## Abstract

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_{jk}D _{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 _{r}σ_{k}S_{r}S_{k}∂ ^{2}V/∂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.

Original language | English (US) |
---|---|

Pages (from-to) | 215-228 |

Number of pages | 14 |

Journal | Applied Mathematical Finance |

Volume | 10 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2003 |

## ASJC Scopus subject areas

- Finance
- Applied Mathematics