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Stochastic Calculus Course

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Stochastic Calculus Course

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Stochastic Calculus Course

Name: Stochastic Calculus Course
Price: 222 USD

Master the rigorous mathematical framework behind modern stochastic calculus, from measure-theoretic foundations to Itô's formula and stochastic differential equations. This course equips quantitative researchers, mathematicians, and financial engineers with the theoretical depth and computational tools demanded by advanced probability and derivatives modeling.

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What you will learn:

You will build a complete, rigorous foundation in stochastic calculus, starting from measure theory and probability spaces and progressing through Brownian motion, the Itô integral, and Itô's formula. You will study continuous-time martingale theory, Girsanov's theorem, and the full theory of stochastic differential equations, including existence, uniqueness, and explicit solution methods. The course connects SDEs to partial differential equations through generator theory and the Feynman-Kac formula. Supplementary material covers numerical methods for SDEs, jump processes and Lévy calculus, mathematical finance applications, stochastic control, and backward SDEs.

How you study in practice Stochastic Calculus Course

How you practice Stochastic Calculus Course

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Course Content

8 Chapters • 40 LessonsDuration between 4 and 360 hours (you decide)

Chapter 1See details

Probability Theory Foundations for Stochastic Calculus

Lesson 1 • Random Variables and Distributions
Defines random variables as measurable functions and characterizes their distributions. Connects abstract measure theory to practical probabilistic computation.
Lesson 2 • Measure Theory and Probability Spaces
Introduces sigma-algebras, measurable sets, and probability measures as the formal basis for randomness. Establishes the language used throughout all subsequent chapters.
Lesson 3 • Expectation and Integration
Develops the Lebesgue integral as the rigorous definition of expectation. Provides tools for computing moments and applying convergence theorems.
Lesson 4 • Modes of Convergence
Distinguishes almost-sure, in-probability, and Lp convergence and their relationships. Equips students to analyze limiting behavior of stochastic sequences.
Lesson 5 • Conditional Expectation
Defines conditional expectation with respect to a sigma-algebra, the cornerstone of martingale theory. Prepares students for filtration-based reasoning in later chapters.

Chapter 2See details

Stochastic Processes and Filtrations

Lesson 1 • Markov Processes and Transition Kernels
Introduces the Markov property and transition kernels as a special class of stochastic processes. Provides context for diffusion processes studied later.
Lesson 2 • Filtrations and Adapted Processes
Formalizes filtrations as increasing families of sigma-algebras representing information over time. Introduces adapted and progressively measurable processes.
Lesson 3 • Martingales in Discrete Time
Develops discrete-time martingales, submartingales, and supermartingales with key inequalities. Builds intuition before the continuous-time extension in later chapters.
Lesson 4 • Gaussian Processes
Characterizes Gaussian processes through their mean and covariance functions. Lays the groundwork for Brownian motion as the canonical Gaussian process.
Lesson 5 • Fundamentals of Stochastic Processes
Defines stochastic processes, sample paths, and finite-dimensional distributions. Establishes the vocabulary for all process-based analysis in the course.

Chapter 3See details

Brownian Motion: Construction and Properties

Lesson 1 • Path Properties of Brownian Motion
Analyzes Holder continuity, nowhere differentiability, and unbounded variation. These properties motivate the need for a new integration theory.
Lesson 2 • Constructing Brownian Motion
Presents Wiener's construction and the Levy-Ciesielski series expansion. Demonstrates existence and provides intuition for the irregular nature of paths.
Lesson 3 • Multidimensional and Transformed Brownian Motion
Extends Brownian motion to multiple dimensions and introduces key transformations. Prepares students for vector-valued stochastic differential equations.
Lesson 4 • Brownian Motion as a Martingale
Establishes Brownian motion and its key transformations as martingales. Connects path properties to the martingale framework developed earlier.
Lesson 5 • Axiomatic Definition of Brownian Motion
States the defining properties of standard Brownian motion and verifies consistency. Anchors all subsequent stochastic calculus in a well-defined process.

Chapter 4See details

The Ito Integral

Lesson 1 • Ito Integral for Simple Processes
Defines the Ito integral for elementary step processes and establishes the isometry. Provides the rigorous base from which the general integral is extended.
Lesson 2 • Properties of the Ito Integral
Derives linearity, adaptedness, continuity, and the quadratic variation of Ito integrals. Equips students with tools for manipulating stochastic integrals in applications.
Lesson 3 • Limitations of Classical Integration
Demonstrates why Riemann-Stieltjes integration fails for Brownian motion paths. Motivates the construction of a new integral suited to unbounded-variation integrators.
Lesson 4 • Ito Integral with Respect to General Martingales
Generalizes the Ito integral beyond Brownian motion to square-integrable martingales. Broadens the framework for applications to jump processes and general semimartingales.
Lesson 5 • Extension to Square-Integrable Integrands
Extends the Ito integral to the full L2 class via density and isometric extension. Establishes the martingale property of the resulting integral.

