Matthias Albrecht Fahrenwaldt

Heat Trace Asymptotics of Subordinate Brownian Motion in Euclidean Space

We derive the heat trace asymptotics of the generator of subordinate Brownian motion on Euclidean space for a class of Laplace exponents. The terms in the asymptotic expansion can be computed to arbitrary order and depend both on the geometry of Euclidean space and the short-time behaviour of the process. If the Blumenthal-Getoor index of the process is rational, then the asymptotics may contain logarithmic terms. The key assumption is the existence of a suitable density for the Lévy measure of the subordinator. The analysis is highly explicit.

Spectral functions of subordinate Brownian motion on closed manifolds

For a class of subordinators, we investigate the spectrum of the infinitesimal generator of subordinate Brownian motion on a closed manifold. We consider three spectral functions of the generator: the zeta function, the heat trace and the spectral action. Each spectral function explicitly yields both probabilistic and geometric information, the latter through the classical heat invariants. All constructions are done with classical pseudodifferential operators and are fully analytically tractable.

Dynamics of solvency risk in life insurance liabilities

We describe the time dynamics of the solvency level of life insurance contracts by representing the solvency level and the underlying risk sources as the solution of a forward–backward system. This leads to an additive decomposition of the total solvency level with respect to time and different risk sources. The decomposition turns out to be an intuitive tool to study risk sensitivities. We study the forward–backward system and discuss two methods to obtain explicit representations: via linear partial differential equations and via a Monte Carlo method based on Malliavin calculus.

Sensitivity of life insurance reserves via Markov semigroups

Abstract Fahrenwaldt M. Sensitivity of life insurance reserves via Markov semigroups. Scandinavian Actuarial Journal. We consider Thiele’s differential equation for the reserve of a multi-state insurance contract with functional dependence on the surplus. In an analytic approach based on semigroups, we obtain existence and uniqueness results and prove growth and regularity properties. Moreover, we investigate the sensitivity of the reserves with respect to the surplus, payment rate, and transition assumptions in terms of uniform and pointwise estimates. The approach can easily be generalized.

Nonrecursive Separation of Risk and Time Preferences

Recursive utility disentangles preferences with respect to time and risk by recursively building up a value function of local increments. This involves certainty equivalents of indirect utility. Instead we disentangle preferences with respect to time and risk by building up a value function as a non-linear aggregation of certainty equivalents of direct utility of consumption. This entails time-consistency issues which are dealt with by looking for an equilibrium control and an equilibrium value function rather than a classical optimal control and a classical optimal value function. We characterize the solution in a general diffusive incomplete market model and find that, in certain special cases of utmost interest, the characterization coincides with what would arise from a recursive utility approach. But also importantly, in other cases, it does not: The two approaches are fundamentally different but match, exclusively but importantly, in the mathematically special case of homogeneity of the value function.

Short-time asymptotic expansions of semilinear evolution equations

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid both for the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudodifferential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to obtain approximate solutions of Markovian backward stochastic differential equations.

A PDE Approach to Option Pricing Under Liquidity Risk

Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.