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BEGIN:VEVENT
SUMMARY:Xin Guo (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20200504T150000Z
DTEND;VALUE=DATE-TIME:20200504T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/1
DESCRIPTION:Title: Connecting Generative adversarial networks with Mean Fiel
d Games\nby Xin Guo (UC Berkeley) as part of Oxford Stochastic Analysi
s and Mathematical Finance Seminar\n\nLecture held in Oxford Mathematical
Institute.\n\nAbstract\nGenerative Adversarial Networks (GANs) have celebr
ated great empirical success\, especially in image generation and processi
ng. Meanwhile\, Mean-Field Games (MFGs)\, as analytically feasible approx
imations for N-player games\, have experienced rapid growth in theory of c
ontrols. In this talk\, we will discuss a new theoretical connections betw
een 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. Mor
eover\, this interpretation helps establish an analytical connection betwe
en GANs and Optimal Transport (OT) problems\, the connection previously un
derstood mostly from the geometric perspective. We will illustrate by nume
rical examples of using GANs to solve high dimensional MFGs\, demonstratin
g its superior performance over existing methodology.\n\nRegistration URL:
\nhttps://zoom.us/meeting/register/tJ0oceGoqDsrH9PwXl9eEUDoA6rGri-Zaf_R\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Schied (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20200511T150000Z
DTEND;VALUE=DATE-TIME:20200511T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/2
DESCRIPTION:Title: Weierstrass bridges\nby Alexander Schied (University
of Waterloo) as part of Oxford Stochastic Analysis and Mathematical Financ
e Seminar\n\nLecture held in Oxford Mathematical Institute.\n\nAbstract\nM
any classical fractal functions\, such as the Weierstrass and Takagi-van d
er Waerden functions\, admit a finite p-th variation along a natural seque
nce of partitions. They can thus serve as integrators in pathwise Itô cal
culus. Motivated by this observation\, we introduce a new class of stochas
tic processes\, which we call Weierstrass bridges. They have continuous sa
mple paths and arbitrarily low regularity and so provide a new example cla
ss of “rough” stochastic processes. We study some of their sample path
properties including p-th variation and moduli of continuity. This talk i
ncludes joint work with Xiyue Han and Zhenyuan Zhang.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan Nourdin (University of Luxembourg)
DTSTART;VALUE=DATE-TIME:20200518T150000Z
DTEND;VALUE=DATE-TIME:20200518T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/3
DESCRIPTION:Title: The functional Breuer-Major theorem\nby Ivan Nourdin
(University of Luxembourg) as part of Oxford Stochastic Analysis and Mathe
matical Finance Seminar\n\nLecture held in Oxford Mathematical Institute.\
n\nAbstract\nLet $X=\\{ X_n\\}_{n\\in \\mathbb{Z}}$ be zero-mean stationar
y Gaussian sequence of random variables with covariance function $\\rho$ s
atisfying $\\rho(0)=1$. Let $\\varphi:\\mathbb{R}\\to\\mathbb{R}$ be a fun
ction such that $E[\\varphi(X_0)^2]<\\infty$ and assume that $\\varphi$ ha
s Hermite rank $d \\geq 1$. The celebrated Breuer-Major theorem asserts th
at\, if $\\sum_{r\\in\\mathbb{Z}} |\\rho(r)|^d<\\infty$ then the finite di
mensional distributions of $\\frac1{\\sqrt{n}}\\sum_{i=0}^{\\lfloor n\\cdo
t\\rfloor-1} \\varphi(X_i)$ converge to those of $\\sigma\\\,W$\, where $W
$ is a standard Brownian motion and $\\sigma$ is some (explicit) constant.
Surprisingly\, and despite the fact this theorem has become over the year
s a prominent tool in a bunch of different areas\, a necessary and suffici
ent condition implying the weak convergence in the space ${\\bf D}([0\,1])
$ of càdlàg functions endowed with the Skorohod topology is still missin
g. Our main goal in this paper is to fill this gap. More precisely\, by us
ing suitable boundedness properties satisfied by the generator of the Orns
tein-Uhlenbeck semigroup\, we show that tightness holds under the sufficie
nt (and almost necessary) natural condition that $E[|\\varphi(X_0)|^{p}]<\
\infty$ for some $p>2$.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabian Harang (Oslo)
DTSTART;VALUE=DATE-TIME:20200525T150000Z
DTEND;VALUE=DATE-TIME:20200525T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/4
DESCRIPTION:Title: Infinitely regularising paths and regularisation by noise
.\nby Fabian Harang (Oslo) as part of Oxford Stochastic Analysis and M
athematical Finance Seminar\n\nLecture held in Oxford Mathematical Institu
te.\n\nAbstract\nWe discuss regularization by noise from a pathwise perspe
ctive using non-linear Young integration\, and discuss the relations with
occupation measures and local times. This methodology of pathwise regulari
zation by noise was originally proposed by Gubinelli and Catellier (2016)\
, who use the concept of averaging operators and non-linear Young integrat
ion to give meaning to certain ill posed SDEs. \nIn a recent work together
with Nicolas Perkowski we show that there exists a class of paths with
exceptional regularizing effects on ODEs\, using the framework of Gubinell
i and Catellier. In particular we prove existence and uniqueness of ODEs p
erturbed by such a path\, even when the drift is given as a Schwartz distr
ibution. Moreover\, the flow associated to such ODEs are proven to be infi
nitely differentiable. Our analysis can be seen as purely pathwise\, and i
s only depending on the existence of a sufficiently regular occupation mea
sure associated to the path added to the ODE. As an example\, we show that
a certain type of Gaussian processes has infinitely differentiable local
times\, whose paths then can be used to obtain the infinitely regularizing
effect on ODEs. This gives insight into the powerful effect that noise ma
y have on certain equations. If time permits\, I will also discuss an ong
oing extension of these results towards regularization of certain PDE/SPDE
s by noise.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frederi Viens (Michigan State University)
DTSTART;VALUE=DATE-TIME:20200601T150000Z
DTEND;VALUE=DATE-TIME:20200601T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/5
DESCRIPTION:Title: A martingale approach for fractional Brownian motions and
related path dependent PDEs\nby Frederi Viens (Michigan State Univers
ity) as part of Oxford Stochastic Analysis and Mathematical Finance Semina
r\n\nLecture held in Oxford Mathematical Institute.\n\nAbstract\nWe study
dynamic backward problems\, with the computation of conditional expectatio
