Solutions to a stationary nonlinear Black-Scholes type equation
type: Item
instance of: scholarly article
title: Solutions to a stationary nonlinear Black-Scholes type equation
zbMATH Open document ID: 1027.60077
DOI: 10.1016/S0022-247X(02)00434-1
author: C. G. Averbuj
author: Pablo Amster
author: Maria C. Mariani
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The authors study the stationary Dirichlet problem \[ \begin{cases} {1\over 2}\sigma^2S^2{\partial^2 f\over\partial S^2}+{1\over 2} \sigma^2 V^2{\partial^2 f\over\partial\sigma^2}+ \rho\sigma^2 VS{\partial^2f\over \partial S\partial\sigma}- {1\over 2}\rho\sigma^2 V{\partial f\over\partial\sigma}= rg(f)f- rS{\partial f\over\partial S}\quad\text{in }\Omega_0,\\ f= h_0\quad\text{on }\partial\Omega_0.\end{cases}\tag{1a} \] It is motivated by a nonlinear partial differential equation arising in an option pricing model, indeed a Black-Scholes model with stochastic volatilities. After some changing of variables and introducing a parameter \(\lambda\) the new differential equation is \[ \begin{cases} \Delta u+ 2\rho{\partial^2u\over\partial x\partial y}= \lambda(\rho{\partial u\over\partial x}+ (1- {2r\over x^2V^2}){\partial u\over\partial y}+ {2rg(u)\over x^2 V^2}u)\quad\text{in }\Omega,\\ u= h\quad\text{on }\partial\Omega.\end{cases}\tag{1b} \] The authors prove by applying a Newton-type iteration method the existence of a solution in the Sobolev space \(H^2(\Omega)\) under some assumption. A second proof under different assumptions uses the upper and lower solution method.
reviewed by: Uwe Küchler
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 91G20
zbMATH DE Number: 1866743
zbMATH Keywords: partial differential equation
zbMATH Keywords: Black-Scholes formula
zbMATH Keywords: Black-Scholes type equation
MaRDI profile type: Publication
cites work: Q4266143
cites work: A boundary value problem for a semilinear second order ODE
cites work: A Risk-Neutral Stochastic Volatility Model
cites work: Q4040640
cites work: Q4794126
cites work: Q3992729
cites work: Q4794267
full work available at URL: s0022-247x(02)00434-1
OpenAlex ID: W2084210474
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description: scientific article; zbMATH DE number 1866743