We consider a version of the Thiele differential equation that allows for the distribution of a surplus to policyholders. This model appears in the actuarial literature and reflects standard insurance products in many countries. The surplus is driven by the value of a risky asset whose behaviour is modelled by an Ito process. The insurance reserve is then modelled through a backward stochastic differential equation, and a corresponding system of PDEs is derived via a Feynman-Kac-type theorem. The solution of this system is typically given as a stochastic representation, i.e. as the expectation of a stochastic process under a suitable probability measure. The key drawback of the conventional stochastic treatment is that it does not lend itself easily to an analysis of growth, regularity and sensitivity of the insurance reserve.

We propose to overcome this difficulty by applying a purely analytic approach based on the study of linear operators between Banach spaces of H"older continuous functions.

Our key results are

(i) Existence, uniqueness and regularity of a solution of the PDE system which gives the reserve as a function of the surplus. The solution lives in a H"older space on the real line and is also a classical solution in the sense of belonging to the class C^1,2.

(ii) Estimates in terms of Hoelder norms of the sensitivity of the market reserve with respect to the surplus, the expected payment stream and the transition assumptions (essentially mortality and disability).

(iii) Pointwise bounds on the first derivative of the reserve with respect to the surplus given in terms of a solution of another PDE system.

(iv) Finally, in the course of the proofs of the above results, a factorization of the reserve function into risk types (financial, mortality/disability, premium) conceptually elucidating the impact of these factors.

Our key contributions to the actuarial literature include the use of evolution systems generated by linear differential operators with unbounded coefficients on non-compact manifolds. The method we apply lends itself naturally to the following next steps

(i) Insurance reserves when the risky asset is driven by a L'evy process leading to pseudodifferential operator techniques.

(ii) Derivation of short-term asymptotics of the reserve through heat kernel methods thus making the dependence of the reserve on individual parameters very explicit.

(iii) Numerical solutions of the PDE system again through a study of the operators.

*Awarded Life Track Prize