Actuarial Mathematics Unit 2 Review: Stochastic Processes & Time Series

Key Concepts and Definitions

• Stochastic process a collection of random variables indexed by time, representing the evolution of a system over time
• State space the set of all possible values that a stochastic process can take at any given time
• Transition probability the probability of moving from one state to another in a given time interval
• Markov property a stochastic process has the Markov property if the future state depends only on the current state and not on the past states
• Stationarity a stochastic process is stationary if its statistical properties do not change over time
  • Strict stationarity the joint probability distribution of the process remains the same under time shifts
  • Weak stationarity the mean and covariance of the process remain constant over time
• Ergodicity a stationary process is ergodic if its time average converges to the ensemble average as the time interval increases
• Martingale a stochastic process where the expected future value, given the current state, is equal to the current value

Probability Theory Foundations

• Probability space $(\Omega, \mathcal{F}, P)$ consists of a sample space $\Omega$, a $\sigma$-algebra $\mathcal{F}$, and a probability measure $P$
• Random variable a measurable function $X: \Omega \rightarrow \mathbb{R}$ that assigns a real number to each outcome in the sample space
• Expectation the average value of a random variable, defined as $E[X] = \int_{\Omega} X(\omega) dP(\omega)$
• Conditional probability the probability of an event $A$ given that another event $B$ has occurred, denoted as $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Independence two events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$
  • Random variables $X$ and $Y$ are independent if their joint distribution factorizes into the product of their marginal distributions
• Bayes' theorem relates conditional probabilities, expressed as $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• Central limit theorem states that the sum of a large number of independent and identically distributed random variables converges to a normal distribution

Types of Stochastic Processes

• Discrete-time processes stochastic processes where the time index takes integer values (random walks, Markov chains)
• Continuous-time processes stochastic processes where the time index takes real values (Brownian motion, Poisson processes)
• Markov chains discrete-time stochastic processes with the Markov property, characterized by a transition probability matrix
  • Absorbing states states in a Markov chain from which the process cannot escape once entered
  • Ergodic Markov chains have a unique stationary distribution, and the long-run proportion of time spent in each state converges to this distribution
• Poisson processes count the number of events occurring in a fixed time interval, with events occurring independently at a constant average rate
• Brownian motion a continuous-time stochastic process with independent, normally distributed increments (used to model stock prices, particle diffusion)
• Renewal processes generalize Poisson processes by allowing the time between events to follow any probability distribution
• Gaussian processes continuous-time processes where any finite collection of random variables has a multivariate normal distribution

Time Series Analysis Basics

• Time series a sequence of observations indexed by time, often exhibiting temporal dependence or seasonality
• Autocovariance measures the linear dependence between observations at different time lags, defined as $\gamma(h) = E[(X_t - \mu)(X_{t+h} - \mu)]$
• Autocorrelation the autocovariance scaled by the variance, given by $\rho(h) = \frac{\gamma(h)}{\gamma(0)}$
  • Sample autocorrelation function (ACF) a plot of the sample autocorrelations at different lags, used to identify temporal dependence
• Partial autocorrelation measures the correlation between observations at different lags, after removing the effect of intermediate observations
  • Sample partial autocorrelation function (PACF) a plot of the sample partial autocorrelations, used to identify the order of autoregressive models
• Trend a long-term increase or decrease in the level of the time series (linear, exponential, or polynomial)
• Seasonality periodic fluctuations in the time series, often related to calendar effects (monthly, quarterly, or yearly)
• Decomposition separating a time series into trend, seasonal, and residual components (additive or multiplicative)

Modeling Techniques

• Autoregressive (AR) models express the current observation as a linear combination of past observations and a random error term
  • AR(p) model $X_t = \phi_1 X_{t-1} + \ldots + \phi_p X_{t-p} + \varepsilon_t$, where $\varepsilon_t$ is white noise
• Moving average (MA) models express the current observation as a linear combination of past error terms
  • MA(q) model $X_t = \varepsilon_t + \theta_1 \varepsilon_{t-1} + \ldots + \theta_q \varepsilon_{t-q}$
• Autoregressive moving average (ARMA) models combine AR and MA terms to capture both short-term and long-term dependencies
  • ARMA(p,q) model $X_t = \phi_1 X_{t-1} + \ldots + \phi_p X_{t-p} + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \ldots + \theta_q \varepsilon_{t-q}$
• Autoregressive integrated moving average (ARIMA) models extend ARMA models to handle non-stationary time series by differencing
  • ARIMA(p,d,q) model applies an ARMA(p,q) model to the d-th difference of the time series
• Seasonal ARIMA (SARIMA) models incorporate seasonal AR, MA, and differencing terms to capture periodic patterns
• Exponential smoothing methods forecast future values as weighted averages of past observations, with exponentially decaying weights
  • Simple exponential smoothing $\hat{X}_{t+1} = \alpha X_t + (1-\alpha) \hat{X}_t$, where $\alpha$ is the smoothing parameter
  • Holt-Winters method extends exponential smoothing to handle trend and seasonality

