D. ILP Examples
probability
insurance
modeling
research
risk
Finally, we reach the subject under discussion!
\[ \def\E{\mathsf E} \def\F{\mathcal F} \def\P{\mathsf P} \def\R{\mathbb R} \]
Introduction | Probability background | Stochastic processes | General examples | •
1 ILP Definition and Motivation
- Cost of capital risk margin
- Actual risk in a calendar year
- Multi-period risk pricing
2 ILP Properties
- Non-negative
- Two options for process:
- Supermartingale in objective probability because booking at time \(t\) must be greater than or equal to the future expected value to allow for a risk margin
- Martingale in risk adjusted probability corresponding to best estimate market value
- The process need not be continuous, e.g., court verdict or test results could provide an instantaneous information jump
- The process is integrable in real world applications, but could be very thick-tailed
- The process is reasonably assumed uniformly integrable in most applications, so we can model it as the conditional expectations of the ultimate loss \(X\), which as we have already seen, can be interpreted as a convergent sum of differences. It can also be modeled directly from the sigma-algebra filtration.
3 ILP Examples
Introduction | Probability background | Stochastic processes | General examples | •