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SUMMARY:Andreas Petersson (University of Oslo)
DTSTART;VALUE=DATE-TIME:20200918T090000Z
DTEND;VALUE=DATE-TIME:20200918T100000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/1
DESCRIPTION:Title: Finite element approximation of Lyapunov equations for
the computation of quadratic functionals of SPDEs\nby Andreas Petersson (U
niversity of Oslo) as part of STAR seminars\n\n\nAbstract\nWe consider the
computation of quadratic functionals of the solution to a linear paraboli
c stochastic partial differential equation (SPDE) with multiplicative Gaus
sian noise on a bounded domain. The functionals are allowed to be path dep
endent and the noise is white in time and may be white in space. An operat
or valued Lyapunov equation\, whose solution admits a deterministic repres
entation of the functional of the SPDE solution\, is used for this purpose
and error estimates are shown in suitable operator norms for a fully disc
rete approximation of this equation. We also use these estimates to derive
weak error rates for a fully discrete approximation of the SPDE itself. I
n the setting of finite element approximations\, a computational complexit
y comparison reveals that approximating the Lyapunov equation allows us to
compute quadratic functionals more cheaply compared to applying Monte Car
lo or covariance-based methods directly to the discretized SPDE. We illust
rate the theoretical results with numerical simulations.\nThis is joint wo
rk with Adam Andersson\, Annika Lang and Leander Schroer.\n
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BEGIN:VEVENT
SUMMARY:Emel Savku (University of Oslo)
DTSTART;VALUE=DATE-TIME:20200925T090000Z
DTEND;VALUE=DATE-TIME:20200925T100000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/2
DESCRIPTION:Title: Optimal investment strategies in a Markov Regime-Switch
ing Market\nby Emel Savku (University of Oslo) as part of STAR seminars\n\
n\nAbstract\nWe discuss two optimal investment problems by using zero-sum
and nonzerosum stochastic game approaches in a continuous-time Markov regi
meswitching jump-diffusion environment. We represent different states of a
n economy by a D-state Markov chain. The first application is a zero-sum g
ame between an investor and the market\, and the second one formulates a n
onzerosum stochastic differential portfolio game as the sensitivity of two
investors’ terminal gains.We derive regime-switching Hamilton–Jacobi
–Bellman–Isaacs equations and obtain explicit optimal portfolio strate
gies.We illustrate our results in a two-state special case and observe the
impact of regime switches by comparative results.\nJoint work with Gerhar
d Wilhem Weber.\n
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BEGIN:VEVENT
SUMMARY:Jasmina Djordjevic (University of Oslo)
DTSTART;VALUE=DATE-TIME:20201002T090000Z
DTEND;VALUE=DATE-TIME:20201002T100000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/3
DESCRIPTION:Title: Perturbation effect on Reflected Backward Stochastic Di
fferential Equations\nby Jasmina Djordjevic (University of Oslo) as part o
f STAR seminars\n\n\nAbstract\nPerturbed stochastic differential equations
\, in general\, are the topic of permanent interest of many authors\, both
theoretically and in applications. Stochastic models of complex phenomena
under perturbations in analytical mechanics\, control theory and populati
on dynamics\, for example\, can be sometimes compared and approximated by
appropriate unperturbed models of a simpler structure. In this way\, the p
roblems can be translated into more simple and familiar cases which are ea
sier to solve and investigate. Problems of perturbed backward stochastic d
ifferential equations (BSDEs) are very interesting because of their applic
ations in economy and finance. The most interesting problem in this field
of perturbations of BSDEs deals with a large class of reflected backward s
tochastic differential equations whose generator\, barrier process and fin
al condition are arbitrarily dependent on a small parameter. The solution
of perturbed equation\, is compared in the L p -sense\, with the solutions
of the appropriate unperturbed equations. Conditions under which the solu
tion of the unperturbed equation is L p -stable are given. It is shown tha
t for an arbitrary η > 0 there exists an interval [t(η)\, T] ⊂ [0\, T]
on which the L p -difference between the solutions of both the perturbed
and unperturbed equations is less than η.\n
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BEGIN:VEVENT
SUMMARY:Leonardo Rydin Gorjão (Institute of Theoretical Physics\, Univers
ity of Cologne)
DTSTART;VALUE=DATE-TIME:20201009T090000Z
DTEND;VALUE=DATE-TIME:20201009T100000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/4
DESCRIPTION:Title: Applications and developments of stochastic processes i
n power-grid frequency measurements: A data-driven study.\nby Leonardo Ryd
in Gorjão (Institute of Theoretical Physics\, University of Cologne) as p
art of STAR seminars\n\n\nAbstract\nPower-grid frequency is a key measurem
ent of stability of power-grid systems. It comprises the balance of power
generation and consumption\, electricity market exchanges\, and control me
chanism. Power-grid frequency\, as stochastic process\, has been scarcely
studied. We will present the developments in power-grid frequency data col
lection\, the design of a N-dimensional non-parametric estimator for time-
continuous Markov processed\, and the design of a computationally efficien
t Multifractal Detrended Fluctuation Analysis (MFDFA) algorithm. Lastly\,
we will report on the design of a surrogate stochastic model for power-gri
d frequency via a fractional Ornstein–Uhlenbeck process\, the applicatio
