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Course Calendar Course Calendar

Intermediate Mathematics: Understanding Stochastic Calculus

Day One

Probability Theory

  • Random variables, independence and conditional independence. Discrete random variables: mass density, expectation and moments calculation
  • Conditional discrete distributions, sums of discrete random variables
  • Continuous random variables; Probability density function, cumulative probability density function; Expectation and moments calculation; Conditional distributions and conditional expectation; Functions of random variables

Examples: Normal distribution, gamma distribution, exponential distribution, Poisson distribution

Exercise: Properties of the gamma distribution and the log-normal distribution

Workshop: Multivariate normal distributions. Linear transformations. Counter-example

  • Generating functions. Moment generating functions. Characteristic functions
  • Convergence theorems: the strong law of large numbers, the central limit theorem

Examples: Characteristic functions of Bernoulli, binomial, exponential distributions

Exercise: Moment generating functions and characteristic functions of Poisson, normal and multivariate normal distributions

Markov Chains

  • Discrete time Markov chains, the Chapman-Kolmogorov equation
  • Recurrence and transience. Invariance
  • Discrete martingales. Martingale representation theorem. Convergence theorems

Examples: Random walks: simple, reflected, absorbed

Workshop: Pricing European options within the Cox-Ross-Rubinstein model

  • Continuous time Markov chains. Generators
  • Forward/backward equations. Generating functions

Example: The Poisson process

Exercise: Superposition of Poisson Processes. Thinning


Day Two

Stochastic Calculus

  • The Wiener process. Path properties. Monte Carlo simulation
  • Gaussian processes. Diffusion processes

Examples: The Wiener process with drift. The Brownian Bridge

Exercise: The Geometric Brownian Motion. Properties of its distribution (moments)

  • Semi-martingales. Stochastic integration
  • Ito's formula. Integration by parts formula

Workshop: Multivariate normal distributions. Linear transformations. Counter-example

Examples: Characteristic functions of Bernoulli, binomial, exponential distributions

Exercises: Moment generating functions and characteristic functions of Poisson, normal and multivariate normal distributions

Stochastic Differential Equations

  • Stochastic differential equations. Existence and uniqueness of solutions. Equations with explicit solutions
  • The Markov property. Girsanov's theorem

Exercise: The Vasicek model. Connection with the O-U process. Mean. Variance. Covariance. Pricing zero-coupon bonds

Workshop: The Cox Ingersoll Ross Model. Connection with the O-U process. Properties of its distribution (mean variance, covariance). Pricing zero-coupon bonds



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