The Problem Sets were due in the lecture sessions listed in the table. 25% of the course grade is based on the problem sets.
| LEC # |
Assignments |
Topics |
| 4 |
Problem Set 1 (PDF) |
Asymptotics of Rayleigh's random walk, Central Limit Theorem, Gram-Charlier expansion
Exact solution for the position of Cauchy's random walk with non-identical steps
Computer simulation of Pearson's random walk to find the fraction of time spent in the right half plane ("arcsine law") and the first quadrant |
| 8 |
Problem Set 2 (PDF) |
Percentile order statistics, asymptotics of the median versus the mean
Computer simulation of the winding angle for Pearson's random walk, logarithmic scaling and limiting distribution
Globally-valid saddle-point asymptotics for a random walk with exponentially distributed displacements
The Void Model for granular drainage, continuum limits for the void density (mean flow profile) and the position a tracer particle, exact similarity solutions for parabolic flow to a point orifice |
| 15 |
Problem Set 3 (PDF) |
Modified Kramers-Moyall expansion for a general discrete Markov process
Black-Scholes formula for a call option, interpretation as risk neutral valuation, put-call parity
Continuum limit of Bouchaud-Sornette theory for options with residual risk (corrections to the Black-Scholes equation) |
| 24 |
Problem Set 4 (PDF) |
Linear polymer structure. Random walk with exponentially decaying correlations, depending on temperature
Polymer surface adsorption. First passage to a plane, Levy flight for adsorption sites, scalings with the chain length
Solution to the Telegrapher's equation. Fourier-Laplace transform, wave and diffusion limits, exact Green function
Inelastic diffusion. Random walk with exponentially decaying steps, approach to the Central Limit Theorem |