Geodesic
The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. I
Kybernetika, Tome 39 (2003) no. 6, pp. 653-680 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 60H10, 91B28, 91G20
Keywords: weak solution and uniqueness in law in the SDE-theory; $(b, \sigma )$-stock price; its Girsanov and DDS-reduction; investment process; option pricing 
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Štěpán, Josef; Dostál, Petr. The $dX(t)=Xb(X)dt+X\sigma(X)dW$ equation and financial mathematics. I. Kybernetika, Tome 39 (2003) no. 6, pp. 653-680. http://geodesic.mathdoc.fr/item/KYB_2003_39_6_a0/

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