investment in knowledge. In the second period, it receives the results of its investments, but these results depend not only on the size of its investments, but also on the total investments of its neighbors in the network. The target function of each agent depends on its consumption in both periods. The concept of "Jacobian" equilibrium in this model is concretized as "Nash equilibrium with externalities". This equilibrium differs from the usual Nash equilibrium in that each agent, when making a decision about the size of its investment, considers the environment (which includes its investment) as exogenously defined. The main results of the basic deterministic model are presented, in particular, the definition of the adjustment dynamics under continuous consideration. In this paper, we consider a stochastic generalization of the basic model. Agent productivity has not only deterministic, but also Brown components. The explicit solution of stochastic differential equations describing the adjustment dynamics is obtained. In previous studies, the transition dynamics between stable equilibrium states in the network was considered in the deterministic case. It turns out that the boundaries of various scenarios of agent behavior (and these scenarios themselves) in the stochastic case change significantly compared to the deterministic case. In conclusion, possible further statements of the problem are considered.