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 equation BS_n+1/2 ∑_{i = 1}ⁿ A _i √∑_{j, k}ⁿ ρ_jkD _ij^S V D_ik^S V = 0 where 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 _rk^S V is an abbreviation of σ _rσ_kS_rS_k∂ ²V/∂S_r∂S_k along with the volatilities σ_r of the rth asset S_r. It is shown that the associated Cauchy problem has a solution for uniformity bounded continuous data if for all i, j, i ≠ j 0 ≤ A_i < 1 and |ρ_ij| ≤ - (A_i + A_j)/2 + √(1-A _i)(1-A_j) Comment is made on the existence, as A _i → 1 for some i, of small and large correlations between returns of assets.