ns as a special objective\, in a framework where the (forward) state proce
ss satisfies a Volterra type SDE\, with fractional Brownian motion as a ty
pical example. Such processes are neither Markov processes nor semimarting
ales\, 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 resul
t is a functional Itô formula\, extending the Functional Ito calculus to
our more general framework. In particular\, unlike in the Functional Ito c
alculus\, where one needs only to consider stopped paths\, here we need to
concatenate the observed path up to the current time with a certain smoot
h observable curve derived from the distribution of the future paths. We
then derive the path dependent PDEs for the backward problems. Finally\, a
n application to option pricing and hedging in a financial market with rou
gh volatility is presented.\n\nJoint work with JianFeng Zhang (USC).\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christina Goldschmidt (Oxford)
DTSTART;VALUE=DATE-TIME:20200608T150000Z
DTEND;VALUE=DATE-TIME:20200608T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/6
DESCRIPTION:Title: The scaling limit of a critical random directed graph
\nby Christina Goldschmidt (Oxford) as part of Oxford Stochastic Analysis
and Mathematical Finance Seminar\n\nLecture held in Oxford Mathematical In
stitute.\n\nAbstract\nWe consider the random directed graph $\\vec{G}(n\,p
)$ with vertex set $\\{1\,2\,\\ldots\,n\\}$ in which each of the $n(n-1)$
possible directed edges is present independently with probability $p$. We
are interested in the strongly connected components of this directed graph
. A phase transition for the emergence of a giant strongly connected compo
nent is known to occur at $p = 1/n$\, with critical window $p= 1/n + \\lam
bda n^{-4/3}$ for $\\lambda \\in \\mathcal{R}$. We show that\, within this
critical window\, the strongly connected components of $\\vec{G}(n\,p)$\,
ranked in decreasing order of size and rescaled by $n^{-1/3}$\, converge
in distribution to a sequence $(\\mathcal{C}_1\,\\mathcal{C}_2\,\\ldots)$
of finite strongly connected directed multigraphs with edge lengths which
are either 3-regular or loops. The convergence occurs the sense of an $\\e
ll^1$ sequence metric for which two directed multigraphs are close if ther
e are compatible isomorphisms between their vertex and edge sets which rou
ghly preserve the edge-lengths. Our proofs rely on a depth-first explorati
on of the graph which enables us to relate the strongly connected componen
ts to a particular spanning forest of the undirected Erdős-Rényi random
graph $G(n\,p)$\, whose scaling limit is well understood. We show that the
limiting sequence $(\\mathcal{C}_1\,\\mathcal{C}_2\,\\ldots)$ contains on
ly finitely many components which are not loops. If we ignore the edge len
gths\, any fixed finite sequence of 3-regular strongly connected directed
multigraphs occurs with positive probability.\n\nThis is joint work with R
obin Stephenson (Sheffield).\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mykhaylo Shkolnikov (Princeton)
DTSTART;VALUE=DATE-TIME:20200615T150000Z
DTEND;VALUE=DATE-TIME:20200615T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/7
DESCRIPTION:Title: Local stochastic volatility and the inverse of the Markov
ian projection\nby Mykhaylo Shkolnikov (Princeton) as part of Oxford S
tochastic Analysis and Mathematical Finance Seminar\n\nLecture held in Oxf
ord Mathematical Institute.\n\nAbstract\nThe calibration problem for local
stochastic volatility models leads to two-dimensional stochastic differen
tial equations of McKean-Vlasov type. In these equations\, the conditional
distribution of the second component of the solution given the first ente
rs the equation for the first component of the solution. While such equati
ons enjoy frequent application in the financial industry\, their mathemati
cal analysis poses a major challenge. I will explain how to prove the stro
ng existence of stationary solutions for these equations\, as well as the
strong uniqueness in an important special case. \nBased on joint work with
Daniel Lacker and Jiacheng Zhang.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Kurtz (University of Wisconsin)
DTSTART;VALUE=DATE-TIME:20200622T150000Z
DTEND;VALUE=DATE-TIME:20200622T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/8
DESCRIPTION:Title: Controlled and constrained martingale problems\nby Th
omas Kurtz (University of Wisconsin) as part of Oxford Stochastic Analysis
and Mathematical Finance Seminar\n\nLecture held in Oxford Mathematical I
nstitute.\n\nAbstract\nMost of the basic results on martingale problems ex
tend to the setting in which the generator depends on a control. The “c
ontrol” could represent a random environment\, or the generator could sp
ecify a classical stochastic control problem. The equivalence between the
martingale problem and forward equation (obtained by taking expectations o
f the martingales) provides the tools for extending linear programming met
hods introduced by Manne in the context of controlled finite Markov chains
to general Markov stochastic control problems. The controlled martingale
problem can also be applied to the study of constrained Markov processes
(e.g.\, reflecting diffusions)\, the boundary process being treated as a c
ontrol. The talk includes joint work with Richard Stockbridge and with Cr
istina Costantini.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ioannis Karatzas (Columbia University)
DTSTART;VALUE=DATE-TIME:20201012T150000Z
DTEND;VALUE=DATE-TIME:20201012T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/9
DESCRIPTION:Title: A trajectorial approach to the gradient flow properties o
f conservative diffusions and Markov chains\nby Ioannis Karatzas (Colu
mbia University) as part of Oxford Stochastic Analysis and Mathematical Fi
nance Seminar\n\nLecture held in Oxford Mathematical Institute.\n\nAbstrac
t\nWe provide a detailed\, probabilistic interpretation for the variationa
l characterization of conservative diffusion as entropic gradient flow. Jo
rdan\, Kinderlehrer\, and Otto showed in 1998 that\, for diffusions of Lan
gevin-Smoluchowski type\, the Fokker-Planck probability density flow minim
izes the rate of relative entropy dissipation\, as measured by the distanc
e traveled in terms of the quadratic Wasserstein metric in the ambient spa
ce of configurations. Using a very direct perturbation analysis we obtain
novel\, stochastic-process versions of such features\; these are valid alo
ng almost every trajectory of the motion in both the forward and\, most tr
ansparently\, the backward\, directions of time. The original results foll