Statistical Inference for Stochastic Processes

• Parameter estimation determining the values of model parameters that best fit the observed data (maximum likelihood, method of moments)
  • Maximum likelihood estimation (MLE) finds parameter values that maximize the likelihood function, which measures the probability of observing the data given the parameters
  • Method of moments estimation equates sample moments (mean, variance) to their theoretical counterparts and solves for the parameters
• Hypothesis testing assessing the validity of assumptions or comparing different models (likelihood ratio tests, Wald tests)
  • Likelihood ratio test compares the likelihood of the data under two nested models, with the test statistic following a chi-square distribution under the null hypothesis
  • Wald test assesses the significance of individual parameters by comparing their estimates to their standard errors, assuming asymptotic normality
• Model selection choosing the best model from a set of candidates based on a trade-off between goodness-of-fit and complexity (AIC, BIC)
  • Akaike information criterion (AIC) $AIC = -2 \log(L) + 2k$, where $L$ is the likelihood and $k$ is the number of parameters
  • Bayesian information criterion (BIC) $BIC = -2 \log(L) + k \log(n)$, where $n$ is the sample size
• Residual analysis examining the differences between observed and fitted values to assess model adequacy (normality, autocorrelation)
  • Normality tests (Shapiro-Wilk, Jarque-Bera) check if the residuals follow a normal distribution
  • Ljung-Box test assesses the presence of autocorrelation in the residuals

Applications in Actuarial Science

• Mortality modeling stochastic processes (Lee-Carter model, Cairns-Blake-Dowd model) to capture the evolution of mortality rates over time
  • Lee-Carter model $\log(m_{x,t}) = a_x + b_x k_t + \varepsilon_{x,t}$, where $m_{x,t}$ is the central death rate at age $x$ and time $t$, $a_x$ is the average log-mortality at age $x$, $b_x$ is the sensitivity of age $x$ to changes in the mortality index $k_t$, and $\varepsilon_{x,t}$ is an error term
• Loss reserving estimating the outstanding liabilities for claims that have occurred but have not been fully settled (chain-ladder method, Bornhuetter-Ferguson method)
  • Chain-ladder method projects ultimate claims by applying development factors to cumulative claims, assuming a stable pattern of claims development over time
  • Bornhuetter-Ferguson method combines the chain-ladder method with an a priori estimate of ultimate claims, giving more weight to the a priori estimate for less developed accident years
• Asset liability management (ALM) using stochastic models to optimize the allocation of assets to meet future liabilities (stochastic programming, dynamic financial analysis)
  • Stochastic programming optimizes decisions under uncertainty by considering multiple scenarios and their probabilities
  • Dynamic financial analysis (DFA) simulates the performance of an insurer under various economic and operating scenarios to assess risk and inform strategic decisions
• Pricing and hedging of financial derivatives applying stochastic calculus to value and manage the risk of options, futures, and other derivatives (Black-Scholes model, Heston model)
  • Black-Scholes model values European options assuming the underlying asset follows a geometric Brownian motion with constant volatility
  • Heston model extends the Black-Scholes model by allowing the volatility to follow a stochastic process (Cox-Ingersoll-Ross process)

Advanced Topics and Current Research

• Stochastic differential equations (SDEs) model the evolution of continuous-time stochastic processes, often driven by Brownian motion (Itô calculus, Stratonovich calculus)
  • Itô calculus a framework for defining and manipulating stochastic integrals with respect to Brownian motion, leading to the Itô formula for the differential of a function of a stochastic process
  • Stratonovich calculus an alternative to Itô calculus that preserves the rules of ordinary calculus, useful for modeling physical systems with noise
• Lévy processes generalize Brownian motion by allowing for jumps and more flexible distributions of increments (Poisson processes, stable processes)
  • Stable processes have increments following a stable distribution, which includes the normal, Cauchy, and Lévy distributions as special cases
• Fractional processes introduce long-range dependence and self-similarity by using fractional calculus (fractional Brownian motion, fractional ARIMA models)
  • Fractional Brownian motion a generalization of Brownian motion with correlated increments, characterized by the Hurst parameter $H \in (0,1)$
  • Fractional ARIMA (FARIMA) models extend ARIMA models by allowing for fractional differencing, capturing long-memory effects
• Copula-based models capture the dependence structure between multiple stochastic processes without specifying their marginal distributions (Gaussian copulas, Archimedean copulas)
  • Gaussian copulas model the dependence using a multivariate normal distribution, with the correlation matrix determining the strength of the dependence
  • Archimedean copulas (Clayton, Gumbel, Frank) provide a parametric family of copulas with various tail dependence properties
• Machine learning techniques for stochastic processes combine statistical models with data-driven approaches to improve prediction and inference (neural networks, deep learning, reinforcement learning)
  • Neural networks can be used to learn complex, non-linear relationships between stochastic processes and their driving factors
  • Deep learning architectures (recurrent neural networks, long short-term memory) are particularly well-suited for modeling temporal dependencies in time series data
  • Reinforcement learning optimizes decision-making in stochastic environments by learning from the consequences of actions through trial and error