Controllability and hedgibility of Black-Scholes equations with \(N\) stocks
The authors discuss the controllability and observability properties of the general Black-Scholes model with \(N\) stocks allowing the volatility dependence over the stock prices. The external input (controller) in the option pricing model is related to the hedging ratio in option pricing of finance. The novelty of the paper is that the dynamic hedgibility which is generally assumed in Black-Scholes theory, is proved in the context of controllability theory. The hedging problem is reduced to some kind of initial boundary value problem with suitable control. The global null controllability is proved for the general Black-Scholes equations in the bounded domain of \(R^N_+\) by establishing a suitable observability estimate for associated adjoint system and certain energy estimates for linear Black-Scholes equations. The local controllability is proved for certain nonlinear Black-Scholes equations for \(N<6 \), when the nonlinear term is of superlinear behavior at infinity. The results are obtained using the reguality of solutions, classical compactness arguments and Kakutani's fixed point theorem.
scientific article; zbMATH DE number 5760434
scientific article