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 BS_n is the n-dimensional Black-Scholes operator, A_i are positive transaction cost numbers, ρ_jk are the correlations between returns of asset S_j and asset S_k and D^S_rkV is an abbreviation of along with the volatilities σ_r of the rth asset S_r. It is shown that the associated Cauchy problem has a solution for uniformily bounded continuous data if for all i, j, i≠j 0≤A_i<1 andComment is made on the existence, as A_i→1 for some i, of small and large correlations between returns of assets.