Stochastic Analysis & Mathematical Finance Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
26 April 2021
16:00
THALEIA ZARIPHOPOULOU
Abstract

 

In my talk, I will introduce a family of human-machine interaction (HMI) models in optimal portfolio construction (robo-advising). Modeling difficulties stem from the limited ability to quantify the human’s risk preferences and describe their evolution, but also from the fact that the stochastic environment, in which the machine optimizes, adapts to real-time incoming information that is exogenous to the human. Furthermore, the human’s risk preferences and the machine’s states may evolve at different scales. This interaction creates an adaptive cooperative game with both asymmetric and incomplete information exchange between the two parties.

As a result, challenging questions arise on, among others, how frequently the two parties should communicate, what information can the machine accurately detect, infer and predict, how the human reacts to exogenous events, how to improve the inter-linked reliability between the human and the machine, and others. Such HMI models give rise to new, non-standard optimization problems that combine adaptive stochastic control, stochastic differential games, optimal stopping, multi-scales and learning.

 

 

  • Stochastic Analysis & Mathematical Finance Seminars
10 May 2021
16:00
ILYA CHEVYREV
Abstract

In this talk, we will consider multidimensional fast-slow dynamical systems in discrete-time with random initial conditions but otherwise completely deterministic dynamics. The question we will investigate is whether the slow variable converges in law to a stochastic process under a suitable scaling limit. We will be particularly interested in the case when the limiting dynamic is superdiffusive, i.e. it coincides in law with the solution of a Marcus SDE driven by a discontinuous stable Lévy process. Under certain assumptions, we will show that generically convergence does not hold in any Skorokhod topology but does hold in a generalisation of the Skorokhod strong M1 topology which we define using so-called path functions. Our methods are based on a combination of ergodic theory and ideas arising from (but not using) rough paths. We will finally show that our assumptions are satisfied for a class of intermittent maps of Pomeau-Manneville type. 

 

  • Stochastic Analysis & Mathematical Finance Seminars
17 May 2021
16:00
FRAYDOUN REZAKHANLOU
Abstract

The flow of a Hamilton-Jacobi PDE yields a dynamical system on the space of continuous functions. When the Hamiltonian function is convex in the momentum variable, and the spatial dimension is one, we may restrict the flow to piecewise smooth functions and give a kinetic description for the solution. We regard the locations of jump discontinuities of the first derivative of solutions as the sites of particles. These particles interact via collisions and coagulations. When these particles are selected randomly according to certain Gibbs measures initially, then the law of particles remains Gibbsian at later times, and one can derive a Boltzmann/Smoluchowski type PDE for the evolution of these Gibbs measures.  In higher dimensions, we assume that the Hamiltonian function is independent of position and  that the initial condition is piecewise linear and convex. Such initial conditions can be identified as (Laguerre) tessellations and the Hamilton-Jacobi evolution  can be described as a billiard on the set of tessellations.

  • Stochastic Analysis & Mathematical Finance Seminars
24 May 2021
16:00
DAVAR KHOSHNEVISAN
Abstract

We consider a reaction-diffusion equation of the type ∂tψ=∂2xψ+V(ψ)+λσ(ψ)W˙on (0,∞)×𝕋, subject to a "nice" initial value and periodic boundary, where 𝕋=[−1,1] and W˙ denotes space-time white noise. The reaction term V:ℝ→ℝ belongs to a large family of functions that includes Fisher--KPP nonlinearities [V(x)=x(1−x)] as well as Allen-Cahn potentials [V(x)=x(1−x)(1+x)], the multiplicative nonlinearity σ:ℝ→ℝ is non random and Lipschitz continuous, and λ>0 is a non-random number that measures the strength of the effect of the noise W˙. The principal finding of this paper is that: (i) When λ is sufficiently large, the above equation has a unique invariant measure; and (ii) When λ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.

This is based on joint work with Carl Mueller (Univ. Rochester) and Kunwoo Kim (POSTECH, S. Korea).

 

  • Stochastic Analysis & Mathematical Finance Seminars
14 June 2021
16:00
Abstract

We present linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba in a previous work. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. 

The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.

Joint work with Yichen Feng and Tomoyuki Ichiba.

  • Stochastic Analysis & Mathematical Finance Seminars
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