Scaled penalization of Brownian motion with drift and the Brownian ascent

We study a scaled version of a two-parameter Brownian penalization model introduced by Roynette-Vallois-Yor in arXiv:math/0511102. The original model penalizes Brownian motion with drift $h\in\mathbb{R}$ by the weight process ${\big(\exp(\nu S_t):t\geq 0\big)}$ where $\nu\in\mathbb{R}$ and $\big(S_t:t\geq 0\big)$ is the running maximum of the Brownian motion. It was shown there that the resulting penalized process exhibits three distinct phases corresponding to different regions of the $(\nu,h)$-plane. In this paper, we investigate the effect of penalizing the Brownian motion concurrently with scaling and identify the limit process. This extends a result of Roynette-Yor for the ${\nu<0,~h=0}$ case to the whole parameter plane and reveals two additional "critical" phases occurring at the boundaries between the parameter regions. One of these novel phases is Brownian motion conditioned to end at its maximum, a process we call the Brownian ascent. We then relate the Brownian ascent to some well-known Brownian path fragments and to a random scaling transformation of Brownian motion recently studied by Rosenbaum-Yor.Comment: 32 pages; made additions to Section

Robust pricing–hedging dualities in continuous time

We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413–1438, 2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing–hedging duality result: the infimum over superhedging prices of an exotic option with payoff G is equal to the supremum of expectations of G under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391–427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous G, the asserted duality holds between limiting values of perturbed problems

The Map of Vilnius Graffiti as an Indicator of Social Urban Change: the Case Study of Naujininkai Neighborhood

This article, theoretically based on socio-spatial concepts of Lefebvre, de Certeau and their further interpreta - tions at the New Urban Sociology school (by Gottdiener, Zukin and others), examines the spread of graffiti in the urban space of Vilnius, the change of the local graffiti map during the years 2010–2013 and the possible social implications of the spotted modification of urban landscape. The qualitative research of Vilnius graffiti – which is understood both as an urban practice and an illicit urban inscription – and the case of Naujininkai neighborhood in particular, is based on data obtained from 1) in-depth interviews with experienced graffiti artists, 2) observation of graffiti in public space and 3) visual urban ethnography. Naujininkai neighborhood was attributed by local graffiti writers to the urban periphery in Vilnius graffiti map in 2010. However in 2010–2013 the visual development of urban landscape in Naujininkai indicates the trend, bringing the neigh- borhood a little closer to the urban core