A PDF copy of the syllabus can be found here: APMA_1930U.pdf

November 24: As a final application, we proved that the value function for a class of stochastic optimal control problems satisfies a Hamilton-Jacobi-Bellman equation.

November 19: We turned our attention to nonparametric statistics and discussed how stochastic differential equations can be employed in the analysis of nonparametric regression problems. See for example:

L. Brown and M. Low, Asymptotic equivalence of nonparametric regression and white noise Annals of Statistics, vol. 24, no. 6, 2384-2398, 1996.

Nov 12 - Nov 17: We introduced the notion of the infinitesimal generator of a process and derived the Feynman-Kac formula (sections 6.1 and 6.2).

The fourth homework assignment has been posted.

Notes: p167-p185.pdf

Nov 5 - Nov 10: We derived the martingale inequalities of section 2.7 and proved the existence and uniqueness theorem of section 5.2.3.

Notes: p139-p166.pdf

October 29: We discussed some constructions of solutions to stochastic differential equations that avoid any invocation of martingale estimates (see sections 5.2.1 and 5.2.2).

October 27: We defined white noise as a generalized (distribution-valued) stochastic process. We also investigated the solutions of certain linear SDEs and introduced the Ornstein-Uhlenbeck process.

The third homework assignment has been posted.

Notes: p119-p138.pdf

Oct 20 - Oct 22: We focused on stochastic differentials and derived Itô's chain and product rules (section 4.3).

Notes: p102-p118.pdf

October 15: We defined the notion of a progressively measurable process and constructed the Itô stochastic integral (section 4.2). A historical note on the Itô integral can be found here:

Philip Protter, The work of Kyosi Itô Notices of the American Mathematical Society, 2007.

October 13: We introduced the Paley-Wiener-Zygmund stochastic integral and derived a special case of the Itô isometry. We also discussed the quadratic variation of Brownian motion (section 4.1).

Notes: p81-p101.pdf

October 8: We proved that sample paths of Brownian motion are with probability one nowhere differentiable (section 3.4.2).

The second homework assignment has been posted.

Notes: p65-p80.pdf

Oct 1 - Oct 6: We outlined the construction of Brownian motion due to Norbert Wiener. For more details, see, e.g., chapter 6 of Richard Bass's book on stochastic processes: construction.pdf

Notes: p59-p64.pdf

September 29: In preparation for Wiener's construction of Brownian motion, we discussed the basics of Gaussian processes and reviewed some material on Fourier series.

Notes: p47-p58.pdf

September 24: We investigated the asymptotic distribution of $LaTeX: W_n\left(t\right)=\frac{S_{\lfloor nt\rfloor}}{\sqrt{n}}$ , where LaTeX: S_n is a simple random walk, and concluded the lecture by defining Brownian motion (section 3.2).

Notes: p36-p46.pdf

September 22: We discussed the abstract definition of conditional expectation (section 2.6).

The first homework assignment has been posted.

September 17: We defined the Lebesgue integral with respect to a probability measure (section 2.2).

Notes: p19-p35.pdf

September 15: We introduced several measure theoretic concepts and defined the notion of a random variable as a measurable function.

Notes: p8-p18.pdf

September 10: We discussed how stochastic differential equations differ from ordinary differential equations, and we introduced the notion of white noise as a generalized function. This was an informal discussion; all concepts will be rigorously defined in later lectures.

My lecture notes can be found here: p1-p7.pdf

If you want to read more about the theory of generalized functions (aka Schwartz distributions), have a look at the first chapter of Robert Strichartz's book on the subject: distribution_theory.pdf