Stochastic Processes and Martingales
A stochastic process is a collection of random variables indexed by time. A martingale is a process whose conditional expectation of the next value, given the past, equals the current value, so it models a fair game. The optional stopping theorem says that under stated conditions the expected value at a stopping time equals the starting value, and the exponential martingale built from the surplus process is the mechanism behind the Lundberg ruin bound.
Processes, Filtrations, And The Martingale Property
A stochastic process is a family of random variables indexed by a time parameter, discrete or continuous. A filtration is the increasing family of information sets available up to each time, and a process is adapted when its value at each time is known given that information.
A martingale is an adapted, integrable process whose conditional expectation of the next value given the current information equals the current value. A supermartingale has conditional expectation less than or equal to the current value, modeling a game that drifts down or stays level, and a submartingale drifts up. Williams, Probability with Martingales, Ch. 10 to 12 develops these definitions\; Ross, Stochastic Processes, 2nd ed., Ch. 6 gives the applied treatment used on CAS MAS-II.
Counting Processes And The Poisson Process
A counting process records the number of events observed up to each time. The Poisson process is the counting process with stationary independent increments and rate lambda, so the number of events in an interval of length t is Poisson with mean lambda t. It is the arrival model inside the compound Poisson aggregate-claims process used across ASTAM and the CAS reserving syllabus.
The compensated Poisson process, obtained by subtracting the mean lambda t from the count, is a martingale. This is the first place an actuarial model meets the martingale property directly.
Optional Stopping And The Ruin Connection
A stopping time is a random time whose occurrence is decided by the information available up to that time, with no look-ahead. The optional stopping theorem states that for a martingale and a stopping time, under boundedness or uniform-integrability conditions, the expected value at the stopping time equals the initial value.
The classical risk model uses this through an exponential martingale. With adjustment coefficient R solving the Lundberg equation, the process exp(-R U(t)) is a martingale. Applying optional stopping at the ruin time and discarding the nonnegative survival contribution gives the Lundberg inequality psi(u) less than or equal to exp(-R u). The full surplus model and the adjustment coefficient are developed on the ruin-theory page.
Worked Example: A Simple Random Walk
Let a gambler start with 5 units and bet 1 unit on fair coin flips, stopping at 0 units or at 10 units. The wealth process is a martingale because each fair flip has zero conditional expected change. Let p be the probability of reaching 10 before 0.
By optional stopping the expected wealth at the stopping time equals the starting wealth: 10 p plus 0 times (1 minus p) equals 5, so p equals 0.5. The same gambler-ruin algebra, with an unfair drift replaced by the premium loading, gives the structure of the insurer survival probability. This is why martingale arguments and ruin theory share one toolkit.