Efficient pricing options under regime switching

In the paper, we propose two new efficient methods for pricing barrier option in wide classes of Lévy processes with/without regime switching. Both methods are based on the numerical Laplace transform inversion formulae and the Fast Wiener-Hopf factorization method developed in Kudryavtsev and Levendorski\v{i} (Finance Stoch. 13: 531--562, 2009). The first method uses the Gaver-Stehfest algorithm, the second one -- the Post-Widder formula. We prove the advantage of the new methods in terms of accuracy and convergence by using Monte-Carlo simulations

On perpetual American put valuation and first-passage in a regime-switching model with jumps

In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching L\'{e}vy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem under a state-dependent level rests on a path transformation and a new matrix Wiener-Hopf factorization result for this class of processes.Comment: 22 pages, 3 figures. Tp appear in Finance and Stochastics

Free energy reconstruction from steered dynamics without post-processing

Various methods achieving importance sampling in ensembles of nonequilibrium trajectories enable to estimate free energy differences and, by maximum-likelihood post-processing, to reconstruct free energy landscapes. Here, based on Bayes theorem, we propose a more direct method in which a posterior likelihood function is used both to construct the steered dynamics and to infer the contribution to equilibrium of all the sampled states. The method is implemented with two steering schedules. First, using non-autonomous steering, we calculate the migration barrier of the vacancy in Fe-alpha. Second, using an autonomous scheduling related to metadynamics and equivalent to temperature-accelerated molecular dynamics, we accurately reconstruct the two-dimensional free energy landscape of the 38-atom Lennard-Jones cluster as a function of an orientational bond-order parameter and energy, down to the solid-solid structural transition temperature of the cluster and without maximum-likelihood post-processing.Comment: Accepted manuscript in Journal of Computational Physics, 7 figure

Characterization of a Low Frequency Power Spectral Density f^(-gamma) in a Threshold Model

his study investigates the modifications of the thermal spectrum, at low frequency, induced by an external damping on a system in heat contact with internal fluctuating impurities. Those impurities can move among locations and their oscillations are associated with a loss function depending on the model. The fluctuation properties of the system are provided by a potential function shaped by wells, in such a way that jumps between the stationary positions are allowed. The power spectral density associated with this dissipation mechanism shows a f^(-gamma)tail. The interest of this problem is that many systems are characterized by a typical f^(-\gamma) spectral tail at low frequency. The model presented in this article is based on a threshold type behaviour and its generality allows applications in several fields. The effects of an external force, introduced to produce damping, are studied by using both analytical techniques and numerical simulations. The results obtained with the present model show that no reduction of the power spectral density is appreciable below the main peak of the spectral density.Comment: 15 pages, 6 figure

Efficient pricing options under regime switching

In the paper, we propose two new efficient methods for pricing barrier option in wide classes of Lévy processes with/without regime switching. Both methods are based on the numerical Laplace transform inversion formulae and the Fast Wiener-Hopf factorization method developed in Kudryavtsev and Levendorski\v{i} (Finance Stoch. 13: 531--562, 2009). The first method uses the Gaver-Stehfest algorithm, the second one -- the Post-Widder formula. We prove the advantage of the new methods in terms of accuracy and convergence by using Monte-Carlo simulations

Regimes, Non-Linearities, and Price Discontinuities in Indian Energy Stocks

We construct a representative index of largest Indian energy companies listed on the National Stock Exchange (NIFTY 50). We test for presence of regimes, non-linearities, and jumps in the price signal. We benchmark performance against alternative models, including single-regime models and models with no jumps. We then benchmark the quality of regime identification against other indices examined in the literature, such as Nikkei 225 and FTSE 100. Overall, find that our regime-switching model performs well in identifying the regimes in this comparative setting. Based on our model selection criteria, we prefer a regime-augmented model to a model that allows no regime identification. But overall, we prefer a model with jumps and regimes over those that do not allow for jump-diffusion and Markov regime-switching

