The study analyses some problems arising in stochastic volatility models by using Ito’s lemma and its applications to boundary Cauchy problem by giving the solution of vanilla option pricing models satisfying the partial differential equation obtained by assuming stochastic volatility in replication problems and risk neutral probability.
Heston, S.L. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-343. http://dx.doi.org/10.1093/rfs/6.2.327
Heath, D., Jarrow, R. and Morton. A. (1992) Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. Econometrica, 60, 77-105. http://dx.doi.org/10.2307/2951677