We study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the Functional Ito calculus to our more general framework. In particular, unlike in the Functional Ito calculus, where one needs only to consider stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths.  We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.

Joint work with JianFeng Zhang (USC).

Many classical fractal functions, such as the Weierstrass and Takagi-van der Waerden functions, admit a finite p-th variation along a natural sequence of partitions. They can thus serve as integrators in pathwise Itô calculus. Motivated by this observation, we
introduce a new class of stochastic processes, which we call Weierstrass bridges. They have continuous sample paths and arbitrarily low regularity and so provide a new example class of “rough” stochastic processes. We study some of their sample path properties
including p-th variation and moduli of continuity. This talk includes joint work with Xiyue Han and Zhenyuan Zhang.

Generative Adversarial Networks (GANs) have celebrated great empirical success, especially in image generation and processing. Meanwhile, Mean-Field Games (MFGs),  as analytically feasible approximations for N-player games, have experienced rapid growth in theory of controls. In this talk, we will discuss a new theoretical connections between GANs and MFGs. Interpreting MFGs as GANs, on one hand, allows us to devise GANs-based algorithm to solve MFGs. Interpreting GANs as MFGs, on the other hand, provides a new and probabilistic foundation for GANs. Moreover, this interpretation helps establish an analytical connection between GANs and Optimal Transport (OT) problems, the connection previously understood mostly from the geometric perspective. We will illustrate by numerical examples of using GANs to solve high dimensional MFGs, demonstrating its superior performance over existing methodology.

For a given vector field $h$ on a manifold $M$ and an initial point $x \in M$, let $t \mapsto \exp th(x)$ denote the solution to the Cauchy problem $y' = h(y)$, $y(0) = x$. Given two vector fields $f$, $g$, the flows $\exp(tf)$, $\exp(tg)$ in general are not commutative. That is, it may happen that, for some initial point $x$,

$$\exp(-tg) \circ \exp(-tf) \circ \exp(tg) \circ \exp(tf) (x) ≠ x,$$

for small times $t ≠ 0$.

         As is well-known, the Lie bracket $[f,g] := Dg \cdot f - Df \cdot g$ measures the local non-commutativity of the flows. Indeed, one has (on any coordinate chart)

$$\exp(-tg) \circ \exp(-tf) \circ \exp(tg) \circ \exp(tf) (x) - x = t^2 [f,g](x) + o(t^2)$$

         The non-commutativity of vector fields lies at the basis of many nonlinear issues, like propagation of maxima for solutions of degenerate elliptic PDEs, controllability sufficient conditions in Nonlinear Control Theory, and higher order necessary conditions for optimal controls. The fundamental results concerning commutativity (e.g. Rashevski-Chow's Theorem, also known as Hörmander's full rank condition, or Frobenius Theorem) assume that the vector fields are smooth enough for the involved iterated Lie brackets to be well defined and continuous: for instance, if the bracket $[f,[g,h]]$ is to be used, one posits $g,h \in C^2$ and $f \in C^{1..}$.

         We propose a notion of (set-valued) Lie bracket (see [1]-[3]), through which we are able to extend some of the mentioned fundamental results to families of vector fields whose iterated brackets are just measurable and defined almost everywhere.

References.

[1]  Rampazzo, F. and Sussmann, H., Set-valued differentials and a nonsmooth version of Chow’s Theorem (2001), Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, 2001 (IEEE Publications, New York), pp. 2613-2618.

[2] Rampazzo F.  and Sussmann, H.J., Commutators of flow maps of nonsmooth vector fields (2007), Journal of Differential Equations, 232, pp. 134-175.

[3] Feleqi, E. and Rampazzo, F., Iterated Lie brackets for nonsmooth vector fields (2017), Nonlinear Differential Equations and Applications NoDEA, 24-6.

We discuss hyperkahler implosion spaces, their relevance to group actions and why they should fit into the symplectic duality picture. For certain groups we present candidates for the symplectic duals of the associated implosion spaces and provide computational evidence. This is joint work with Amihay Hanany and Frances Kirwan.
 

I will talk about a novel idea on the Swampland program that uses consistency of what lives on the string probes in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the supergravity theories based on unitarity of the worldsheet CFT. I will show some of the theories with 8 or 16 supersymmetries, which are otherwise consistent looking, belong to the Swampland.