Spatiotemporal Effects of Transport and Network Topology in Biological Systems

Emergent biological phenomena, although observed experimentally, are often not easily characterized or understood. Biological systems are often comprised of many interacting components, which may yield highly complex dynamics. A thorough understanding of these systems often requires a multi-faceted approach involving both experimental and computational techniques. Computer simulation allows for precise definition of system components and facilitates a wider exploration of the system parameter space, often leading to accelerated scientific discovery. In this thesis, we apply stochastic simulation methods to characterize the spatiotemporal behavior of three distinct biological systems. We first explore the role of spatial confinement and diffusion in a bistable reaction network with positive feedback. We find that confined systems with high molecular mobility promote the active steady state, and stochastic switching occurs unidirectionally by nucleation and growth of single active clusters. The results provide a general framework for studying geometry and diffusion in positive feedback networks, and suggest that confinement can be used to initiate the formation of localized active clusters of molecules that then propagate to activate a system. Next, we study transport properties of single molecular motors traversing cytoskeletal networks with random filament configuration. We find that systems containing few, long filaments exhibit slow and highly variable transport. Particular filaments are capable of having an outsized influence on first-passage times by acting as lynchpins that transport motors to and from regions of the system that act as traps that promote extended occupancy. Finally, we use two distinct models to explore the dynamics of protein organization along an actomyosin ring. We find that a positive feedback circuit can be used to establish and maintain polarized protein distributions, and clustering is suppressed by endocytosis and fast diffusion. In the absence of positive feedback and dissociation from the ring, we find that slow association of large patches leads to clustered distributions of higher variability. These results suggest that homogeneous spatial distribution of proteins in mature actomyosin rings may depend on frequent association of small protein clusters. Taken collectively, these findings suggest that stochastic computational modeling can facilitate the elucidation of key mechanistic features of emergent biological phenomena

Non-Linear diffusion processes and applications

Diffusion models are useful tools for quantifying the dynamics of continuously evolving processes. Using diffusion models it is possible to formulate compact descriptions for the dynamics of real-world processes in terms of stochastic differential equations. Despite the exibility of these models, they can often be extremely difficult to work with. This is especially true for non-linear and/or time-inhomogeneous diffusion models where even basic statistical properties of the process can be elusive. As such, we explore various techniques for analysing non-linear diffusion models in contexts ranging from conducting inference under discrete observation and solving first passage time problems, to the analysis of jump diffusion processes and highly non-linear diffusion processes. We apply the methodology to a number of real-world ecological and financial problems of interest and demonstrate how non-linear diffusion models can be used to better understand such phenomena. In conjunction with the methodology, we develop a series of software packages that can be used to accurately and efficiently analyse various classes of non-linear diffusion models

Doctor of Philosophy

dissertationActive transport of cargoes is critical for cellular function. To accomplish this, networks of cytoskeletal filaments form highways along which small teams of mechanochemical enzymes (molecular motors) take steps to pull associated cargoes. The robustness of this transport system is juxtaposed by the stochasticity that exists at several spatial and temporal scales. For instance, individual motors stochastically step, bind, and unbind while the cargo undergoes nonnegligible thermal fluctuations. Experimental advances have produced rich quantitative measurements of each of these stochastic elements, but the interaction between them remains elusive. In this thesis, we explore the roles of stochasticity in motor-mediated transport with four specific projects at different scales. We first construct a mean-field model of a cargo transported by two teams of opposing motors. This system is known to display bidirectionality: switching between phases of transport in opposite directions. We hypothesize that thermal fluctuations of the cargo drive the switching. From our model, we predict how cargo size influences the switching time, an experimentally measurable quantity to verify the hypothesis. In the second work, we investigate the force dependence of motor stepping, formulated as a state-dependent jump-diffusion model. We prove general results regarding the computation of the statistics of this process. From this framework, we find that thermal fluctuations may provide a nonmonotonic influence on the stepping rate of motors. The remaining projects investigate the behavior of nonprocessive motors, which take few steps before detaching. In collaboration with experimentalists, we study seemingly diffusive data of motor-mediated transport. Using a jump-diffusion model, the active and passive portions of the diffusivity are disentangled, and curious higher order statistics are explained as a sampling issue. Lastly, we construct a model of cooperative transport by nonprocessive motors, which we study using reward-renewal theory. The theory provides predictions about measured quantities such as run length, which suggest that geometric effects have a large influence on the transport ability of these motors