ow then simply by “aggregating”\, i.e.\, taking expectations. As a bon
us\, the HWI inequality of Otto and Villani relating relative entropy\, Fi
sher information\, and Wasserstein distance\, falls in our lap\; and with
it the celebrated log-Sobolev\, Talagrand and Poincare inequalities of fun
ctional analysis. Similar ideas work in the context of continuous-time Mar
kov Chains\; but now both the functional analysis and the geometry are con
siderably more involved.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Shreve (Carnegie Mellon University)
DTSTART;VALUE=DATE-TIME:20201026T160000Z
DTEND;VALUE=DATE-TIME:20201026T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/10
DESCRIPTION:Title: Diffusion Limit of Poisson Limit-Order Book Models\n
by Steve Shreve (Carnegie Mellon University) as part of Oxford Stochastic
Analysis and Mathematical Finance Seminar\n\nLecture held in Oxford Mathem
atical Institute.\n\nAbstract\nTrading of financial instruments has largel
y moved away from floor trading and onto electronic exchanges. Orders to
buy and sell are queued at these exchanges in a limit-order book. While a
full analysis of the dynamics of a limit-order book requires an understand
ing of strategic play among multiple agents\, and is thus extremely comple
x\, so-called zero-intelligence Poisson models have been shown to capture
many of the statistical features of limit-order book evolution. These mod
els can be addressed by traditional queueing theory techniques\, including
Laplace transform analysis. In this work\, we demonstrate in a simple se
tting that another queueing theory technique\, approximating the Poisson m
odel by a diffusion model identified as the limit of a sequence of scaled
Poisson models\, can also be implemented. We identify the diffusion limit
\, find an embedded semi-Markov model in the limit\, and determine the sta
tistics of the embedded semi-Markov model. Along the way\, we introduce an
d study a new type of process\, a generalization of skew Brownian motion t
hat we call two-speed Brownian motion.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christa Cuchiero (University of Vienna)
DTSTART;VALUE=DATE-TIME:20201019T150000Z
DTEND;VALUE=DATE-TIME:20201019T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/11
DESCRIPTION:Title: Deep neural networks\, generic universal interpolation a
nd controlled ODEs\nby Christa Cuchiero (University of Vienna) as part
of Oxford Stochastic Analysis and Mathematical Finance Seminar\n\nLecture
held in Oxford Mathematical Institute.\n\nAbstract\nA recent paradigm vie
ws deep neural networks as discretizations of certain controlled ordinary
differential equations\, sometimes called neural ordinary differential equ
ations. We make use of this perspective to link expressiveness of deep net
works to the notion of controllability of dynamical systems. Using this co
nnection\, we study an expressiveness property that we call universal inte
rpolation\, and show that it is generic in a certain sense. The universal
interpolation property is slightly weaker than universal approximation\, a
nd disentangles supervised learning on finite training sets from generaliz
ation properties. We also show that universal interpolation holds for cert
ain deep neural networks even if large numbers of parameters are left untr
ained\, and are instead chosen randomly. This lends theoretical support to
the observation that training with random initialization can be successfu
l even when most parameters are largely unchanged through the training. Ou
r results also explore what a minimal amount of trainable parameters in ne
ural ordinary differential equations could be without giving up on express
iveness.\n\nJoint work with Martin Larsson\, Josef Teichmann.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julien Dubedat (Columbia University)
DTSTART;VALUE=DATE-TIME:20201102T160000Z
DTEND;VALUE=DATE-TIME:20201102T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/12
DESCRIPTION:Title: Stochastic Ricci Flow on surfaces\nby Julien Dubedat
(Columbia University) as part of Oxford Stochastic Analysis and Mathemati
cal Finance Seminar\n\nLecture held in Oxford Mathematical Institute.\n\nA
bstract\nThe Ricci flow on a surface is an intrinsic evolution of the metr
ic converging to a constant curvature metric within the conformal class. I
t can be seen as an (infinite-dimensional) gradient flow. We introduce a n
atural 'Langevin' version of this flow\, thus constructing an SPDE with in
variant measure expressed in terms of Liouville Conformal Field Theory.\n\
nJoint work with Hao Shen (Wisconsin).\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Massimiliano Gubinelli (Bonn)
DTSTART;VALUE=DATE-TIME:20201116T160000Z
DTEND;VALUE=DATE-TIME:20201116T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/13
DESCRIPTION:Title: Elliptic stochastic quantisation and supersymmetry\n
by Massimiliano Gubinelli (Bonn) as part of Oxford Stochastic Analysis and
Mathematical Finance Seminar\n\nLecture held in Oxford Mathematical Insti
tute.\n\nAbstract\nStochastic quantisation is\, broadly speaking\, the use
of a stochastic differential equation to construct a given probability di
stribution. Usually this refers to Markovian Langevin evolution with given
invariant measure. However we will show that it is possible to construct
other kind of equations (elliptic stochastic partial differential equation
s) whose solutions have prescribed marginals. This connection was discover
ed in the '80 by Parisi and Sourlas in the context of dimensional reductio
n of statistical field theories in random external fields. This purely pro
babilistic results has a proof which depends on a supersymmetric formulati
on of the problem\, i.e. a formulation involving a non-commutative random
field defined on a non-commutative space. \nThis talk is based on joint wo
rk with S. Albeverio and F. C. de Vecchi.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Beatrice Acciaio (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201130T160000Z
DTEND;VALUE=DATE-TIME:20201130T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/14
DESCRIPTION:Title: Model-independence in a fixed-income market and weak opt
imal transport\nby Beatrice Acciaio (ETH Zurich) as part of Oxford Sto
chastic Analysis and Mathematical Finance Seminar\n\nLecture held in Oxfor
d Mathematical Institute.\n\nAbstract\nI consider model-independent prici
ng problems in a stochastic interest rates framework. In this case the usu
al tools from Optimal Transport and Skorokhod embedding cannot be applied.
