In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))$ with $\mathbb{Q}$ being the risk-neutral measure. For example, one now would like to value a pure endowment product, i.e. at a fixed time $T$ the prelevant stock price $S(T)$ is paid out if the policyholder is alive at time $T$ else there is no payout. Additionally, such products often contain some further financial guarantee, i.e. minimal payout. But this is not essential for my questions.
Therefore, one also has to model mortality. For this, one often considers $T_{x}$ the future lifespan of a $x$-year old and sets $\mathcal{G}_{t}:=\sigma(\mathbb{1}_{\{T_{x}\leq s \}}\vert s\leq t)$, which defines the "insurance filtration" $\mathbb{G}=(\mathcal{G}_{t})$. Then the one considers the enlarged filtration $\mathbb{H}=\mathbb{F}\vee\mathbb{G}$ and works on the filtered space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{H})$. The survival probability is then defined as $p_{x+t}(t,T):=\mathbb{Q}(T_{x}>T\vert \mathcal{H}_{t})$.
Unfortunately, I never found a general good and formal account of this. Many things seem implicitly assumed. My questions:
- Are there any good references for this general modeling approach?
- Why can the risk-neutral measure even be extended to the enlarged space and in particular be used to measure mortality?
- Or, are they any special conditions needed?
- If we assume that mortality is independent from financial markets, do we need any of this anyways?
Thanks alot for the help.