Chapter 5See details

Ito's Formula and Stochastic Calculus Rules

Lesson 1 • Multidimensional Ito's Formula
Extends Ito's formula to functions of multiple Ito processes. Enables differentiation of vector-valued processes and cross-variation terms.
Lesson 2 • Ito Processes and Stochastic Differentials
Defines Ito processes as solutions to stochastic differential equations in integral form. Introduces the differential notation used throughout applied stochastic calculus.
Lesson 3 • Integration by Parts and Stochastic Product Rule
Derives the stochastic integration-by-parts formula and its applications. Provides tools for transforming and simplifying stochastic integral expressions.
Lesson 4 • Ito's Formula in One Dimension
Derives the one-dimensional Ito formula via Taylor expansion and quadratic variation. Establishes the fundamental calculus rule for stochastic processes.
Lesson 5 • Stratonovich Calculus and Conversion
Introduces the Stratonovich integral and its chain rule, and derives the conversion formula to Ito form. Clarifies when each convention is preferred in applications.

Chapter 6See details

Martingale Theory and Change of Measure

Lesson 1 • Local Martingales and Semimartingales
Defines local martingales via localization sequences and introduces semimartingales. Broadens the class of processes for which stochastic integration is defined.
Lesson 2 • Continuous-Time Martingales
Extends martingale theory to continuous time, covering regularity and decomposition results. Provides the theoretical backbone for stochastic calculus in continuous time.
Lesson 3 • Girsanov's Theorem
Derives Girsanov's theorem for changing the drift of a Brownian motion via measure change. Enables transformation of SDEs to simpler forms under an equivalent measure.
Lesson 4 • Applications of Girsanov's Theorem
Applies measure changes to risk-neutral pricing, filtering, and SDE analysis. Demonstrates the practical power of Girsanov's theorem across multiple domains.
Lesson 5 • Doob-Meyer Decomposition
Decomposes submartingales into a martingale and a predictable increasing process. Introduces the compensator concept central to semimartingale theory.

Chapter 7See details

Stochastic Differential Equations

Lesson 1 • Dependence on Initial Conditions and Parameters
Analyzes how SDE solutions vary with initial data and parameters, including flow properties. Prepares students for sensitivity analysis and control applications.
Lesson 2 • Formulation and Examples of SDEs
Defines SDEs in integral form and presents canonical examples including geometric Brownian motion. Connects the abstract framework to concrete modeling scenarios.
Lesson 3 • Existence and Uniqueness of Strong Solutions
Proves existence and uniqueness under Lipschitz and linear growth conditions. Establishes the theoretical foundation for trusting SDE solutions in applications.
Lesson 4 • Weak Solutions and Martingale Problems
Introduces weak solutions and the martingale problem formulation of SDEs. Extends the solution concept to cases where strong solutions may not exist.
Lesson 5 • Explicit Solution Techniques
Applies Ito's formula and integrating factors to solve linear and separable SDEs explicitly. Builds computational fluency for the most common SDE classes.

Chapter 8See details

Diffusion Processes and Connections to PDEs

Lesson 1 • Feynman-Kac Formula
Derives the Feynman-Kac formula linking parabolic PDEs to conditional expectations. Provides a probabilistic tool for solving PDEs and pricing financial derivatives.
Lesson 2 • Generators of Diffusion Processes
Defines the infinitesimal generator of a diffusion and computes it for standard processes. Connects the generator to the drift and diffusion coefficients of the SDE.
Lesson 3 • Ergodicity and Long-Run Behavior
Analyzes ergodic properties of diffusions and convergence to stationary distributions. Provides tools for long-run simulation and statistical inference on diffusions.
Lesson 4 • Kolmogorov Forward and Backward Equations
Derives the Kolmogorov backward and Fokker-Planck forward equations for transition densities. Establishes the PDE perspective on diffusion dynamics.
Lesson 5 • Boundary Value Problems and Exit Times
Solves Dirichlet and Poisson problems using probabilistic representations via harmonic functions. Connects stopping times to boundary behavior of diffusions.

Certification

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This course is for you:

Applied mathematicians: seeking a rigorous graduate-level probability and calculus treatment.
Quantitative analysts: wanting theoretical grounding behind the models they already use.
PhD students: entering research areas where stochastic analysis is a core language.
Financial engineers: aiming to move beyond formulas and understand their mathematical origins.
Statisticians: expanding into continuous-time stochastic modeling and diffusion processes.
Self-taught programmers: who simulate SDEs but lack the formal theory to back them up.

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