🔀Stochastic Processes Unit 11 – Stochastic calculus

Key Concepts and Definitions

• Stochastic process: A collection of random variables indexed by time, representing the evolution of a system with randomness
• Brownian motion: A continuous-time stochastic process that models random motion of particles, with independent increments and normally distributed displacements
• Martingale: A stochastic process whose expected future value, given the current information, is equal to its current value
  • Submartingale: A stochastic process whose expected future value is greater than or equal to its current value
  • Supermartingale: A stochastic process whose expected future value is less than or equal to its current value
• Filtration: An increasing sequence of σ-algebras that represents the information available at each point in time
• Adapted process: A stochastic process whose value at any time t depends only on the information available up to time t
• Itô integral: A stochastic integral that extends the Riemann-Stieltjes integral to stochastic processes, allowing integration with respect to Brownian motion
• Itô's lemma: A formula for computing the differential of a function of a stochastic process, generalizing the chain rule to stochastic calculus
• Stochastic differential equation (SDE): A differential equation driven by a stochastic process, typically Brownian motion, used to model systems with randomness

Probability Theory Foundations

• Probability space: A mathematical construct consisting of a sample space Ω, a σ-algebra F, and a probability measure P, used to model random phenomena
• Random variable: A measurable function from the sample space to the real numbers, assigning a numerical value to each outcome of a random experiment
• Expectation: The average value of a random variable, calculated as the integral of the variable with respect to the probability measure
• Conditional expectation: The expected value of a random variable given the information available up to a certain time or event
• Convergence concepts: Different notions of convergence for sequences of random variables, such as almost sure convergence, convergence in probability, and convergence in distribution
• Central limit theorem: A fundamental result stating that the sum of a large number of independent and identically distributed random variables converges to a normal distribution
• Law of large numbers: A theorem that describes the long-term stability of the average of a sequence of random variables, converging to the expected value as the number of variables increases

Introduction to Stochastic Processes

• Markov process: A stochastic process whose future state depends only on the current state, not on the past history
  • Markov chain: A discrete-time Markov process with a countable state space
  • Markov property: The memoryless property of a Markov process, where the future state depends only on the current state
• Poisson process: A counting process that models the occurrence of rare events, with independent increments and exponentially distributed inter-arrival times
• Renewal process: A generalization of the Poisson process, where the inter-arrival times are independently and identically distributed, but not necessarily exponential
• Stationary process: A stochastic process whose joint probability distribution does not change when shifted in time
• Ergodicity: A property of a stochastic process where the time average of a single realization converges to the ensemble average as the time horizon increases
• Gaussian process: A stochastic process where any finite collection of random variables has a multivariate normal distribution
• Continuity and differentiability: The properties of sample paths of a stochastic process, such as continuity and differentiability with respect to time

Brownian Motion and Wiener Processes

• Standard Brownian motion: A continuous-time stochastic process $W_t$ with independent and normally distributed increments, satisfying $W_0 = 0$, $\mathbb{E}[W_t] = 0$, and $\text{Var}(W_t) = t$
• Wiener process: Another name for Brownian motion, often used in the context of stochastic calculus
• Properties of Brownian motion:
  • Continuous sample paths: Almost surely, the sample paths of Brownian motion are continuous functions of time
  • Independent increments: For any non-overlapping time intervals, the increments of Brownian motion are independent random variables
  • Normally distributed increments: The increments of Brownian motion over an interval of length t are normally distributed with mean 0 and variance t
• Quadratic variation: A measure of the variability of a stochastic process, defined as the limit of the sum of squared increments over a partition of the time interval, as the mesh of the partition goes to zero
  • For Brownian motion, the quadratic variation over the interval $[0, t]$ is equal to t
• Lévy characterization: A theorem stating that a continuous, adapted process with independent and stationary increments is a Brownian motion if and only if it has continuous sample paths and quadratic variation equal to t
• Geometric Brownian motion: A stochastic process obtained by exponentiating Brownian motion, used to model stock prices in financial mathematics
• Fractional Brownian motion: A generalization of Brownian motion with correlated increments, characterized by the Hurst parameter $H \in (0, 1)$, where $H = 1/2$ corresponds to standard Brownian motion

