Stochastic calculus, Fall 2014

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, filters 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 policy

A student will not be allowed to take this class unless she or he agrees, in writing (see part 1 of Assignment 1), to follow the following academic integrity policies for the class. These policies are consistent with the policies of the Graduate School of Arts and Sciences of NYU, the Department of Mathematics of the Courant Institute of NYU, and the Program in Mathematics in Finance.

  1. Students are strongly encouraged to:
    • Collaborate with each other to understand the material
    • Discuss assigned exercises, both analytical and programming, with fellow students and others.
    • Seek help from the instructor or the teaching assistant during office hours and problem sessions.
    • Consult outside reference materials including web-based resources.
    • Exchange information or seek help using the class message board.
    • Inform the instructor or seek extensions to deadlines if the assignment is too long of if reasonable circumstances prevent getting it done on time.
  2. Students may not copy solutions to exercises from other students or from web-based resources. Every word and formula a student hands in must be written or composed by that student.
  3. Students may not copy computer code from other students. Students may copy a few lines of code from web-based resources for non-mathematical things such as output formatting or procedure specification. Students may not copy mathematical code that involve formulas, random number generation, or data visualization.
  4. Students may not share their work in ways that would allow other students to violate guidlines (2) or (3) above. Students may not share completed assignments or code with other students.
  5. Students who are suspected of violating guidelines (2)-(4) may receive warnings or punishments at the discression of the instructor. Punishments may include lowering a homework or course grade, a note in the student's academic record, or, for severe or repeated violations, expulsion from NYU (very rare).
  6. A student has the right to be informed in righting if she or he is being punished. A student has the right to appeal to the department chair and the Dean of the Graduate School.