Kybernetika 39 no. 6, 653-680, 2003

The dX(t)=Xb(X)dt+Xσ(X)dW equation and financial mathematics. I

Josef Štěpán and Petr Dostál


The existence of a weak solution and the uniqueness in law are assumed for the equation, the coefficients $b$ and $\sigma$ being generally $C({\mathbb R}^+)$-progressive processes. Any weak solution $X$ is called a $(b,\sigma)$-stock price and Girsanov Theorem jointly with the DDS Theorem on time changed martingales are applied to establish the probability distribution $\mu_\sigma$ of $X$ in $C({\mathbb R}^+)$ in the special case of a diffusion volatility $\sigma(X,t)=\tilde\sigma(X(t)).$ A martingale option pricing method is presented.