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 BSn+1/2 ∑i = 1n A i √∑j, kn ρjkD ijS V DikS V = 0 where BS n 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 D rkS V is an abbreviation of σ rσkSrSk∂ 2V/∂Sr∂Sk along with the volatilities σr of the rth asset Sr. It is shown that the associated Cauchy problem has a solution for uniformity bounded continuous data if for all i, j, i ≠ j 0 ≤ Ai < 1 and |ρij| ≤ - (Ai + Aj)/2 + √(1-A i)(1-Aj) Comment is made on the existence, as A i → 1 for some i, of small and large correlations between returns of assets.
ASJC Scopus subject areas
- Applied Mathematics