IFoA CS2 key terms

A stochastic process whose next state depends only on the current state, not on the sequence of states visited before it; states and transition probabilities are summarised in a transition matrix.

The memoryless conditional independence property that, given the present state, future evolution is independent of the past; it justifies using only the current state to forecast a Markov process.

A square matrix whose (i,j) entry gives the probability of moving from state i to state j in one step; each row is non-negative and sums to one.

Relations expressing multi-step transition probabilities as sums of products of one-step probabilities, equivalent to multiplying the transition matrix by itself; they underpin matrix powers for n-step transitions.

A probability vector that is unchanged when multiplied by the transition matrix; if a chain starts in it, the state distribution remains the same for all future steps.

The long-run probability distribution of a Markov chain's state as the number of steps tends to infinity, provided such a limit exists and is independent of the starting state.

A property of a Markov chain in which every state can be reached from every other state with positive probability, possibly over several steps; it prevents permanent trapping in a subset of states.

A state that, once entered, is revisited with probability one; in finite irreducible chains all states are recurrent, and expected return times are finite in the positive recurrent case.

The property of an irreducible, aperiodic, positive recurrent Markov chain whereby long-run averages converge to the stationary distribution regardless of the initial state.

A counting process in which events occur singly, with independent increments and a constant rate lambda, so that the number of events in an interval of length t is Poisson distributed with mean lambda times t.

A continuous distribution of waiting times with constant hazard rate and the memoryless property; it is the standard model for inter-arrival times in a Poisson process.

A process whose value at time t is the sum of a random number, Poisson distributed, of independent claim-size random variables; it models aggregate claims in collective risk theory.

A continuous-time process that stays in a state for an exponentially distributed holding time and then jumps to another state according to transition probabilities, described by a generator matrix.

Differential equations for the transition probabilities of a continuous-time Markov chain, obtained from the generator matrix, describing how probabilities evolve forward in time.

A continuous-path process with independent normally distributed increments, zero mean and variance proportional to elapsed time; it is the fundamental building block of continuous-time diffusion models.

A process whose logarithm follows Brownian motion with drift, giving strictly positive values; it is widely used to model asset prices and other quantities that cannot go negative.

An equation describing the evolution of a random process through a drift term and a diffusion term driven by Brownian motion, often written with differentials for state and time.

A chain rule for functions of stochastic processes, including an extra second-order term from quadratic variation, used to derive dynamics of transformed processes such as log prices.

A process whose expected future value, conditional on the information available now, equals its current value; it formalises fair-game dynamics and underlies risk-neutral pricing.

A probability measure under which discounted asset prices are martingales; it allows derivative prices to be computed as discounted expected payoffs, independent of real-world risk preferences.

The assumption that no portfolio can make a riskless profit with zero net investment; it guarantees the existence of a risk-neutral measure and consistent derivative pricing.

A continuous-time framework assuming geometric Brownian motion for the underlying and constant volatility, yielding closed-form European option prices via a partial differential equation.

A discrete-time lattice model in which the underlying moves up or down each period with specified probabilities, allowing backward induction to price options and American early exercise.

A probabilistic model for the time until an event such as death or decrement, described by a survival function, hazard rate or force of mortality and used in life-contingent calculations.

The instantaneous rate of mortality at a given age, equal to the hazard rate of the lifetime distribution; it determines survival probabilities through integration over age.

A tabulation of mortality experience by age, typically giving numbers surviving, deaths and expected future lifetime, used to compute survival and annuity factors.

A model in which several competing causes can remove an individual from a state, each with its own cause-specific rate; it distinguishes dependent and independent decrement rates.

The instantaneous probability of an event occurring at time t given survival to t; it fully characterises a continuous failure-time distribution together with the survival function.

A non-parametric estimator of the survival function from censored data, computed as a product of conditional survival factors at each observed event time.

A short-term insurance model in which aggregate claims over a period are the sum of a random number of individual claim amounts, typically analysed via the compound Poisson process.

The probability that an insurer's surplus, starting from a given initial capital, ever falls below zero under a specified surplus process and claim model.

An upper bound on ruin probability of exponential form involving the adjustment coefficient, derived from the surplus process and used to set solvency capital requirements.

A classical risk process in which premiums arrive continuously at a constant rate and claims follow a compound Poisson process; ruin theory and the adjustment coefficient are developed within it.

A numerical method that generates pseudo-random scenarios from a model and averages the resulting outcomes to estimate expectations, probabilities or risk measures.

A set of techniques such as antithetic variates, control variates or importance sampling that increase the precision of simulation estimates for a given number of scenarios.

An algorithm producing a deterministic sequence of numbers that behaves statistically like independent uniform variates, forming the input to simulation and scenario generation.

A risk measure giving the loss threshold that is exceeded with a specified small probability over a defined horizon; it is a quantile of the loss distribution.

The conditional average loss given that the loss exceeds the Value at Risk threshold; it is a coherent risk measure and captures tail severity beyond a quantile.

A risk measure satisfying monotonicity, translation invariance, positive homogeneity and subadditivity; subadditivity ensures that combining risks does not increase measured risk.