Abstract
In this paper, we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process by using Malliavin calculus. Similar to the celebrated Black–Scholes formula, we aim to express the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of stop-loss contracts, the building block is given by the distribution function of the terminal cumulated loss taken at the Value at Risk when computing the expected shortfall risk measure.
Highlights
Risk analysis in the context of insurance or reinsurance is often based on the study of properties of a so-called cumulative loss process L := (Lt )t∈[0,T ] over a period of time [0, T ] where T > 0 denotes the maturity of a contract
The one we propose here relies on Malliavin computation in order to provide a decomposition formula into a building block
The reinsurance stop-loss contract will be activated on the basis of the total damages L T h (LT) on the houses, whereas the effective damages L T h (L T) will include all other insured belongings
Summary
Risk analysis in the context of insurance or reinsurance is often based on the study of properties of a so-called cumulative loss process L := (Lt )t∈[0,T ] over a period of time [0, T ] where T > 0 denotes the maturity of a contract. We provide an exact formula for (3) in terms of the building block x → E [h(LT + x)] (or of a related quantity for the more general situation (3), see (11) for a precise statement) This goal will be achieved by using Malliavin calculus available for jump processes. This paper proposes a general framework of dependencies: we do not assume a Markovian setting, nor specify explicit dependencies among inter-arrival times and the claim sizes. We would like to stress that the aforementioned structural account of the loss process provided by the Malliavin derivative calls for a precise description of the probability space on which the Cox process is defined It appears that very few explicit and complete descriptions are presented in the literature.
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