On parabolic equations with gauge function term and applications to the multidimensional Leland equation

Sufficient conditions for existence and a closed form probabilistic representation are obtained for solutions of nonlinear parabolic equations with gauge function term. In particular, the result applies to the generalized Leland equationwhere BSn is the n-dimensional Black-Scholes operator, Ai are positive transaction cost numbers, ρjk are the correlations between returns of asset Sj and asset Sk and DSrkV is an abbreviation of along with the volatilities σr of the rth asset Sr. It is shown that the associated Cauchy problem has a solution for uniformily bounded continuous data if for all i, j, i≠j 0≤Ai<1 and [image omitted] [image omitted]Comment is made on the existence, as Ai→1 for some i, of small and large correlations between returns of assets.