n of a Hurst index and a volatility estimator\, and the limitations due to
multifractional and time-and-space coloured noise.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Sanz-Solé (University of Barcelona)
DTSTART;VALUE=DATE-TIME:20201016T090000Z
DTEND;VALUE=DATE-TIME:20201016T100000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/5
DESCRIPTION:Title: Stochastic wave equations with super-linear coefficient
s\nby Marta Sanz-Solé (University of Barcelona) as part of STAR seminars\
n\n\nAbstract\nWe consider a stochastic wave equation on R^d \, d ∈ {1\,
2\, 3}\, driven by a Gaussian noise in (t\, x)\, white in time. We assume
that the free terms b and σ are such that\, for |x| → ∞\, \n|σ(x)|
≤ σ_1 + σ2_|x| (ln_+(|x|))^a \, |b(x)| ≤ θ_1 + θ_2|x| (ln_+(|x|))^
δ \, (1) \nwhere θ_2\, σ_2 > 0\, δ\, a > 0\, with b dominating over σ
. For any fixed time horizon T > 0 and with a suitable constraints on the
parameters a\, δ\, σ_2 and θ_2\, we prove existence of a random field s
olution to the equation and that this solution is unique\, and bounded in
time and in space a.s. The research is motivated by the article [R. Dalang
\, D. Khoshnevisan\, T. Zhang\, AoP\, 2019] on a 1-d reaction-diffusion eq
uation with coefficients satisfying conditions similar to (1). We see that
the L^∞- method used by these authors can be successfully implemented i
n the case of wave equations. This is joint work with A. Millet (U. Paris
1\, Panthéon-Sorbonne).\n
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BEGIN:VEVENT
SUMMARY:Samy Tindel (Purdue University)
DTSTART;VALUE=DATE-TIME:20201023T090000Z
DTEND;VALUE=DATE-TIME:20201023T100000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/6
DESCRIPTION:Title: A coupling between Sinai’s random walk and Brox diffu
sion\nby Samy Tindel (Purdue University) as part of STAR seminars\n\n\nAbs
tract\nSinai’s random walk is a standard model of 1-dimensional random w
alk in random environment. Brox diffusion is its continuous counterpart\,
that is a Brownian diffusion in a Brownian environment. The convergence in
law of a properly rescaled version of Sinai’s walk to Brox diffusion ha
s been established 20 years ago. In this talk\, I will explain a strategy
which yields the convergence of Sinai’s walk to Brox diffusion thanks to
an explicit coupling. This method\, based on rough paths techniques\, ope
ns the way to rates of convergence in this demanding context. Notice that
I’ll try to give a maximum of background about the objects I’m manipul
ating\, and will keep technical considerations to a minimum.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yaozhong Hu (University of Alberta)
DTSTART;VALUE=DATE-TIME:20201106T100000Z
DTEND;VALUE=DATE-TIME:20201106T110000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/7
DESCRIPTION:Title: Functional central limit theorems for stick-breaking pr
iors\nby Yaozhong Hu (University of Alberta) as part of STAR seminars\n\n\
nAbstract\nWe obtain the empirical strong law of large numbers\, empirical
\nGlivenko-Cantelli theorem\, central limit theorems\, \nfunctional centr
al limit theorems for various nonparametric Bayesian priors\nwhich includ
e the Dirichlet process with general stick-breaking weights\, \nthe P
oisson-Dirichlet process\, the normalized inverse Gaussian \nprocess\,
the normalized generalized gamma \nprocess\, and the generalized Di
richlet process. \nFor the Dirichlet process with general stick-breaking
weights\, \nwe introduce two general conditions such that the central limi
t theorem holds. \nExcept in the case of generalized Dirichlet process\,
since the finite dimensional \ndistributions of these processes are either
hard to obtain or are \ncomplicated to use even they are available\, \nw
e use the general moment method to obtain the convergence results. \nFor
the generalized Dirichlet process we use its finite dimensional marginal
distributions to obtain the asymptotics although \nthe computations are
highly technical.\n
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BEGIN:VEVENT
SUMMARY:Rama Cont (University of Oxford)
DTSTART;VALUE=DATE-TIME:20201113T100000Z
DTEND;VALUE=DATE-TIME:20201113T110000Z
DTSTAMP;VALUE=DATE-TIME:20201101T013121Z
UID:STochastics_And_Risk/8
DESCRIPTION:Title: Excursion risk\nby Rama Cont (University of Oxford) as
part of STAR seminars\n\n\nAbstract\nA broad class of dynamic trading stra
tegies may be characterized in terms of excursions of the market price o
f a portfolio away from a reference level. We propose a mathematical fram
ework for the risk analysis of such strategies\, based on a description i
n terms of price excursions\, first in a pathwise setting\, without probab
ilistic assumptions\, then in a probabilistic setting\, when the price is
modelled as a Markov process.\n\nWe introduce the notion of δ-excursion\,
defined as a path which deviates by δ from a reference level before ret
urning to this level. We show that every continuous path has a unique deco
mposition into such δ-excursions\, which turn out to be useful for the sc
enario analysis of dynamic trading strategies\, leading to simple expressi
ons for the number of trades\, realized profit\, maximum loss and drawdown
. \nWhen the underlying asset follows a Markov process\, we combine these
results with Ito's excursion theory to obtain a tractable decomposition of
the process as a concatenation of independent δ-excursions\, whose distr
ibution is described in terms of Ito's excursion measure. We provide analy
tical 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 excursions match those observed in a data set.\n\nThis is joint work
with: Anna Ananova and RenYuan Xu.\n
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