Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.
Fahrenwaldt, M. A., & Roch, A. F. (2017). Option prices under liquidity risk as weak solutions of semilinear diffusion equations. Nonlinear Differential Equations and Applications, 24(2), . https://doi.org/10.1007/s00030-017-0435-0