Well-Posedness Analysis of Leland's Nonlinear Black-Scholes Equation with Transaction Costs

This paper establishes the well-posedness of Leland’s nonlinear Black-Scholes equation incorporating transaction costs. The model, derived from the classical Black-Scholes framework with proportional transaction costs and discrete rebalancing assumptions, results in a nonlinear parabolic partial differential equation with nonlinear volatility. Under the condition 0 1, where Le is the Leland constant representing transaction cost intensity, we prove existence and uniqueness of weak solutions using an energy method and Galerkin approximation, based on a transformed version of the original model. Uniqueness is established via energy estimates combined with Poincaré’s inequality and Gronwall’s lemma, while existence follows from monotone operator theory and the Minty-Browder lemma applied to the Galerkin approximation scheme. The analysis confirms the mathematical robustness of Leland’s model for practical financial applications under moderate transaction cost regimes, providing a rigorous foundation for numerical implementations.