Discrete and continuous-time stochastic models, including Markov chains and Poisson processes. Examines stationarity, autocorrelation, and transition probabilities to analyze systems evolving over time.
The sequence concludes with an introduction to stochastic sequences, simulating random walks to model stock price movements.
Students analyze bonds as series of cash flows, using differentiation to calculate Duration and assess interest rate risk.
This lesson explores the valuation of infinite horizons, applying geometric series convergence to price perpetuities and the Dividend Discount Model.
Learners model loan payments and savings plans using finite geometric series, deriving amortization formulas for mortgages and annuities.
Students derive the compound interest formulas as geometric sequences, exploring the impact of compounding frequency and the limit as it approaches infinity (continuous compounding).
Introduction to Martingales and the Optional Stopping Theorem, applying these concepts to fair games and boundary crossing probabilities.
Defines conditional expectation as a random variable measurable with respect to a sub-sigma-algebra, utilizing the Radon-Nikodym theorem.
Analysis of Monotone Convergence, Fatou's Lemma, and Dominated Convergence Theorems to determine when limits and expectations commute.
Focuses on the derivation and application of Markov, Chebyshev, Jensen, Hölder, and Minkowski inequalities to bound expected values.
Students define expectation using the Lebesgue integral, moving from simple functions to non-negative random variables and addressing the limitations of Riemann integration.
Exploring Stochastic Gradient Descent (SGD) and its role in navigating high-dimensional, non-convex landscapes in machine learning.
Solving optimization problems under constraints using the method of Lagrange multipliers, focusing on the alignment of gradient vectors.
This sequence bridges the gap between discrete mathematics and quantitative finance, focusing on the application of geometric series to asset valuation, loan amortization, and risk management. Graduate students will develop the mathematical foundations for pricing complex financial instruments and understanding market dynamics.
A graduate-level exploration of expected value through the lens of measure theory, covering Lebesgue integration, fundamental inequalities, convergence theorems, and conditional expectation using Sigma-algebras.
A graduate-level sequence exploring the gradient vector as the foundational tool for modern optimization. Students move from the geometric interpretation of multivariate derivatives to the implementation of stochastic algorithms used in machine learning.
An advanced graduate-level exploration of stochastic processes, covering discrete and continuous-time Markov chains, Poisson processes, and queueing theory. The sequence bridges theoretical rigor with computational application through simulations and real-world modeling.
An advanced graduate-level sequence exploring the mathematical foundations and computational applications of stochastic processes, from discrete-time Markov chains to Monte Carlo simulations.
A comprehensive introduction to Time Series Analysis for 12th-grade students, focusing on random processes, autocorrelation, stationarity, and smoothing techniques. Students move from basic random walks to understanding complex dependencies in temporal data.
A project-based exploration of stochastic modeling, focusing on Queueing Theory and Monte Carlo simulations. Students design and build computational models to optimize real-world systems like traffic flow and service lines.
A 12th-grade statistics sequence exploring Poisson processes, transitioning from discrete counts to continuous time intervals and waiting times. Students will investigate arrival rates, the exponential distribution, and the unique memoryless property through inquiry and simulation.
A high-level exploration of stochastic processes, focusing on how random systems reach equilibrium. Students will master Markov chains, steady-state algebra, and real-world applications like Google's PageRank algorithm.
A comprehensive sequence for 12th-grade students on discrete-time Markov chains, covering state diagrams, transition matrices, and n-step probability calculations using matrix algebra.
A graduate-level sequence exploring continuous-time stochastic processes through the lens of computational simulation. Students transition from discrete to continuous time models, focusing on Poisson processes, CTMCs, and queuing theory with a strong emphasis on empirical validation and theoretical rigor.
A graduate-level exploration of the mathematical foundations of discrete-time Markov chains, focusing on state classification, limiting behavior, and time reversibility. This sequence emphasizes formal derivation, proofs, and the application of linear algebra to stochastic systems.
Student worksheet for Lesson 5, focusing on manual random walk simulations and the mathematical application of the stochastic price equation.
Teacher facilitation guide for Lesson 5, detailing the introduction to stochastic sequences, the drift/volatility model, and the concept of Monte Carlo simulations.
Slide deck for Lesson 5 introducing stochastic sequences, simulations of random walks, and their application to stock price modeling.
Student worksheet for Lesson 4, covering bond price calculations and the derivation of Macaulay and Modified duration to assess interest rate sensitivity.
Teacher facilitation guide for Lesson 4, detailing the bond pricing series derivation and the calculus-based sensitivity analysis (duration).
Slide deck for Lesson 4 exploring bond pricing as a finite series and the derivation of Macaulay duration to measure interest rate risk.
Student worksheet for Lesson 3, covering the pricing of perpetuities and the mathematical derivation/application of the Dividend Discount Model.
Teacher facilitation guide for Lesson 3, outlining the mathematical derivation of perpetuity valuation and the Dividend Discount Model convergence conditions.
Slide deck for Lesson 3 exploring the valuation of perpetuities and the Dividend Discount Model through the lens of infinite geometric series convergence.
Student worksheet for Lesson 2, covering the calculation of future value for savings and the derivation of monthly mortgage payments using finite geometric sums.
Teacher facilitation guide for Lesson 2, outlining the derivation of the annuity formula and providing guidance for the amortization simulation.
Slide deck for Lesson 2 focusing on finite geometric series and their application in calculating annuity values and loan amortization.