I will show how some pricing problems in a fixed-income market can be ref
ormulated as Weak Optimal Transport (WOT) problems as introduced by Gozlan
et al. I will present a super-replication theorem that follows from an ex
tension of WOT results to the case of non-convex cost functions.\n\nThis t
alk is based on joint work with M. Beiglboeck and G. Pammer.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:RenYuan Xu (University of Oxford)
DTSTART;VALUE=DATE-TIME:20201123T160000Z
DTEND;VALUE=DATE-TIME:20201123T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/15
DESCRIPTION:Title: Excursion risk\nby RenYuan Xu (University of Oxford)
as part of Oxford Stochastic Analysis and Mathematical Finance Seminar\n\
nLecture held in Oxford Mathematical Institute.\n\nAbstract\nThe risk and
return profiles of a broad class of dynamic trading strategies\, including
pairs trading and other statistical arbitrage strategies\, may be charact
erized in terms of excursions of the market price of a portfolio away from
a reference level. We propose a mathematical framework for the risk analy
sis of such strategies\, based on a description in terms of price excursio
ns\, first in a pathwise setting\, without probabilistic assumptions\, the
n in a Markovian setting.\n\nWe introduce the notion of δ-excursion\, def
ined as a path which deviates by δ from a reference level before returnin
g to this level. We show that every continuous path has a unique decomposi
tion into δ-excursions\, which is useful for scenario analysis of dynamic
trading strategies\, leading to simple expressions for the number of trad
es\, realized profit\, maximum loss and drawdown. As δ is decreased to ze
ro\, properties of this decomposition relate to the local time of the path
.\n\nWhen the underlying asset follows a Markov process\, we combine these
results with Ito's excursion theory to obtain a tractable decomposition o
f the process as a concatenation of independent δ-excursions\, whose dist
ribution is described in terms of Ito's excursion measure. We provide anal
ytical results for linear diffusions and give new examples of stochastic p
rocesses for flexible and tractable modeling of excursions. Finally\, we d
escribe a non-parametric scenario simulation method for generating paths w
hose excursion properties match those observed in empirical data.\n\nJoint
work with Anna Ananova and Rama Cont.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diyora Salimova (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201109T160000Z
DTEND;VALUE=DATE-TIME:20201109T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/16
DESCRIPTION:Title: Space-time deep neural network approximations for high-d
imensional partial differential equations\nby Diyora Salimova (ETH Zur
ich) as part of Oxford Stochastic Analysis and Mathematical Finance Semina
r\n\nLecture held in Oxford Mathematical Institute.\n\nAbstract\nIt is one
of the most challenging issues in applied mathematics to approximately so
lve high-dimensional partial differential equations (PDEs) and most of the
numerical approximation methods for PDEs in the scientific literature suf
fer from the so-called curse of dimensionality (CoD) in the sense that the
number of computational operations employed in the corresponding approxim
ation scheme to obtain an approximation precision 𝜀>0 grows exponentia
lly in the PDE dimension and/or the reciprocal of 𝜀. Recently\, certain
deep learning based approximation methods for PDEs have been proposed an
d various numerical simulations for such methods suggest that deep neural
network (DNN) approximations might have the capacity to indeed overcome th
e CoD in the sense that the number of real parameters used to describe th
e approximating DNNs grows at most polynomially in both the PDE dimension
𝑑∈\n and the reciprocal of the prescribed approximation accuracy
𝜀>0. There are now also a few rigorous mathematical results in the scie
ntific literature which substantiate this conjecture by proving that DNN
s overcome the CoD in approximating solutions of PDEs. Each of these resu
lts establishes that DNNs overcome the CoD in approximating suitable PDE s
olutions at a fixed time point 𝑇>0 and on a compact cube [𝑎\,𝑏]
𝑑 but none of these results provides an answer to the question whether
the entire PDE solution on [0\,𝑇]×[𝑎\,𝑏]𝑑 can be approximated
by DNNs without the CoD. \nIn this talk we show that for every 𝑎∈\\R
\, 𝑏∈(𝑎\,∞) solutions of suitable Kolmogorov PDEs can be appro
ximated by DNNs on the space-time region [0\,𝑇]×[𝑎\,𝑏]𝑑 witho
ut the CoD.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Cheridito (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20201207T160000Z
DTEND;VALUE=DATE-TIME:20201207T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/17
DESCRIPTION:Title: Efficient approximation of high-dimensional functions wi
th neural networks\nby Patrick Cheridito (ETH Zurich) as part of Oxfor
d Stochastic Analysis and Mathematical Finance Seminar\n\nLecture held in
Oxford Mathematical Institute.\n\nAbstract\nWe develop a framework for sho
wing that neural networks can overcome the curse of dimensionality in diff
erent high-dimensional approximation problems. Our approach is based on th
e notion of a catalog network\, which is a generalization of a standard ne
ural network in which the nonlinear activation functions can vary from lay
er to layer as long as they are chosen from a predefined catalog of functi
ons. As such\, catalog networks constitute a rich family of continuous fun
ctions. We show that under appropriate conditions on the catalog\, catalog
networks can efficiently be approximated with ReLU-type networks and prov
