💹Financial Mathematics Unit 4 – Stochastic Processes

Key Concepts and Definitions

• Stochastic process represents a system that evolves over time with an element of randomness or uncertainty
• State space defines the set of all possible values or states that a stochastic process can take at any given time
• Transition probabilities quantify the likelihood of moving from one state to another in a stochastic process
• Filtration captures the information available up to a certain point in time, used to model the evolution of knowledge in a stochastic process
• Martingale property characterizes a stochastic process where the expected future value, given the current information, equals the current value
• Stationarity implies that the statistical properties of a stochastic process remain constant over time
• Ergodicity suggests that the time average of a stochastic process converges to the ensemble average as the observation time increases

Probability Theory Foundations

• Probability space $(\Omega, \mathcal{F}, P)$ provides a mathematical framework for modeling random phenomena
  • Sample space $\Omega$ represents the set of all possible outcomes
  • $\sigma$-algebra $\mathcal{F}$ defines a collection of events (subsets of $\Omega$) to which probabilities can be assigned
  • Probability measure $P$ assigns probabilities to events in $\mathcal{F}$, satisfying axioms of non-negativity, normalization, and countable additivity
• Random variables map outcomes from the sample space to real numbers, allowing for quantitative analysis of stochastic processes
• Expectation $\mathbb{E}[X]$ represents the average value of a random variable $X$, calculated as the weighted sum of possible values weighted by their probabilities
• Conditional probability $P(A|B)$ measures the probability of event $A$ occurring given that event $B$ has already occurred
• Independence implies that the occurrence of one event does not affect the probability of another event
  • Two events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$
  • Two random variables $X$ and $Y$ are independent if their joint distribution equals the product of their marginal distributions
• Law of large numbers states that the sample average converges to the expected value as the sample size increases, providing a connection between theoretical probabilities and empirical observations

Types of Stochastic Processes

• Discrete-time stochastic processes have a countable index set (e.g., integers), representing the time points at which the process is observed
  • Examples include random walks and autoregressive processes
• Continuous-time stochastic processes have an uncountable index set (e.g., real numbers), allowing for observations at any point in time
  • Examples include Brownian motion and Poisson processes
• Markov processes exhibit the Markov property, where the future state depends only on the current state, not on the past history
  • Markov chains are discrete-time Markov processes with a finite or countable state space
• Gaussian processes have joint distributions that follow a multivariate normal distribution, characterized by mean and covariance functions
• Lévy processes are continuous-time stochastic processes with independent and stationary increments, generalizing Brownian motion and Poisson processes
• Point processes model the occurrence of events in time or space, such as the arrival of customers or the occurrence of earthquakes
• Branching processes describe the evolution of a population where individuals independently produce offspring according to a probability distribution

Markov Chains and Their Applications

• Markov chains model systems that transition between states based on transition probabilities
• Transition matrix $P$ contains the probabilities $p_{ij}$ of moving from state $i$ to state $j$ in one step
• Chapman-Kolmogorov equations allow for the calculation of multi-step transition probabilities by matrix multiplication
• Stationary distribution $\pi$ represents the long-run proportion of time spent in each state, satisfying $\pi P = \pi$
• Ergodic Markov chains converge to a unique stationary distribution regardless of the initial state distribution
• Absorbing Markov chains have one or more absorbing states that, once entered, cannot be left
  • Fundamental matrix $N = (I - Q)^{-1}$ captures the expected number of visits to transient states before absorption, where $Q$ is the submatrix of transient states
• Applications of Markov chains include modeling customer behavior, machine maintenance, genetic mutations, and language processing

Brownian Motion and Wiener Processes

• Brownian motion, also known as a Wiener process, is a continuous-time stochastic process with independent and normally distributed increments
• Wiener process $W_t$ satisfies the following properties:
  • $W_0 = 0$ (starts at zero)
  • $W_t - W_s \sim N(0, t-s)$ for $t > s$ (normally distributed increments)
  • $W_t - W_s$ is independent of $W_u - W_v$ for disjoint time intervals $[s, t]$ and $[v, u]$ (independent increments)
  • Sample paths of $W_t$ are continuous but nowhere differentiable
• Itô integral $\int_0^T f(t, W_t) dW_t$ extends the notion of integration to stochastic processes, allowing for the integration of random functions with respect to Brownian motion
• Itô's lemma provides a formula for the differential of a function of a stochastic process, generalizing the chain rule of ordinary calculus
• Geometric Brownian motion models the exponential growth or decay of a quantity subject to random fluctuations, commonly used in financial modeling for asset prices
• Ornstein-Uhlenbeck process is a mean-reverting stochastic process that combines Brownian motion with a drift term pulling the process towards a long-term mean

Stochastic Calculus Basics

• Stochastic calculus extends the tools of ordinary calculus to stochastic processes, enabling the analysis and modeling of systems with randomness
• Stochastic differential equations (SDEs) describe the evolution of a stochastic process by combining a drift term (deterministic) and a diffusion term (stochastic)
  • General form: $dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$, where $\mu$ is the drift coefficient, $\sigma$ is the diffusion coefficient, and $W_t$ is a Wiener process
• Itô's formula allows for the calculation of the differential of a function of a stochastic process, accounting for the quadratic variation of the process
  • For a function $f(t, x)$, Itô's formula states: $df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x}dW_t$
• Girsanov's theorem allows for the change of probability measure in a stochastic process, enabling the simplification of calculations and the derivation of risk-neutral pricing formulas
• Feynman-Kac formula establishes a connection between SDEs and partial differential equations (PDEs), providing a probabilistic solution to certain PDEs
• Numerical methods for SDEs, such as the Euler-Maruyama scheme and the Milstein scheme, approximate the solution of an SDE by discretizing time and simulating sample paths

Financial Modeling with Stochastic Processes

• Stochastic processes are widely used in financial modeling to capture the uncertainty and dynamics of financial markets
• Black-Scholes-Merton model uses geometric Brownian motion to describe the evolution of asset prices and derives a closed-form solution for pricing European options
  • Assumes constant volatility, no transaction costs, and continuous trading
• Stochastic volatility models, such as the Heston model and the SABR model, incorporate time-varying and stochastic volatility to better capture the observed market dynamics
• Interest rate models, such as the Vasicek model and the Cox-Ingersoll-Ross (CIR) model, describe the evolution of interest rates using mean-reverting stochastic processes
• Credit risk models, such as the Merton model and the reduced-form models, assess the probability of default and the loss given default for credit-sensitive instruments
• Portfolio optimization techniques, such as the Markowitz model and the Black-Litterman model, use stochastic processes to model asset returns and optimize investment strategies based on risk and return preferences
• Monte Carlo simulation is a powerful tool for pricing complex financial derivatives and assessing risk by generating numerous sample paths of the underlying stochastic processes

Real-World Applications and Case Studies

• Option pricing and hedging strategies in financial markets, using models like Black-Scholes-Merton and stochastic volatility models
• Portfolio management and asset allocation, incorporating stochastic models for asset returns and risk factors
• Interest rate modeling and fixed income valuation, using short-rate models and term structure models
• Credit risk assessment and pricing of credit derivatives, such as credit default swaps (CDS) and collateralized debt obligations (CDOs)
• Insurance and actuarial applications, modeling claim arrivals, claim sizes, and policy liabilities using stochastic processes
• Energy and commodity markets, modeling price dynamics and valuing real options in production and investment decisions
• Queueing theory and operations research, analyzing waiting times, system capacity, and resource allocation in service systems and manufacturing processes
• Epidemiology and public health, modeling the spread of infectious diseases and assessing intervention strategies using stochastic compartmental models