Stochastic calculus, Fall 2015

Course Details and Syllabus

Source materials

See the Resources page.

Detailed course outline

The course is a sequence of modules which take from one to three weeks. Material may be cut, or added (less likely) depending on time constraints.

  1. Discrete time Gaussian processes. Review of multivariate normals and the role of linear algebra and matrix computation. Linear Gaussian recurrence relations, stability, and limiting distributions. Paths as multivariate normals. Filtering and prediction using matrix Ricatti equations. Maximum likelihood parameter identification.
  2. Discrete Markov chains, path space, transtion probabilities, evolution of probability, value functions, backward and forward equations. The path space formalism of stochastic processes in the discrete case: probabilities in path space, conditional expectation with partial information, set algebras and filtrations representing partial information, the tower property, progressively measurable functions. Various random walks as examples.
  3. Brownian motion. Continuous paths in continuous time. Convergence of discrete time processes to Brownian motion. Independent increments property, scaling of increments. Gaussian transition probabilities and the semi-group property. Relationship to the heat equation. Heat equation methods for hitting probabilities and hitting times, the method of images, probability flux.
  4. The Ito calculus for Brownian motion. The Ito integral, convergence of Riemann sums for non-anticipating integrands. Ito's lemma for functions of Brownian motion. Relationship to the heat equation. Heat equation methods for hitting probabilities and hitting times, the method of images, probability flux. Geometric Brownian motion.
  5. General Ito and diffusion processes. Infinitesimal mean and variance, quadratic variation. Modeling with stochastic differential equations. Weak solutions, and solutions as functions of Brownian motion (strong solutions). Ito integral and Ito's lemma for general processes. Computer simulation of diffusion processes. Martingales and functions of martingales.
  6. Partial differential equations and diffusion processes. Derivation of backward and forward equations, Feynman Kac. Qualitative properties of solutions, well posedness, positivity, bounds, regularity. Finite difference methods for approximate solution. Special exponential ansatz solutions.
  7. Change of measure, Girsanov theory. Radon Nikodym and likelihood ratio function representation of one measure in terms of another. Girsanov formula for changing drift. Applications in finance and Monte Carlo.

Academic integrity (cheating)

Please the NYU academic integrity policy. All those rules apply to this class. Unless explicitly stated in writing on the assignment, all homework in this class is individual. Students may not hand in work they have copied from another source. Students are forbidden to allow their homework to be copied for the purpose of cheating. If assignments from different students have similarities that show one was copied from the other, both students will be penalized. This applies to written work and coding.

The instructor (I) will try to create an environment that does not encourage academic integrity violations. I will ensure that the work load is managable by an individual student working independently. I will work with the grader and TA to identify violations. This will minimize the benefit of cheating and ensure that those who don't cheat are not at a disadvantage. I will listen to anyone's thoughts or complaints on this issue. Please let me know if the work load is unmanageable or if you suspect others of cheating.