State tameness: a new approach for credit constrains.

Londoño, Jaime A.

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The $d X (t) = X b (X) d t + X σ (X) d W$ equation and financial mathematics. I

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