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Keywords:
weak solution and uniqueness in law in the SDE-theory; $(b,\sigma )$-stock price; its Girsanov and DDS-reduction; investment process; option pricing
weak solution and uniqueness in law in the SDE-theory; $(b,\sigma )$-stock price; its Girsanov and DDS-reduction; investment process; option pricing
Summary:
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.
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