Finite Difference Approximation of a Historical Empirical Distribution

Stochastic Process for Merton Default Modelling

We require a filtered probability space as the default probability is conditional on information we have accumulated at time $t$. It is somewhat intuitive that as our standard sigma algebra contains the probabilities of all times $t$, we require a filtered sigma-algebra that will contain all probabilities within our predefined time frame. Our predefined time frame is composed of a terminal time horizon $T$. Hence we define filtered probability space $$\Omega, \mathcal{F}, (\mathcal{F}_t)_{0 \leq t \leq T}, P$$ where $P$ is a subjective probability measure.

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