Convergence of stochastic integrals and their applications

DATES: 3 - 7 October 2022 (5 sessions)
TIME: 09:00-11:00 (a total of 10 hours)
LOCATION: BCAM Seminar Room and Online


ABSTRACT:
Most physical, chemical, biological and economic phenomena present an intrinsic degree of randomness. These are typically modeled by stochastic differential equations (SDEs). SDEs are typically constructed with respect to the Wiener process (i.e. Brownian motion). In some cases, the evaluation of the integral may depend on how the stochastic integral is constructed, i.e. right-hand limit, left hand limit, Simpson's rule, etc. The aim of this course is to give an overview of how these construction issues of stochastic integrals arise in applications and how to prove theorems using a powerful tool from Kurtz and Protter (1991). This tool arises by considering convergence of stochastic integrals with respect to sequences of stochastic processes. In addition to convergence of stochastic integrals, this course will provide motivation for theoretical mathematics and physics in proving theorems for a variety of applications in physics, biology and atmospheric science


CONTENTS:
Day 1: Introduction to stochastic processes and stochastic integrals
Day 2: The Langevin equation and homogenization
Day 3: Convergence of Stochastic Integrals
Day 4: Applications of convergence to physics and biology
Day 5: Sliding dynamics and applications in atmospheric science



REFERENCES:
[- Day 1:
Aksendal, Bernt. "Stochastic differential equations." Stochastic differential equations. Springer, Berlin, Heidelberg, 2003. 65-84.
Karatzas, Ioannis, and Steven Shreve. Brownian motion and stochastic calculus. Vol. 113. Springer Science & Business Media, 2012.

- Day 2:
Gardiner, Crispin. Stochastic methods. Vol. 4. Berlin: Springer, 2009.
Coffey, William, and Yu P. Kalmykov. The Langevin equation: with applications to stochastic problems in physics, chemistry and electrical engineering. Vol. 27. World Scientific, 2012.
Pavliotis, Grigoris, and Andrew Stuart. Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008.
Hottovy, Scott, Giovanni Volpe, and Jan Wehr. "Noise-induced drift in stochastic differential equations with arbitrary friction and diffusion in the Smoluchowski-Kramers limit." Journal of Statistical Physics 146.4 (2012): 762-773.

- Day 3:
Wong, Eugene, and Moshe Zakai. "On the convergence of ordinary integrals to stochastic integrals." The Annals of Mathematical Statistics 36.5 (1965): 1560-1564.
Kurtz, Thomas G., and Philip Protter. "Weak limit theorems for stochastic integrals and stochastic differential equations." The Annals of Probability (1991): 1035-1070.

- Day 4:
Hottovy, Scott, et al. "The Smoluchowski-Kramers limit of stochastic differential
equations with arbitrary state-dependent friction." Communications in Mathematical
Physics 336.3 (2015): 1259-1283.
Pesce, Giuseppe, et al. "Stratonovich-to-It" transition in noisy systems with multiplicative feedback." Nature communications 4.1 (2013): 1-7.
Hottovy, Scott, Austin McDaniel, and Jan Wehr. "A small delay and correlation time limit of stochastic differential delay equations with state-dependent colored noise." Journal of Statistical Physics 175.1 (2019): 19-46.

- Day 5:
Simpson, David JW, and Rachel Kuske. "The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise." Stochastics and Dynamics 14.04 (2014): 1450010.
Hottovy, Scott, and Samuel N. Stechmann. "Threshold models for rainfall and convection: Deterministic versus stochastic triggers." SIAM Journal on Applied Mathematics 75.2 (2015): 861-884.
Hottovy, Scott, and Samuel N. Stechmann. "Convergence of Rain Process Models to Point Processes." arXiv preprint arXiv:2011.14247 (2020).


*Registration is free, but mandatory before 27 September 2022. To sign-up go to https://forms.gle/5mDRwJMAchdUCAmJA and fill the registration form.

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