270 research outputs found
On small time asymptotics for rough differential equations driven by fractional Brownian motions
We survey existing results concerning the study in small times of the density
of the solution of a rough differential equation driven by fractional Brownian
motions. We also slightly improve existing results and discuss some possible
applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of
Peter Laurenc
An Optimal Execution Problem with Market Impact
We study an optimal execution problem in a continuous-time market model that
considers market impact. We formulate the problem as a stochastic control
problem and investigate properties of the corresponding value function. We find
that right-continuity at the time origin is associated with the strength of
market impact for large sales, otherwise the value function is continuous.
Moreover, we show the semi-group property (Bellman principle) and characterise
the value function as a viscosity solution of the corresponding
Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of
the optimal strategies change completely, depending on the amount of the
trader's security holdings and where optimal strategies in the Black-Scholes
type market with nonlinear market impact are not block liquidation but gradual
liquidation, even when the trader is risk-neutral.Comment: 36 pages, 8 figures, a modified version of the article "An optimal
execution problem with market impact" in Finance and Stochastics (2014
Light--like Wilson loops and gauge invariance of Yang--Mills theory in 1+1 dimensions
A light-like Wilson loop is computed in perturbation theory up to for pure Yang--Mills theory in 1+1 dimensions, using Feynman and
light--cone gauges to check its gauge invariance. After dimensional
regularization in intermediate steps, a finite gauge invariant result is
obtained, which however does not exhibit abelian exponentiation. Our result is
at variance with the common belief that pure Yang--Mills theory is free in 1+1
dimensions, apart perhaps from topological effects.Comment: 10 pages, plain TeX, DFPD 94/TH/
Electroweak Gauge-Boson Production at Small q_T: Infrared Safety from the Collinear Anomaly
Using methods from effective field theory, we develop a novel, systematic
framework for the calculation of the cross sections for electroweak gauge-boson
production at small and very small transverse momentum q_T, in which large
logarithms of the scale ratio M_V/q_T are resummed to all orders. These cross
sections receive logarithmically enhanced corrections from two sources: the
running of the hard matching coefficient and the collinear factorization
anomaly. The anomaly leads to the dynamical generation of a non-perturbative
scale q_* ~ M_V e^{-const/\alpha_s(M_V)}, which protects the processes from
receiving large long-distance hadronic contributions. Expanding the cross
sections in either \alpha_s or q_T generates strongly divergent series, which
must be resummed. As a by-product, we obtain an explicit non-perturbative
expression for the intercept of the cross sections at q_T=0, including the
normalization and first-order \alpha_s(q_*) correction. We perform a detailed
numerical comparison of our predictions with the available data on the
transverse-momentum distribution in Z-boson production at the Tevatron and LHC.Comment: 34 pages, 9 figure
Drell-Yan production at small q_T, transverse parton distributions and the collinear anomaly
Using methods from effective field theory, an exact all-order expression for
the Drell-Yan cross section at small transverse momentum is derived directly in
q_T space, in which all large logarithms are resummed. The anomalous dimensions
and matching coefficients necessary for resummation at NNLL order are given
explicitly. The precise relation between our result and the
Collins-Soper-Sterman formula is discussed, and as a by-product the previously
unknown three-loop coefficient A^(3) is obtained. The naive factorization of
the cross section at small transverse momentum is broken by a collinear
anomaly, which prevents a process-independent definition of x_T-dependent
parton distribution functions. A factorization theorem is derived for the
product of two such functions, in which the dependence on the hard momentum
transfer is separated out. The remainder factors into a product of two
functions of longitudinal momentum variables and x_T^2, whose
renormalization-group evolution is derived and solved in closed form. The
matching of these functions at small x_T onto standard parton distributions is
calculated at O(alpha_s), while their anomalous dimensions are known to three
loops.Comment: 32 pages, 2 figures; version to appear in Eur. Phys. J.
Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach
We consider the problem of soft gluon resummation for gauge theory amplitudes
and cross sections, at next-to-eikonal order, using a Feynman diagram approach.
At the amplitude level, we prove exponentiation for the set of factorizable
contributions, and construct effective Feynman rules which can be used to
compute next-to-eikonal emissions directly in the logarithm of the amplitude,
finding agreement with earlier results obtained using path-integral methods.
