Optimal Liquidation Behaviour Analysis for Stochastic Linear and Nonlinear Systems of Self-Exciting Model with Decay

When the market environment changes, we extend the self-exciting price impact model and further analysis of investors’ liquidation behaviour. It is assumed that the model is accompanied by an exponential decay factor when the temporary impact and its coefficient are linear and nonlinear. Using the optimal control method, we obtain that the optimal liquidation behaviours satisfy the second-order nonlinear ODEs with variable coefficients in the case of linear and nonlinear temporary impact. Next, we solve the ODEs and get the form of the investors’ optimal liquidation behaviour in four cases. Furthermore, we prove the decreasing properties of the optimal liquidation behaviour under the linear temporary impact. Through numerical simulation, we further explain the influence of the changed parameters ρ, a, b, x, and α on the investors’ liquidation strategy Xt in twelve scenarios. Some interesting properties have been found.