A PDF copy of the course program can be found here: IMT_3800.pdf

December 1: As a final application, we derived the Black-Scholes equation and discussed the problem of options pricing in financial engineering (section 6.4).

November 29: We proved that the value function for a class of stochastic optimal control problems satisfies a Hamilton-Jacobi-Bellman equation. We also discussed the application of these ideas to the problem of optimal portfolio selection in financial engineering. See, e.g., sections 7.5 & 7.6 of these notes

Nov 22 - Nov 24: We turned our attention to nonparametric statistics and discussed how stochastic differential equations can be employed in the analysis of nonparametric regression problems. See also the first chapter of:

Alexandre Tsybakov, Introduction to Nonparametric Estimation Springer, 2009.

Notes: nonparametric_regression.pdf

Nov 15 - 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).

Notes: p167-p185.pdf

Nov 8 - Nov 10: We derived the martingale inequalities of section 2.7 and proved the existence and uniqueness theorem of section 5.2.3.

The fourth homework assignment has been posted.

Nov 3 - Nov 5: We defined the notion of a martingale and analyzed the martingale betting strategy. We concluded the lecture by proving the optional sampling theorem.

Due to technical problems during last Wednesday's lecture, there will be an extra lecture this Friday (Nov 5) at 10 am.

Notes: p139-p166.pdf

October 27: 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 25: We investigated the solutions of certain linear stochastic differential equations and introduced the Ornstein-Uhlenbeck process.

Notes: p119-p138.pdf

Oct 6 - Oct 13: We defined white noise as a generalized (distribution-valued) stochastic process. For more details on the theory of generalized functions (aka Schwartz distributions), see chapter 7 of Paul Sacks's book on functional analysis: distribution_theory.pdf

The third homework assignment has been posted.

Sep 29 - Oct 4: We focused on stochastic differentials and derived Itô's chain and product rules (section 4.3).

Notes: p102-p118.pdf

Sep 22 - Sep 27: 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.

September 20: We discussed the significance of quadratic variation in the theory of integration, and we computed the quadratic variation of Brownian motion.

September 15: We introduced the Paley-Wiener-Zygmund stochastic integral and derived a special case of the Itô isometry (section 4.1).

The second homework assignment has been posted.

Notes: p81-p101.pdf

Sep 8 - Sep 13: We proved that sample paths of Brownian motion are with probability one nowhere differentiable (section 3.4.2).

Notes: p65-p80.pdf

September 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

September 1: In preparation for Wiener's construction of Brownian motion, we discussed the basics of Gaussian processes and reviewed some material on Fourier series.

Notes: p40-p64.pdf

August 30: 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).

The first homework assignment has been posted.

Aug 23 - 25: We discussed the abstract definition of conditional expectation (section 2.6).

Notes: p27-p39.pdf

Aug 16 - Aug 18: We introduced several measure theoretic concepts and defined the Lebesgue integral with respect to a probability measure (sections 2.1 & 2.2).

My lecture notes can be found here: p1-p26.pdf

August 16: First day of classes.