Jump-diffusion path-dependent option compute under fixed proportional cost

A problem of pricing jump-diffusion path-dependent option is considered in the market with transaction costs. The problem is transformed to a stochastic control problem with two control variables. Integral-differential inequality in corresponding with the value function which the stock price follows jump-diffusion is given. Based on the Markov chain to the discrete form, it is shown that the discrete form is the constraint viscosity solution of the variational inequality.