ide precise estimates on the number of parameters needed for a given appro
ximation accuracy. As special cases of the general results\, we obtain dif
ferent classes of functions that can be approximated with ReLU networks wi
thout the curse of dimensionality.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Donghan Kim (Columbia University)
DTSTART;VALUE=DATE-TIME:20210125T160000Z
DTEND;VALUE=DATE-TIME:20210125T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/18
DESCRIPTION:Title: Open Markets\nby Donghan Kim (Columbia University) a
s part of Oxford Stochastic Analysis and Mathematical Finance Seminar\n\nL
ecture held in Oxford Mathematical Institute.\n\nAbstract\nAn open market
is a subset of a larger equity market\, composed of a certain fixed number
of top‐capitalization stocks. Though the number of stocks in the open m
arket is fixed\, their composition changes over time\, as each company's r
ank by market capitalization fluctuates. When one is allowed to invest als
o in a money market\, an open market resembles the entire “closed” equ
ity market in the sense that the market viability (lack of arbitrage) is e
quivalent to the existence of a numéraire portfolio (which cannot be outp
erformed). When access to the money market is prohibited\, the class of po
rtfolios shrinks significantly in open markets\; in such a setting\, we di
scuss how to construct functionally generated stock portfolios and the con
cept of the universal portfolio.\nThis talk is based on joint work with Io
annis Karatzas.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathieu Lauriere (Princeton)
DTSTART;VALUE=DATE-TIME:20210118T160000Z
DTEND;VALUE=DATE-TIME:20210118T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/19
DESCRIPTION:Title: Machine Learning for Mean Field Games\nby Mathieu La
uriere (Princeton) as part of Oxford Stochastic Analysis and Mathematical
Finance Seminar\n\nLecture held in Oxford Mathematical Institute.\n\nAbstr
act\nMean field games (MFG) and mean field control problems (MFC) are fram
eworks to study Nash equilibria or social optima in games with a continuum
of agents. These problems can be used to approximate competitive or coope
rative situations with a large finite number of agents. They have found a
broad range of applications\, from economics to crowd motion\, energy prod
uction and risk management. Scalable numerical methods are a key step towa
rds concrete applications. In this talk\, we propose several numerical met
hods for MFG and MFC. These methods are based on machine learning tools su
ch as function approximation via neural networks and stochastic optimizati
on. We provide numerical results and we investigate the numerical analysis
of these methods by proving bounds on the approximation scheme. If time p
ermits\, we will also discuss model-free methods based on extensions of th
e traditional reinforcement learning setting to the mean-field regime.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Titus Lupu (Sorbonne Universite)
DTSTART;VALUE=DATE-TIME:20210201T160000Z
DTEND;VALUE=DATE-TIME:20210201T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/20
DESCRIPTION:Title: Extremal distance and conformal radius of a $CLE_4$ loop
.\nby Titus Lupu (Sorbonne Universite) as part of Oxford Stochastic An
alysis and Mathematical Finance Seminar\n\nLecture held in Oxford Mathemat
ical Institute.\n\nAbstract\nThe $CLE_4$ Conformal Loop Ensemble in a 2D s
imply connected domain is a random countable collection of fractal Jordan
curves that satisfies a statistical conformal invariance and appears\, or
is conjectured to appear\, as a scaling limit of interfaces in various sta
tistical physics models in 2D\, for instance in the double dimer model. Th
e $CLE_4$ is also related to the 2D Gaussian free field. Given a simply
connected domain D and a point z in D\, we consider the $CLE_4$ loop that
surrounds z and study the extremal distance between the loop and the bound
ary of the domain\, and the conformal radius of the interior surrounded by
the loop seen from z. Because of the conformal invariance\, the joint law
of this two quantities does not depend (up to a scale factor) on the choi
ce of the domain D and the point z in D. The law of the conformal radius a
lone has been known since the works of Schramm\, Sheffield and Wilson. We
complement their result by deriving the joint law of (extremal distance\,
conformal radius). Both quantities can be read on the same 1D Brownian pat
h\, by tacking a last passage time and a first hitting time. This joint la
w\, together with some distortion bounds\, provides some exponents related
to the $CLE_4$. \n\nThis is joint work with Juhan Aru and Avelio Sepulve
da.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Del Moral (INRIA (France))
DTSTART;VALUE=DATE-TIME:20210308T160000Z
DTEND;VALUE=DATE-TIME:20210308T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/21
DESCRIPTION:Title: A backward Ito-Ventzell formula with an application to s
tochastic interpolation\nby Pierre Del Moral (INRIA (France)) as part
of Oxford Stochastic Analysis and Mathematical Finance Seminar\n\nLecture
held in Oxford Mathematical Institute.\n\nAbstract\nWe discuss a novel bac
kward Ito-Ventzell formula and an extension of the Aleeksev-Gröbner inter
polating formula to stochastic flows. We also present some natural spectra
l conditions that yield direct and simple proofs of time uniform estimates
of the difference between the two stochastic flows when their drift and d
iffusion functions are not the same\, yielding what seems to be the first
results of this type for this class of anticipative models.\n\nWe illustr
ate the impact of these results in the context of diffusion perturbation t
