Backward stochastic differential equations in dynamics of life insurance solvency risk

Abstract

In this thesis we describe the dynamics of solvency level in life insurance contracts. We do this by representing the underlying sources of risk and the solvency level as the solution to a forward-backward stochastic differential equation system. We start by introducing Brownian motion, stochastic integration, stochastic differential equations, and backward stochastic differential equations. With these notions described we can start constructing the model for solvency risk. Afterwards we also give a link to partial differential equation theory and a Monte Carlo example for obtaining explicit representations for the processes involved. We will denote the net value of the contract by a process N, which will depend on underlying economic and demographic variables. We say that the contract is solvent at time t if Nt ≥ 0. We can express the change in solvency probability at the expiry time T as P(NT ≥ 0|Ft) − P(NT ≥ 0|F0) = Z t 0 U ⊤ r dMX r = Z t 0 Z ⊤ r dBr, where the filtration (Ft)t≥0 describes the information available at time t, MX r is the martingale part from Doob’s decomposition of the process X. Furthermore, the pro gressively measurable processes U and Z represent the contributions of the aforemen tioned underlying variables to the overall solvency risk, and the effects the Brownian driver B has on the solvency level, respectively. More technically, the forward-backward system we study is of the form ( d(Xs, V − s ) ⊤ = ˜µ(s, Xs, V − s )ds + ˜σ(s, Xs)dBs, (Xt , V − t ) ⊤ = (v, x) ⊤ −dYs = −Z ⊤ s dBs, YT = Ψ X (t,x) T , V −(t,x,v) T

, where ˜µ and ˜σ are used in defining the process X and contain the information on actuarial assumptions, V − is the retrospective reserve, which describes the present value of assets that belong to the insurance contract at each time t, and Ψ is a ter minal condition, which in our case is not continuous. Under some Lipschitz, bound edness and continuity conditions it will yield a unique, square integrable solution (Xs, V − s , Ys, Zs) s∈[t,T] which we use for the description of solvency level in two differ ent viewpoints; one considering the effects of the underlying demographic variables and the other studying the contributions of the Brownian driver