Asymptotic analysis of a Stackelberg differential game for insurance under model ambiguity

ABSTRACT

We consider the problem of to which extent a diffusion process serves as a valid approximation of the classical Cramér-Lundberg (CL) risk process for a Stackelberg differential game between a buyer and a seller of insurance. We show that the equilibrium for the diffusion approximation equals the limit of the equilibrium for the scaled CL process, and it is nearly optimal for the pre-limit problem. Specifically, if the loss process follows a CL risk process and ambiguity is measured via entropic divergence, then the Stackelberg equilibrium of the diffusion approximation with squared-error divergence approximates the equilibrium for the former model to order $\mathcal{O}(\frac{1}{\sqrt{n}})$, in which we scale the CL model via n, as in Cohen and Young [(2020). Rate of convergence of the probability of ruin in the Cramér-Lundberg model to its diffusion approximation. Insurance: Mathematics and Economics 93: 333–340].

KEYWORDS:

• Stackelberg differential game
• insurance
• ambiguity
• diffusion approximation
• random time horizon

AMS 2020 SUBJECT CLASSIFICATIONS:

• 91A05
• 91A10
• 91A23
• 91A25
• 91B52
• 91G05
• 90C59

JEL SUBJECT CLASSIFICATIONS::

• C61
• C72
• C73
• D81
• G22

Acknowledgments

V. R. Young thanks the Cecil J. and Ethel M. Nesbitt Professorship for financial support of her research.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Note that we could consider more general, time-dependent indemnity strategies $\left\{I_t\right(z\left)\right\}$, but the equilibrium strategy is static, as shown in Cao et al. (2022a), so we assume a static strategy from the outset.

2 Cao et al. (2022a) only require that $\varphi >-1$, but the ϕ chosen by the buyer and the seller are automatically non-negative (as they are for the more general problem in this paper), so we impose the non-negativity condition without loss of generality.

3 We require the condition in (7(7) ${\int }_0^{\mathrm{\infty }}(1+\theta \left(z\right))z\mathrm{d}{F}_Z\left(z\right)<\mathrm{\infty }.$(7) ) so that the premium rate is finite for all $I\in \mathcal{I}$.

4 ${\mathbb{E}}_n\left(g\right(Z_n\left)\right)={\int }_0^{\mathrm{\infty }}g\left(z\right)\mathrm{d}{F}_{Z_n}\left(z\right)={\int }_0^{\mathrm{\infty }}g\left(z\right)\mathrm{d}{F}_Z\left(\sqrt{n}z\right)={\int }_0^{\mathrm{\infty }}g\left(t/\sqrt{n}\right)\mathrm{d}{F}_Z\left(t\right)=\mathbb{E}\left(g\right(Z/\sqrt{n}\left)\right).$