heory\, interacting diffusions and discrete time approximations.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Olla (Paris Dauphine)
DTSTART;VALUE=DATE-TIME:20210215T160000Z
DTEND;VALUE=DATE-TIME:20210215T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/22
DESCRIPTION:Title: Thermal boundaries for energy superdiffusion\nby Ste
fano Olla (Paris Dauphine) as part of Oxford Stochastic Analysis and Mathe
matical Finance Seminar\n\nLecture held in Oxford Mathematical Institute.\
n\nAbstract\nhttps://www.maths.ox.ac.uk/node/38174\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Larsson (Carnegie Mellon)
DTSTART;VALUE=DATE-TIME:20210208T160000Z
DTEND;VALUE=DATE-TIME:20210208T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/24
DESCRIPTION:Title: Finance and Statistics: Trading Analogies for Sequential
Learning\nby Martin Larsson (Carnegie Mellon) as part of Oxford Stoch
astic Analysis and Mathematical Finance Seminar\n\nLecture held in Oxford
Mathematical Institute.\n\nAbstract\nThe goal of sequential learning is to
draw inference from data that is gathered gradually through time. This is
a typical situation in many applications\, including finance. A sequentia
l inference procedure is `anytime-valid’ if the decision to stop or cont
inue an experiment can depend on anything that has been observed so far\,
without compromising statistical error guarantees. A recent approach to an
ytime-valid inference views a test statistic as a bet against the null hyp
othesis. These bets are constrained to be supermartingales - hence unprofi
table - under the null\, but designed to be profitable under the relevant
alternative hypotheses. This perspective opens the door to tools from fina
ncial mathematics. In this talk I will discuss how notions such as superma
rtingale measures\, log-optimality\, and the optional decomposition theore
m shed new light on anytime-valid sequential learning. \n\nThis talk is ba
sed on joint work with Wouter Koolen (CWI)\, Aaditya Ramdas (CMU) and Joha
nnes Ruf (LSE).\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Röckner (Bielefeld)
DTSTART;VALUE=DATE-TIME:20210301T160000Z
DTEND;VALUE=DATE-TIME:20210301T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/25
DESCRIPTION:Title: Nonlinear Fokker-Planck equations with measures as initi
al data and McKean-Vlasov equations\nby Michael Röckner (Bielefeld
) as part of Oxford Stochastic Analysis and Mathematical Finance Seminar\n
\nLecture held in Oxford Mathematical Institute.\n\nAbstract\nThis talk is
about joint work with Viorel Barbu. We consider a class of nonlinear Fokk
er-Planck (- Kolmogorov) equations of type \n∂𝑡𝑢(𝑡\,𝑥)−Δ
𝑥𝛽(𝑢(𝑡\,𝑥))+div(𝐷(𝑥)𝑏(𝑢(𝑡\,𝑥))𝑢(𝑡\,
𝑥))=0\,𝑢(0\,⋅)=𝜇\,\nwhere (𝑡\,𝑥)∈[0\,∞)×ℝ𝑑\,
𝑑≥3 and 𝜇 is a signed Borel measure on ℝ𝑑 of bounded variatio
n. In the first part of the talk we shall explain how to construct a solut
ion to the above PDE based on classical nonlinear operator semigroup theor
y on 𝐿1(ℝ𝑑) and new results on 𝐿1−𝐿∞ regularization of t
he solution semigroups in our case. In the second part of the talk we shal
l present a general result about the correspondence of nonlinear Fokker-Pl
anck equations (FPEs) and McKean-Vlasov type SDEs. In particular\, it is s
hown that if one can solve the nonlinear FPE\, then one can always constru
ct a weak solution to the corresponding McKean-Vlasov SDE. We would like t
o emphasize that this\, in particular\, applies to the singular case\, whe
re the coefficients depend "Nemytski-type" on the time-marginal law of the
solution process\, hence the coefficients are not continuous in the measu
re-variable with respect to the weak topology on probability measures. Thi
s is in contrast to the literature in which the latter is standardly assum
ed. Hence we can cover nonlinear FPEs as the ones above\, which are PDEs f
or the marginal law densities\, realizing an old vision of McKean.\n\nRefe
rences V. Barbu\, M. Röckner: From nonlinear Fokker-Planck equations to s
olutions of distribution dependent SDE\, Ann. Prob. 48 (2020)\, no. 4\, 19
02-1920. V. Barbu\, M. Röckner: Solutions for nonlinear Fokker-Planck equ
ations with measures as initial data and McKean-Vlasov equations\, J. Func
t. Anal. 280 (2021)\, no. 7\, 108926.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Bouchard (Paris Dauphine)
DTSTART;VALUE=DATE-TIME:20210315T160000Z
DTEND;VALUE=DATE-TIME:20210315T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/26
DESCRIPTION:Title: Ito formula for C1 functionals and path-dependent applic
ations in mathematical finance\nby Bruno Bouchard (Paris Dauphine) as
part of Oxford Stochastic Analysis and Mathematical Finance Seminar\n\nLec
ture held in Oxford Mathematical Institute.\n\nAbstract\nWe will discuss s
everal versions of Ito’s formula in the case where the function is path
dependent and only concave or C1 in the sense of Dupire. In particular\, w
e will show that it can be used to solve (super) hedging problems\, in the
context of market impact or under volatility uncertainty.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Fehrman (Oxford)
DTSTART;VALUE=DATE-TIME:20210222T160000Z
DTEND;VALUE=DATE-TIME:20210222T170000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/27
DESCRIPTION:Title: Non equilibrium fluctuations in interactive particle sys
tems and conservative Stochastic PDEs\nby Benjamin Fehrman (Oxford) as
part of Oxford Stochastic Analysis and Mathematical Finance Seminar\n\nLe
cture held in Oxford Mathematical Institute.\n\nAbstract\nInteracting part
icle systems have found diverse applications in mathematics and several re
lated fields\, including statistical physics\, population dynamics\, and m
achine learning. We will focus\, in particular\, on the zero range proces
