A Logistic Black-Scholes Partial Differential Equation with Stochastic Volatility and Transaction Costs

In this research, a formula for the asset price at time t, which  satisfies an extended version of a logistic geometric Brownian motion,  is derived. The extended differential form extends an original logistic  geometric Brownian motion by assuming a stochastic volatility. In  addition, a new logistic Black-Scholes partial differential equations based on the extended price process are formulated. In this differential  equation, transaction costs are also considered