Stochastic Processes for Actuarial Risk Modeling

Poisson Processes

The homogeneous Poisson process models claim arrivals with constant rate lambda. Key properties: N(t) ~ Poisson(lambda*t), interarrival times are i.i.d. Exponential(lambda), and the process has independent and stationary increments. The nonhomogeneous Poisson process has rate lambda(t), giving N(s,t) ~ Poisson(integral from s to t of lambda(u)du). The compound Poisson process adds random claim sizes: S(t) = sum of X_i for i=1 to N(t). E[S(t)] = lambda*t*E[X] and Var(S(t)) = lambda*t*E[X^2]. This is the continuous-time analog of the collective risk model.

Brownian Motion and Random Walks

Brownian motion (Wiener process) W(t) has independent, stationary, normally distributed increments: W(t)-W(s) ~ N(0, t-s). It models the continuous diffusion component of surplus processes. The Brownian motion with drift models insurer surplus: U(t) = u + c*t minus S(t), where u is initial surplus, c is premium rate, and S(t) is aggregate claims. Ruin probability analysis studies P(U(t) < 0 for some t). The random walk is the discrete-time analog. Exam MAS-I covers Poisson process properties and basic Brownian motion concepts.