[Paper Review] A characterization of martingale-equivalent compound mixed Poisson process

Demetrios P. Lyberopoulos, N. D. Macheras|arXiv (Cornell University)|May 18, 2019

Probability and Risk ModelsDecision Sciences13 references3 citations

TL;DR

This paper characterizes all progressively equivalent probability measures $ Q $ under which a compound mixed Poisson process (CMPP) under $ P $ remains a CMPP with improved properties, generalizing Delbaen & Haezendonck (1989). The key result, Theorem 4.3, provides a characterization via regular conditional probabilities and Radon-Nikodým derivatives, enabling applications in actuarial pricing and markets with no free lunch with vanishing risk.

ABSTRACT

If a given aggregate process $S$ is a compound mixed Poisson process under a probability measure $P$, a characterization of all probability measures $Q$ on the domain of $P$, such that $P$ and $Q$ are progressively equivalent and $S$ remains a compound mixed Poisson process with improved properties, is provided. This result generalizes earlier work of Delbaen & Haezendonck (1989). Implications related to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.

Motivation & Objective

• To generalize Delbaen & Haezendonck's (1989) characterization of martingale-equivalent measures for compound Poisson processes to the broader class of compound mixed Poisson processes (CMPPs).
• To establish conditions under which a CMPP under a probability measure $ P $ remains a CMPP under a progressively equivalent measure $ Q $, using regular conditional probabilities.
• To provide a framework for computing premium calculation principles in insurance markets satisfying the no-free-lunch-with-vanishing-risk (NFLVR) condition.
• To extend the applicability of martingale-equivalent measure theory to non-classical risk theory settings involving random intensity and claim size distributions.

Proposed method

• Characterizes CMPPs using regular conditional probabilities, reducing them to ordinary compound Poisson processes under disintegrating measures.
• Derives the Radon-Nikodým derivative of $ Q $ with respect to $ P $, enabling the construction of progressively equivalent measures.
• Applies Proposition 3.4 to characterize the density process of $ Q $-martingales in terms of the underlying CMPP structure.
• Uses Theorem 4.3 to identify all measures $ Q $ that preserve the CMPP property under progressive equivalence.
• Applies the results to construct a wide class of canonical stochastic processes inducing progressively equivalent martingale measures (PEMMs).
• Establishes the NFLVR condition via Theorem 5.3, ensuring absence of arbitrage in the constructed market model.

Experimental results

Research questions

• RQ1 Under what conditions does a compound mixed Poisson process under $ P $ remain a CMPP under a progressively equivalent measure $ Q $?
• RQ2 How can the Radon-Nikodým derivative of $ Q $ with respect to $ P $ be characterized in terms of regular conditional probabilities for CMPPs?
• RQ3 What class of stochastic processes induces progressively equivalent martingale measures that preserve the CMPP structure?
• RQ4 How do changes of measure affect the expected number of claims and claim sizes in a CMPP framework?
• RQ5 What implications does the preservation of the CMPP property under $ Q $ have for premium calculation principles in markets satisfying NFLVR?

Key findings

• Theorem 4.3 provides a complete characterization of all progressively equivalent measures $ Q $ such that a $ P $-CMPP remains a CMPP under $ Q $, generalizing Delbaen & Haezendonck (1989).
• The Radon-Nikodým derivative of $ Q $ with respect to $ P $ is fully determined by the regular conditional distributions of the claim number and claim size processes.
• In Example 6.2, the change of measure increases both the expected number of claims and the expected claim size, with $ p(Q) = 810 $ and $ p(P) = 5 $, satisfying condition (13) when $ \theta > 1/2 $.
• In Example 6.3, $ \mathbb{E}_{Q_{\theta}}[X_1] = 2 $ and $ \mathbb{E}_{Q}[N_1] = (c+1)^2 J(c) $, with $ J(c) \in (0,\infty) $, ensuring $ p(Q) < \infty $, and condition (14) holds under a quadratic inequality in $ \theta $.
• The process $ V_t(\varTheta) = S_t - 10t\varTheta^2 $ satisfies the NFLVR condition under Theorem 5.3, confirming absence of arbitrage in the constructed market.
• The pair $ (g, Q) $ constructed in Corollary 4.6 ensures $ Q_{\theta} $ is a proper compound Poisson process, with $ \mathbb{E}_{Q_{\theta}}[N_1] = g(\theta) $ and $ \mathbb{E}_{Q_{\theta}}[X_1] = 2 $, validating the CMPP structure under $ Q $.