$\tau$ follows an exponential law: $${\mathbb{P}}\{t_0<\tau\leq t|{\mathcal{F}\/}(t)\}=1-e^{-\lambda (t-t_0)}. 
\label{PP}$$ Note that (\[PP\]) identifies $\lambda$ as the inverse of the mean value of the transition waiting time interval $\tau-t_0$, ${\mathbb{E}}[\tau-t_0|{\mathcal{F}\/}(t_0)]=\lambda^{-1}$. In the previous expressions and hereafter ${\mathcal{F}\/}(t)$ represents [*all*]{} the available information up to time $t$.

The asset behaviour may be easily visualized when we express it in terms of returns, $R(t;t_0)\equiv \log(S(t)/S(t_0))$, instead of spot prices. The return will follow a drifted Wiener process with parameters $(\mu_a,\sigma_a)$ up to time $\tau$. After that time the initial Brownian motion freezes and a second drifted Wiener process drives the subsequent evolution of the return: $$R(t;t_0)=R_a(\min(t,\tau);t_0)+R_b(\max(t,\tau);\tau),
$$ with $R_{{a,b}}(t;t')=\sigma_{{a,b}} W_{{a,b}}(t-t')+\tilde{\theta}_{{a,b}}(t-t')$, and $\tilde{\theta}_{{a,b}}=\mu_{{a,b}}-\sigma_{{a,b}}^2/2$.

In the most common situation the stock $S$ and its derivatives, contracts whose price depends upon the value of this underlying, are the only securities affected by the actual value of $\tau$. When part of the risk is not directly traded in the market, the market may be incomplete: we will not be able to reproduce the behaviour of some assets by means of a [*replicating*]{} portfolio. In our case, if we want to hedge the market exposure of derivatives to volatility or dividend risk we must also include in the 