Doctoral Theses (Mathematical & Computer Sciences)http://hdl.handle.net/10399/372020-01-25T01:03:00Z2020-01-25T01:03:00ZOn weak and strong convergence rate for the Heston stochastic volatility modelZheng, Chaohttp://hdl.handle.net/10399/41062020-01-24T22:01:37Z2016-06-01T00:00:00ZOn weak and strong convergence rate for the Heston stochastic volatility model
Zheng, Chao
The Heston stochastic volatility model is one of the most fundamental models in mathematical ﬁnance. Recently, many numerical schemes have been developed for the Heston model. However, in the literature, there is no weak or strong convergence rate obtained for the full parameter regime. In this PhD thesis, we shall focus on the numerical scheme that simulates the variance process exactly and applies the stochastic trapezoidal rule to approximate the time integral of the variance process in the SDE of the logarithmic asset process. Our goal is to obtain the weak and strong convergence rates of such a numerical scheme for the Heston model. The weak convergence rate is of traditional interest, because it is an important measure on how fast the bias of a numerical scheme decays. We prove that the numerical scheme we consider converges at rate two for the whole parameter regime, and the test function can be any polynomial of the logarithmic asset process. The rate is consistent with the standard rate of the stochastic trapezoidal rule, although the Lipschitz assumption is not satisﬁed. The strong convergence analysis is meaningful in the framework of Multi-level Monte Carlo (MLMC). The MLMC can be regarded as a variance reduction technique for numerical schemes on SDEs, as long as there is a MLMC estimator with a good strong convergence rate. We establish efﬁcient MLMC estimators, separately for the path-independent and path-dependent simulations. We are able to provide the strong convergence rates in both situations.
2016-06-01T00:00:00ZStatistical modelling of the consistency of symptoms reported during hypoglycaemia for individual patientsZulkaﬂi, Hani Syahidahttp://hdl.handle.net/10399/40952020-01-20T22:01:29Z2017-05-01T00:00:00ZStatistical modelling of the consistency of symptoms reported during hypoglycaemia for individual patients
Zulkaﬂi, Hani Syahida
In this thesis, we use Bayesian methodology and Markov chain Monte Carlo tech-
niques to construct logistic-type latent variable statistical models for estimating the
consistency of hypoglycaemic symptoms experienced by individual diabetic patients.
Consistency in reporting experienced symptoms of hypoglycaemia is related to early
detection of symptoms and is therefore important for fast corrective action. Based
on a model developed by Zammit et al. (2011) we classify symptoms into diﬀerent
groups and consider between-groups variability. Our work also explores a number of
possible symptom-experiencing thresholds that can be used in the consistency model.
To evaluate the performance of each consistency model, we develop ideas based on
Bayesian latent residuals (Streftaris and Gibson, 2012) to check on the models’ ﬁt and
utilise posterior predictive checking methodology (Gelman et al., 1996 and Streftaris
et al., 2013) to assess relevant performance. The impact of using data from hypo-
glycaemic episodes occurring within 24 hours from an earlier episode is also explored
using various approaches, as previous work claims that such episodes might lead to di-
minished intensity of the episodes. Using generalised linear-type model methodology,
we investigate how various factors such as age, gender, type and duration of diabetes,
body mass index, retinopathy and others, or their interaction, can aﬀect patients’
consistency. Additionally, we develop a hierarchical model that is able to estimate
consistency and identify factors aﬀecting it in a single setting. Finally, we work on
determining the best sets of variables for a predictive model. For this purpose, we
use Gibbs variable selection and a stepwise regression procedure. Due to model un-
certainty, we apply Bayesian model averaging to a number of selected models given
by Gibbs variable selection.
2017-05-01T00:00:00ZTopics in coagulation-fragmentation equationsAlghamdi, Matabhttp://hdl.handle.net/10399/40452019-10-18T21:01:38Z2014-11-01T00:00:00ZTopics in coagulation-fragmentation equations
Alghamdi, Matab
In this thesis we study the mathematics of a model for the dynamics of cluster growth.
The sizes of the clusters change in time as the clusters undergo coagulation and
fragmentation events. The equations are for j = 1,2,...
c′ j =1 2j−1 X k=1[aj−k,kcj−kck −bj−k,kcj]−∞ X k=1[aj,kcjck −bj,kcj+k] (1.1)
where cj(t) is the concentration of clusters of size j and aj,k,bj,k are the constant rates
of coagulation and fragmentation.
Chapter 1 reviews some results on (1.1) and introduces some mathematical tools
used in the thesis. It also introduces the concept of gelation, which is the formation
of an inﬁnite cluster leading to the loss of mass conservation.
In Chapter 2 we study gelation in (1.1) and discuss ﬁnite dimensional approxima
tions which are used for numerical studies. We explain why a certain ﬁnite dimen
sional system which does not conserve density is suitable for numerical studies of (1.1)
including gelation.
All solutions of the ﬁnite dimensional system converge to zero and Chapter 3
deals with the asymptotic behaviour. For the case in which the coagulation and
fragmentation terms are non zero and satisfy a detailed balance condition, we obtain
a general result on the asymptotic decay. However, for the pure coagulation case
(bj,k = 0), we show that a wide variety of asymptotics is possible.
Chapter 4 is concerned with a model for the treatment of Alzheimer’s disease.
The model is a modiﬁed form of (1.1). We prove some mathematical results for the
system and obtain an approximate formula for the decay rate.
Chapter 5 deals with numerical approximations to the continuous version of (1.1).
We consider a piecewise constant in space approximation in both collocation and the
Galerkin formulation. Numerical results indicate that the Galerkin ﬁnite element
method has second order accuracy. These approximations of the continuous problem
are themselves discrete systems like (1.1).
2014-11-01T00:00:00ZAn investigation of service degradation in long-term human-robot interaction with a particular reference to recharge behaviourDeshmukh, Amol Arunhttp://hdl.handle.net/10399/40442019-10-18T21:01:41Z2015-12-01T00:00:00ZAn investigation of service degradation in long-term human-robot interaction with a particular reference to recharge behaviour
Deshmukh, Amol Arun
Autonomous long-term operation of social robots has always been a challenge in Human robot-interaction. Social mobile robots acting as companions or assistants will need to operate over a long-term period of time (days, weeks or even months) to perform daily tasks and interact with users. Therefore they should be capable of operating with a great degree of autonomy and will require sustainable social intelligence. Social robots are fallible and have their own limitations with the service they provide. One of the most important limitations of mobile robots is power constraints and the need for frequent recharging. Social mobile robots generally draw power from batteries carried on the robot in order to operate various sensors, actuators and perform tasks. However, batteries have a limited power life and take a long time to recharge via a power source. While the recharge behaviour is active, which may impede human-robot interaction and lead to service degradation. This thesis raises some important issues related to recharge behaviour of social mobile robots which appear to have been overlooked in social robotics research. This work investigated service degradation in long-term interaction due to recharge behaviour of autonomous social mobile robots and proposes an approach to manage service degradation due to recharge. First we performed a long-term study to investigate the service degradation caused by the recharging behaviour of a social robot. Second we conducted a more focused social study which helped to understand user’s attitudes towards a mobile robot with respect to recharge activity. We explored a social strategy by modifying the robot’s verbal behaviour to manage service degradation during recharge. The results obtained from our social study indicates the use of verbal strategies (transparency, apology, politeness) made the robot more acceptable to the users during recharge. We believe that social mobile robots should behave in a socially intelligent manner while managing service degradation. We also provide some recommendations for social mobile robots to manage their recharge behaviour in this thesis.
2015-12-01T00:00:00Z