s and the symmetric simple exclusion process. The large-scale behavior of
these systems is essentially deterministic\, and is described in terms of
a hydrodynamic limit. However\, the particle process does exhibit large
fluctuations away from its mean. Such deviations\, though rare\, can have
significant consequences---such as a concentration of energy or the appea
rance of a vacuum---which make them important to understand and simulate.\
n\nIn this talk\, which is based on joint work with Benjamin Gess\, I will
introduce a continuum model for simulating rare events in the zero range
and symmetric simple exclusion process. The model is based on an approxim
ating sequence of stochastic partial differential equations with nonlinear
\, conservative noise. The solutions capture to first-order the central l
imit fluctuations of the particle system\, and they correctly simulate rar
e events in terms of a large deviations principle.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Davar Khoshnevisan (University of Utah)
DTSTART;VALUE=DATE-TIME:20210524T150000Z
DTEND;VALUE=DATE-TIME:20210524T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/28
DESCRIPTION:Title: Phase Analysis for a family of stochastic reaction-diffu
sion equations\nby Davar Khoshnevisan (University of Utah) as part of
Oxford Stochastic Analysis and Mathematical Finance Seminar\n\nLecture hel
d in Oxford Mathematical Institute.\n\nAbstract\nWe consider a reaction-di
ffusion equation of the type\n∂tψ=∂2xψ+V(ψ)+λσ(ψ)W˙on (0\,∞)
×𝕋\,\nsubject 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--KP
P 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 t
he strength of the effect of the noise W˙.\nThe 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 p
articular infinite. This proves an earlier prediction of Zimmerman et al.
(2000). Our methods also say a great deal about the structure of these inv
ariant measures.\n\nThis is based on joint work with Carl Mueller (Univ. R
ochester) and Kunwoo Kim (POSTECH\, S. Korea).\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Pierre Fouque (University of California Santa Barbara)
DTSTART;VALUE=DATE-TIME:20210614T150000Z
DTEND;VALUE=DATE-TIME:20210614T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/29
DESCRIPTION:Title: Linear-Quadratic Stochastic Differential Games on Direct
ed Chain Networks\nby Jean-Pierre Fouque (University of California San
ta Barbara) as part of Oxford Stochastic Analysis and Mathematical Finance
Seminar\n\nLecture held in Oxford Mathematical Institute.\n\nAbstract\nWe
present linear-quadratic stochastic differential games on directed chains
inspired by the directed chain stochastic differential equations introduc
ed by Detering\, Fouque\, and Ichiba in a previous work. We solve explicit
ly for Nash equilibria with a finite number of players and we study more g
eneral finite-player games with a mixture of both directed chain interacti
on and mean field interaction. We investigate and compare the correspondin
g games in the limit when the number of players tends to infinity. \nThe l
imit is characterized by Catalan functions and the dynamics under equilibr
ium is an infinite-dimensional Gaussian process described by a Catalan Mar
kov chain\, with or without the presence of mean field interaction.\n\nJoi
nt work with Yichen Feng and Tomoyuki Ichiba.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thaleia Zariphopoulou (University of Texas\, Austin)
DTSTART;VALUE=DATE-TIME:20210426T150000Z
DTEND;VALUE=DATE-TIME:20210426T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/30
DESCRIPTION:Title: Human-machine interaction models and robo-advising\n
by Thaleia Zariphopoulou (University of Texas\, Austin) as part of Oxford
Stochastic Analysis and Mathematical Finance Seminar\n\nLecture held in Ox
ford Mathematical Institute.\n\nAbstract\nI will introduce a family of hum
an-machine interaction (HMI) models in optimal portfolio construction (rob
o-advising). Modeling difficulties stem from the limited ability to quanti
fy the human’s risk preferences and describe their evolution\, but also
from the fact that the stochastic environment\, in which the machine optim
izes\, adapts to real-time incoming information that is exogenous to the h
uman. Furthermore\, the human’s risk preferences and the machine’s sta
tes may evolve at different scales. This interaction creates an adaptive c
ooperative game with both asymmetric and incomplete information exchange b
etween the two parties.\n\nAs a result\, challenging questions arise on\,
among others\, how frequently the two parties should communicate\, what in
formation can the machine accurately detect\, infer and predict\, how the
human reacts to exogenous events\, how to improve the inter-linked reliabi
lity between the human and the machine\, and others. Such HMI models give
rise to new\, non-standard optimization problems that combine adaptive sto
chastic control\, stochastic differential games\, optimal stopping\, multi
-scales and learning.\n\nhttps://zoom.us/meeting/register/tJEudOysqDktEtRY
1O1qvMurCmzAEkP0c91V\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fraydoun Rezakhanlou (University of California\, Berkeley)
DTSTART;VALUE=DATE-TIME:20210517T150000Z
DTEND;VALUE=DATE-TIME:20210517T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/31
DESCRIPTION:Title: Kinetic Theory for Hamilton-Jacobi PDEs\nby Fraydoun
Rezakhanlou (University of California\, Berkeley) as part of Oxford Stoch
astic Analysis and Mathematical Finance Seminar\n\nLecture held in Oxford
Mathematical Institute.\n\nAbstract\nThe flow of a Hamilton-Jacobi PDE yie
lds a dynamical system on the space of continuous functions. When the Hami
