Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Probability and Statistics ↓


Stochastic Processes


Stochastic processes are a fundamental concept in probability and statistics, widely used in various fields such as finance, economics, biology, and physics. They provide a framework for modeling systems that evolve in a random manner over time. Understanding stochastic processes allows us to make predictions and decisions based on incomplete information.

Basic concept

A stochastic process can be thought of as a collection of random variables that represent the evolution of a system of random values over time. Each random variable in the collection corresponds to the state of the system at a particular point in time. Formally, a stochastic process is usually defined as a family of random variables {X(t), t ∈ T}, where T is an index set, often representing time.

Time domain

The time domain T can be discrete or continuous. If T is countable, the stochastic process is called a discrete-time process. If T is uncountable (for example, the interval of real numbers), the process is called a continuous-time process.

Time domain examples

  • Discrete time: T = {0, 1, 2, 3, ...}
  • Continuous-time: T = [0, ∞)

State location

The state space of a stochastic process is the set of all possible values that the random variable can take. Like the time domain, the state space can also be discrete or continuous.

Examples of state spaces

  • Discrete state space: {0, 1, 2, ...}
  • Continuous state space: [0, 1]

Classification of stochastic processes

Stochastic processes can be classified according to various criteria:

Based on the time domain

  • Discrete-time stochastic process: A process where the time index set T is discrete.
  • Continuous-time stochastic process: A process where T is continuous.

Based on state location

  • Discrete-state stochastic process: The state space is countable.
  • Continuous-state stochastic process: The state space is infinite.

Examples of stochastic processes

Bernoulli process

The Bernoulli process is a simple type of stochastic process used to model sequences of binary outcomes, such as tossing a coin. The outcome of each trial is either a success or a failure.

    Let Xn be the outcome of the nth trial (0 for failure, 1 for success).
    The Bernoulli process {X n } is a sequence of independent random variables, where for each X n, P(X n =1) = p and P(X n =0) = 1 - p.

Random walk

Random walk is another important type of stochastic process that is often used in physics and finance. It describes a path consisting of a sequence of random steps.

    Consider a simple case of a random walk on a number line starting at 0. 
    At each step, you move one unit to the right with probability p and one unit to the left with probability 1-p.

Markov process

A Markov process is a specific type of stochastic process characterized by the "memoryless" property. This means that the future state depends only on the current state, not on the sequence of events before it.

    A discrete-time Markov chain is a stochastic process { Xn } with a finite or countable state space S such that:
    P(x n+1 = j | x 0 = i0, ..., x n = in) = P(x n+1 = j | x n = i).

Brownian motion

Brownian motion, also known as the Wiener process, is a continuous-time stochastic process used in finance and physics. It is used to model random behavior in a variety of contexts.

    The properties of Brownian motion {B(t), t ≥ 0} include:
    - B(0) = 0 almost surely
    - The paths of B are continuous but nowhere differentiable
    - B has independent increase
    – Increment B(t)-B(s) ~ N(0, ts) for 0 ≤ s < t
b(t)

Poisson process

The Poisson process is a stochastic process that models the random occurrence of events over time, having applications in queuing theory, telecommunications, etc.

    A Poisson process {N(t), t ≥ 0} with rate λ is characterized by:
    - N(0) = 0 almost surely
    - Increment is independent
    - The number of events in a time interval of length t has a Poisson distribution with parameter λt, that is, P(N(t)=n) = e -λt (λt) n /n!

Applications of stochastic processes

Stochastic processes are used in a variety of fields because of their ability to model real-world phenomena that are subject to randomness and uncertainty.

Finance

In finance, stochastic processes are used to model stock prices, interest rates, and other economic indicators. The well-known Black-Scholes model uses geometric Brownian motion to price European options.

Economics

Economists use stochastic processes to analyze economic phenomena that evolve over time, such as inflation, exchange rates, and economic cycles. For example, Markov processes are often used in modeling economic growth.

Biology and medicine

In biology and medicine, stochastic processes model population dynamics, the spread of diseases, neuron firing, and much more. The Hodgkin-Huxley model uses stochastic differential equations to describe action potentials in neurons.

Mathematical formulation of stochastic processes

Stochastic processes often require detailed mathematical formulations for analysis. Probability theory and measure theory provide a foundation. Here we will discuss some of the concepts frequently used in this context.

Probability space

Probability space is a mathematical structure that provides a formal model of a random experiment. It consists of three elements:

  • The sample space Ω, the set of all possible outcomes
  • The σ-algebra F, a set of events
  • Probability measure P, a function assigning probabilities to events

Random variables

A random variable is a measurable function from the sample space Ω to the real numbers. For a stochastic process {X(t), t ∈ T}, each X(t) is a random variable associated with the process.

Filtration

Filtering is a concept used to model the information available over time for a stochastic process. It is a growing family of σ-algebras {F_t} that represent information up to time t.

    Filtration satisfies: 
    F_s ⊆ F_t for all 0 ≤ s ≤ t

Conclusion

Stochastic processes are a rich and diverse field of study in probability and statistics. They provide essential tools for modeling complex systems that evolve under uncertainty. From simple models such as random walks to sophisticated applications in finance and biology, understanding stochastic processes is important for both theoretical research and practical implementation.

By exploring different types of processes such as Bernoulli processes, Markov chains, Brownian motion and Poisson processes, we can appreciate the wide range of applications. The mathematical intricacies of probability spaces, random variables and filtration emphasize the rigorous foundation needed to advance knowledge and practical skills in this field.

Further studies could delve deeper into numerical methods for simulating stochastic processes, computational tools for analysis, and interdisciplinary approaches that apply stochastic models to new and emerging areas.


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