Purba Das

King's College London | KCL · Mathematics

We study the concept of (generalized) $p$-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the $p$-th variation of a given function is closely related to the finiteness of $\ell^p$-norm of the coefficients along a Schauder basis, similar to the fact that H\"ol...

We investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent $$H< 0.5$$ H < 0.5 for modeling the volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized p -th varia...

We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a càdlàg path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized n...

Since the classical work of L\'evy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this characterization have primarily focused on Markovian or martingale settings, this work presents a highly anticipated extension to fractional Brownian motion -- a promin...

We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. Our construction method is based on Schauder representation along a general sequence of partitions and has two ramifications. We show that the variation index of a process along a given partition sequence (t...

In this paper, we consider a zero-sum undiscounted stochastic game which has finite state space and finitely many pure actions. Also, we assume the transition probability of the undiscounted stochastic game is controlled by one player and all the optimal strategies of the game are strictly positive. Under all the above assumptions, we show that the...

We investigate the statistical evidence for the use of 'rough' fractional processes with Hurst exponent H < 0.5 for the modeling of volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized p-th variation alon...

We present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic variation with respect to the sequence of partitions for these constructions. We identify a class of paths whose...

We present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic variation with respect to the sequence of partitions for these constructions. We identify a class of paths whose...

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the quadratic roughness of a path along a partition sequence and show that, for Holder-continuous paths satisfying this roughness condition, the quadratic variation along...

In this paper, we consider a zero-sum undiscounted stochastic game which has finite state space and finitely many pure actions. Also, we assume the transition probability of the undiscounted stochastic game is controlled by one player and all the optimal strategies of the game are strictly positive. Under all the above assumptions, we show that the...

In this article, we considered the problem of sea ice cover is melting. Considering the 'satellite passive microwave remote sensing data' as functional data, we studied daily observation of sea ice cover of each year as a smooth continuous function of time. We investigated the mean function for the sea ice area for following decades and computed th...

In this article, we considered the problem of sea ice cover is melting. Considering the `satellite passive microwave remote sensing data' as functional data, we studied daily observation of sea ice cover of each year as a smooth continuous function of time. We investigated the mean function for the sea ice area for following decades and computed th...