ltonian function is convex in the momentum variable\, and the spatial dime
nsion is one\, we may restrict the flow to piecewise smooth functions and
give a kinetic description for the solution. We regard the locations of ju
mp discontinuities of the first derivative of solutions as the sites of pa
rticles. These particles interact via collisions and coagulations. When th
ese particles are selected randomly according to certain Gibbs measures in
itially\, then the law of particles remains Gibbsian at later times\, and
one can derive a Boltzmann/Smoluchowski type PDE for the evolution of thes
e Gibbs measures. In higher dimensions\, we assume that the Hamiltonian f
unction is independent of position and that the initial condition is piec
ewise linear and convex. Such initial conditions can be identified as (Lag
uerre) tessellations and the Hamilton-Jacobi evolution can be described a
s a billiard on the set of tessellations.\n\nhttps://zoom.us/meeting/regis
ter/tJMtce6vrzojHd0_w6e6eOTwrgM1AL7v6GT9\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Chevyrev (University of Edinburgh)
DTSTART;VALUE=DATE-TIME:20210510T150000Z
DTEND;VALUE=DATE-TIME:20210510T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/32
DESCRIPTION:Title: Superdiffusive limits for deterministic fast-slow dynami
cal systems\nby Ilya Chevyrev (University of Edinburgh) as part of Oxf
ord Stochastic Analysis and Mathematical Finance Seminar\n\nLecture held i
n Oxford Mathematical Institute.\n\nAbstract\nWe consider multidimensional
fast-slow dynamical systems in discrete-time with random initial conditio
ns but otherwise completely deterministic dynamics. The question we will i
nvestigate is whether the slow variable converges in law to a stochastic p
rocess under a suitable scaling limit. We will be particularly interested
in the case when the limiting dynamic is superdiffusive\, i.e. it coincide
s in law with the solution of a Marcus SDE driven by a discontinuous stabl
e Lévy process. Under certain assumptions\, we will show that generically
convergence does not hold in any Skorokhod topology but does hold in a ge
neralisation of the Skorokhod strong M1 topology which we define using so-
called path functions. Our methods are based on a combination of ergodic t
heory and ideas arising from (but not using) rough paths. We will finally
show that our assumptions are satisfied for a class of intermittent maps o
f Pomeau-Manneville type.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuchong Zhang (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210607T150000Z
DTEND;VALUE=DATE-TIME:20210607T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/33
DESCRIPTION:Title: Risk-Taking Contest and its Mean Field Approximation
\nby Yuchong Zhang (University of Toronto) as part of Oxford Stochastic An
alysis and Mathematical Finance Seminar\n\nLecture held in Oxford Mathemat
ical Institute.\n\nAbstract\nIn the risk-taking model of Seel and Strack\,
n players decide when to stop privately observed Brownian motions with dr
ift and absorption at zero. They are then ranked according to their level
of stopping and paid a rank-dependent reward. We study the optimal reward
design where a principal is interested in the average performance and the
performance at a given rank. While the former can be related to reward ine
quality in the Lorenz sense\, the latter can have a surprising shape. Next
\, I will present the mean-field version of this problem. A particular fea
ture of this game is to be tractable without necessarily being smooth\, wh
ich turns out to offer a cautionary tale. We show that the mean field equi
librium induces n-player ε-Nash equilibria for any continuous reward func
tion— but not for discontinuous ones. We also analyze the quality of the
mean field design (for maximizing the median performance) when used as a
proxy for the optimizer in the n-player game. Surprisingly\, the quality d
eteriorates dramatically as n grows. We explain this with an asymptotic si
ngularity in the induced n-player equilibrium distributions.\n\nJoint work
with M. Nutz.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jin Ma (University of Southern California)
DTSTART;VALUE=DATE-TIME:20210621T150000Z
DTEND;VALUE=DATE-TIME:20210621T160000Z
DTSTAMP;VALUE=DATE-TIME:20210613T000759Z
UID:OxfordStochasticAnalysis/34
DESCRIPTION:Title: Set-valued Backward SDEs and Set-valued Stochastic Analy
sis\nby Jin Ma (University of Southern California) as part of Oxford S
tochastic Analysis and Mathematical Finance Seminar\n\nLecture held in Oxf
ord Mathematical Institute.\n\nAbstract\nWe establish an analytic framewor
k for studying Set-Valued Backward Stochastic Differential Equations (SVBS
DE for short)\, motivated largely by the current studies of dynamic set-va
lued risk measures for multi-asset or network-based financial models. Our
framework will be based on the notion of Hukuhara difference between sets\
, in order to compensate the lack of “inverse” operation of the tradit
ional Minkowski addition\, whence the vector space structure\, in traditio
nal set-valued analysis. We shall examine and establish a useful foundatio
n of set-valued stochastic analysis under this algebraic framework\, inclu
ding some fundamental issues regarding Aumann-Ito integrals\, especially w
hen it is connected to the martingale representation theorem. We shall ide
ntify some fundamental challenges and propose some extensions of the exist
ing theory that are necessary to study the SVBSDEs.\n\nThis talk is based
on the joint work with Cagın Ararat and Wenqian Wu.\n
LOCATION:https://researchseminars.org/talk/OxfordStochasticAnalysis/34/
END:VEVENT
END:VCALENDAR