For cross sections, we also consider sub-eikonal corrections to the phase space
for multiple soft-gluon emissions, which contribute to next-to-eikonal
logarithms. To clarify the discussion, we examine a class of log(1 - x) terms
in the Drell-Yan cross-section up to two loops. Our results are the first steps
towards a systematic generalization of threshold resummations to
next-to-leading power in the threshold expansion.Comment: 66 pages, 19 figure
Non-global Structure of the O({\alpha}_s^2) Dijet Soft Function
High energy scattering processes involving jets generically involve matrix
elements of light- like Wilson lines, known as soft functions. These describe
the structure of soft contributions to observables and encode color and
kinematic correlations between jets. We compute the dijet soft function to
O({\alpha}_s^2) as a function of the two jet invariant masses, focusing on
terms not determined by its renormalization group evolution that have a
non-separable dependence on these masses. Our results include non-global single
and double logarithms, and analytic results for the full set of non-logarithmic
contributions as well. Using a recent result for the thrust constant, we
present the complete O({\alpha}_s^2) soft function for dijet production in both
position and momentum space.Comment: 55 pages, 8 figures. v2: extended discussion of double logs in the
hard regime. v3: minor typos corrected, version published in JHEP. v4: typos
in Eq. (3.33), (3.39), (3.43) corrected; this does not affect the main
result, numerical results, or conclusion
Instanton Contribution to the Quark Form Factor
The nonperturbative effects in the quark form factor are considered in the
Wilson loop formalism. The properties of the Wilson loops with cusp
singularities are studied taking into account the perturbative and
nonperturbative contributions, where the latter are considered within the
framework of the instanton liquid model. For the integration path corresponding
to this form factor -- the angle with infinite sides -- the explicit expression
for the vacuum expectation value of the Wilson operator is found to leading
order. The calculations are performed in the weak-field limit for the instanton
vacuum contribution and compared with the one- and two-loop order results for
the perturbative part. It is shown that the instantons produce the powerlike
corrections to the perturbative result, which are comparable in magnitude with
the perturbative part at the scale of order of the inverse average instanton
size. It is demonstrated that the instanton contributions to the quark form
factor are exponentiated to high orders in the small instanton density
parameter.Comment: Version coincident with the journal publication. LaTeX, 15 pages, 1
figur
Factorization Properties of Soft Graviton Amplitudes
We apply recently developed path integral resummation methods to perturbative
quantum gravity. In particular, we provide supporting evidence that eikonal
graviton amplitudes factorize into hard and soft parts, and confirm a recent
hypothesis that soft gravitons are modelled by vacuum expectation values of
products of certain Wilson line operators, which differ for massless and
massive particles. We also investigate terms which break this factorization,
and find that they are subleading with respect to the eikonal amplitude. The
results may help in understanding the connections between gravity and gauge
theories in more detail, as well as in studying gravitational radiation beyond
the eikonal approximation.Comment: 35 pages, 5 figure
Quality control for multiple breath washout tests in multicentre bronchiectasis studies:Experiences from the BRONCH-UK clinimetrics study
Multiple Breath Washout (MBW) to measure Lung Clearance Index (LCI) is increasingly being used as a secondary endpoint in multicentre bronchiectasis studies. LCI data quality control or “over-reading” is resource intensive and the impact is unclear. Objectives: To assess the proportion of MBW tests deemed unacceptable with over-reading, and to assess the change in LCI (number of turnovers), LCI coefficient of variation (CV%) and tidal volume (VT) CV% results after over-reading. Methods: Data were analysed from 250 MBW tests (from 98 adult bronchiectasis patients) collected as part of the Bronch-UK Clinimetrics study in 5 UK centres. Each MBW test was over-read centrally using pre-defined criteria. MBW tests with <2 technically valid and repeatable trials were deemed unacceptable to include in analysis. In accepted tests, values for LCI, LCI CV% and VT CV% before and after over-reading, were compared. Results: Insufficient data was collected in 10/250 tests. With over-reading, 30/240 (12%) were deemed unacceptable to include in analysis. In those accepted tests, overall the change in LCI, LCI CV% and VT CV% with over-reading was not statistically significant. When MBW new sites were compared to MBW expert sites, the change in LCI with over-reading was significantly greater in MBW new sites (p = 0.047). Data suggests that over-reading could be important up to at least 12 months post initiation of MBW activity. Conclusion: MBW over-reading was important in this study as 12% of tests were considered unacceptable. Over-reading improved test result accuracy in sites new to MBW
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