# Lecture Series

## Lecture series on rough path

Lecturer: Peter Friz

- Monday, March 23, 3:00 pm - 4:00 pm
- Tuesday, March 24, 11:00 am - 12:30 pm
- Wednesday, March 25, 11:00 am - 12:30 pm
- Wednesday, March 25, 3:00 pm - 4:00 pm

Abstract: We will cover the basics of rough path integration, - differential equations and some recent applications to stochastic analysis.

Suggested reading:

- Terry J. Lyons, Michael Caruana, and Thierry Lévy. Differential equations driven by rough paths, volume 1908 of Lecture Notes in Mathematics. Springer, Berlin, 2007.
- P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths, volume 120 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
- Peter K. Friz and Martin Hairer. A Course on Rough Paths: With an Introduction to Regularity Structures. Springer Universitext. Springer, 2014.

## Lecture series on game theoretic probability

Lecturer: Vladimir Vovk

- Monday, March 30, 10:00 am - 12:00 noon
- Tuesday, March 31, 10:00 am - 12:00 noon
- Wednesday, April 1, 10:00 am - 12:00 noon

Abstract: The informal notion of probability is extremely rich, and historically there have been more than one approach to formalizing it. At this time the dominant approach is measure-theoretic probability, but in this series of lectures I will concentrate on another very old approach, which I will refer to as game-theoretic probability. The main difference between the two approaches is that the notion of game-theoretic probability is derivative: it is defined via other more fundamental notions (for example, via the security prices in the context of financial markets). This makes it possible to apply the theory of game-theoretic probability in the absence of any stochastic assumptions. In the first part of the series I will state familiar laws of measure-theoretic probability, such as the strong law of large numbers, zero-one laws, and the central limit theorem in terms of game-theoretic probability. They then become worst-case results about perfect-information games and guarantee the existence of a winning strategy for one of the players. In the second part I will discuss applications of game-theoretic probability to an idealized continuous-time financial market. Considering an idealized financial security with continuous price path and without making any stochastic assumptions, this part will explore the general phenomenon “a continuous price path should look like Brownian motion”, starting from simple arguments showing how a speculator with a unit initial capital can become infinitely rich without risking bankruptcy when some basic properties of Brownian motion are violated for the price path. I will then show that typical price paths possess quadratic variation, where “typical” is understood in the same non-stochastic sense: there exists a trading strategy that does not risk more than one monetary unit and earns infinite capital when the process of quadratic variation does not exist. Replacing time by the quadratic variation process, we will see that the price path becomes Brownian motion. This is essentially the same conclusion as in the Dubins-Schwarz result, except that the probabilities (constituting the Wiener measure) emerge instead of being postulated.

## Lecture series on super-Ricci flow for metric measure spaces

Lecturer: Karl-Theodor Sturm

- Monday, March 30, 2:30 pm - 4:00 pm
- Wednesday, April 1, 2:30 pm - 4:00 pm
- Thursday, April 2, 10:00 am - 11:30 am