PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Probability and Statistics ↓


Stochastic Processes


Stochastic processes are a powerful tool used in probability and statistics to model systems or phenomena that evolve over time under the influence of randomness. These processes are often found in various fields such as finance, biology, physics, and engineering. A thorough understanding of stochastic processes is important as it lays the foundation for various complex models and analyses.

Introduction to stochastic processes

Basically, a stochastic process is a collection of random variables indexed by time or space. Let us first understand this:

- Random variables: These are variables that take on different values, each of which has a particular probability. For example, the roll of a fair dice is a random variable that takes on integer values between 1 and 6.

- Index sets: these are often time-based, meaning that the random variable is described in terms of time (e.g. time step 1, 2, 3, ...). However, they can also be based on other dimensions, such as distance or area.

When talking about stochastic processes, two concepts often emerge:

- State Space: The set of all possible states in which the process can be.

- Sample path: The sample path is the realization of the stochastic process; it is like watching the random behavior evolve step by step over time.

Examples of stochastic processes

1. Random walk

Probably the simplest and most intuitive example of a stochastic process is a random walk. Imagine you are standing on a number line at position 0. Every second, you make one move: one step to the right or one step to the left, each with equal probability.

01234

The above diagram shows a simplified number line, and the red dots indicate the positions where the pedestrian might take a step. If each step is taken based on a fair coin toss (heads for a step to the right), then after a certain number of steps, your series of positions forms a "random walk" on the number line.

2. Brownian motion

Brownian motion models the random motion of particles suspended in a fluid. The concept is similar to random walk, but it is continuous in time and space. It is often used in financial mathematics to model stock price movements.

B(t) = B(0) + ∑ Z_i Δt_i where Z_i ~ N(0,1)

Brownian motion can be thought of as the limit of random walk where the steps become extremely small or occur extremely quickly.

Basic properties

There are several key properties of stochastic processes:

1. Stability

A stochastic process is said to be stationary if its statistical properties do not change over time. In other words, for any change in time, the behaviour of the process in terms of mean and variance remains stationary.

2. Markov property

The Markov property describes "memorylessness". A process has this property when the future state depends only on the current state and not on how the process got there. A popular example of this is the Markov chain.

AB

In the above graphical representation, the process moves from state A to state B, which is indicated by the arrow. The transition does not depend on how we reached state A - it is simply a transition from A to B.

Types of stochastic processes

1. Discrete-time processes

These are processes observed at discrete intervals. Random walks and Markov chains are classic examples. Think of observing the temperature every hour.

2. Continuous-time processes

Here, the observations occur continuously over time, such as in Brownian motion. Continuous-time processes are more complex to handle, but they accurately represent systems where changes can occur spontaneously at any time.

3. Discrete-space processes

In this case, the state space is countable. A random walk on a lattice, where every position lies on a discrete number line, is a typical example.

4. Continuous-space processes

In contrast, continuous space allows state variables to take any values within a range. Such processes are heavily used in physics and financial modeling.

Application

1. Finance

Stochastic processes accurately encode the unpredictable behavior of a financial asset. The random and continuous paths of an asset price are often captured by stochastic differential equations. The Black-Scholes model, a well-known model for pricing options, is based on stochastic processes.

2. Biology

Biological systems often involve variation and randomness. Modelling of population growth, the spread of diseases or even gene expression can be achieved using stochastic processes.

3. Engineering and physics

In engineering, stochastic models are often used to filter out noise in signal processing. Similarly, in physics, the modeling of heat diffusion and molecular motion often relies on stochastic processes.

Mathematical basis

Let us examine some mathematical aspects of stochastic processes:

1. Expectation and variation

For any stochastic process X(t), its expected value and variance at any time t are given by:

E[X(t)] = ∫ xp(x,t) dx Var(X(t)) = E[(X(t) - E[X(t)])^2]

The function p(x,t) is the probability density function of X at time t.

2. Correlation

Correlation functions measure how two variables are related. The autocorrelation function for a process at two different times s and t is defined as:

R_X(s,t) = E[(X(s) - E[X(s)])(X(t) - E[X(t)])]

3. Martingales

The martingale is a special type of stochastic process, particularly meaningful in financial contexts. Its defining feature is that the future expectation is equal to the current observation. Formally,

E[X(t+k) | X(t)] = X(t) for all k ≥ 0

Conclusion

Stochastic processes provide a rich framework for modeling complex, time-varying systems under uncertainty. While they can be mathematically sophisticated, breaking them down into fundamental principles helps us understand their nature and reveals their essential applications across various domains. As we appreciate their intricacies, leveraging stochastic processes helps make insightful predictions and informed decisions across a spectrum of challenging scenarios.


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Stochastic Processes

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