Itô Calculus

• Itô integral: A stochastic integral that extends the Riemann-Stieltjes integral to stochastic processes, allowing integration with respect to Brownian motion
  • Definition: For a suitable stochastic process $X_t$, the Itô integral $\int_0^t X_s dW_s$ is defined as the limit in probability of the sum $\sum_{i=1}^n X_{t_{i-1}} (W_{t_i} - W_{t_{i-1}})$ over a partition of $[0, t]$, as the mesh of the partition goes to zero
  • Properties: The Itô integral is a martingale, has zero mean, and its quadratic variation is given by $\int_0^t X_s^2 ds$
• Itô's lemma: A formula for computing the differential of a function of a stochastic process, generalizing the chain rule to stochastic calculus
  • For a twice continuously differentiable function $f(t, x)$ and a stochastic process $X_t$ satisfying $dX_t = \mu_t dt + \sigma_t dW_t$, Itô's lemma states that $df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma_t \frac{\partial f}{\partial x} dW_t$
• Itô isometry: A formula relating the expected value of the square of an Itô integral to the expected value of the integrated process
  • For a suitable stochastic process $X_t$, the Itô isometry states that $\mathbb{E}\left[\left(\int_0^t X_s dW_s\right)^2\right] = \mathbb{E}\left[\int_0^t X_s^2 ds\right]$
• Change of variables formula: A generalization of Itô's lemma to multidimensional stochastic processes and vector-valued functions
• Girsanov's theorem: A result that allows for changing the probability measure of a stochastic process, effectively transforming a process with drift into a martingale
• Feynman-Kac formula: A theorem connecting the solution of a partial differential equation to the expectation of a functional of a stochastic process

Stochastic Differential Equations

• Stochastic differential equation (SDE): A differential equation driven by a stochastic process, typically Brownian motion, used to model systems with randomness
  • 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 Brownian motion
• Existence and uniqueness of solutions: Conditions under which an SDE has a unique solution, such as the Lipschitz continuity and linear growth conditions on the coefficients
• Explicit solutions: SDEs that can be solved analytically, such as the geometric Brownian motion and the Ornstein-Uhlenbeck process
• Numerical methods: Techniques for approximating the solution of an SDE when an explicit solution is not available, such as the Euler-Maruyama method and the Milstein method
• Stability analysis: Investigating the long-term behavior of the solutions of an SDE, such as the existence of stationary distributions and the convergence of numerical schemes
• Stochastic control: The problem of finding the optimal control strategy for a system described by an SDE, in order to minimize a cost functional or maximize a reward functional
• Stochastic filtering: The problem of estimating the state of a system described by an SDE, based on noisy observations of the system
• Backward stochastic differential equations (BSDEs): A class of SDEs where the terminal condition is given, and the solution is adapted to the filtration generated by the driving Brownian motion

Applications in Finance and Physics

• Financial mathematics:
  • Option pricing: Using stochastic calculus to derive the Black-Scholes formula for pricing European options, and extensions to more complex options (American, Asian, barrier)
  • Portfolio optimization: Applying stochastic control techniques to find the optimal investment strategy for a portfolio of assets, considering risk and return trade-offs
  • Interest rate models: Modeling the dynamics of interest rates using SDEs, such as the Vasicek model and the Cox-Ingersoll-Ross model
  • Credit risk: Assessing the probability of default and the loss given default for financial instruments, using stochastic models for the underlying assets and the credit events
• Physics:
  • Diffusion processes: Modeling the random motion of particles in a medium using Brownian motion and related stochastic processes, such as the Ornstein-Uhlenbeck process
  • Stochastic thermodynamics: Extending the laws of thermodynamics to systems at the nanoscale, where thermal fluctuations play a significant role, using stochastic models and Itô calculus
  • Quantum mechanics: Describing the evolution of quantum systems using stochastic processes, such as quantum Brownian motion and quantum stochastic differential equations
  • Turbulence: Modeling the chaotic and unpredictable behavior of fluid flows using stochastic partial differential equations, such as the stochastic Navier-Stokes equations
• Other fields:
  • Biology: Modeling the dynamics of populations, the spread of epidemics, and the evolution of species using stochastic processes and SDEs
  • Engineering: Analyzing the reliability and performance of complex systems, such as power grids and communication networks, using stochastic models and control techniques
  • Machine learning: Incorporating stochastic processes and SDEs into learning algorithms, such as stochastic gradient descent and Bayesian inference, to handle uncertainty and improve generalization

Problem-Solving Techniques

• Transformation methods: Simplifying an SDE by applying a change of variables, such as the logarithmic transformation for geometric Brownian motion
• Moment generating functions: Deriving the moments (mean, variance, etc.) of the solution of an SDE by computing the moment generating function and its derivatives
• Fokker-Planck equation: A partial differential equation that describes the time evolution of the probability density function of the solution of an SDE
  • For an SDE $dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dW_t$, the Fokker-Planck equation for the probability density function $p(t, x)$ is $\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}(\mu p) + \frac{1}{2} \frac{\partial^2}{\partial x^2}(\sigma^2 p)$
• Kolmogorov equations: A system of partial differential equations that characterize the transition probabilities of a stochastic process, consisting of the forward equation (Fokker-Planck) and the backward equation
• Monte Carlo simulation: Approximating the solution of an SDE or the expectation of a functional of the solution by generating a large number of sample paths and computing the empirical average
• Martingale representation theorem: A result stating that any square-integrable martingale can be represented as an Itô integral with respect to a Brownian motion, which can be used to solve BSDEs and stochastic control problems
• Duality methods: Exploiting the relationship between an SDE and its associated backward SDE or partial differential equation to solve problems in stochastic control and filtering
• Asymptotic analysis: Studying the behavior of the solution of an SDE in the limit of small or large parameters, such as the noise intensity or the time horizon, using perturbation methods and scaling arguments