This work develops some simple models to study risky corporate debt using first passage-time approach. Analytical valuation expression derived from different models as functions of firm’s values and the short-term interest rate with time-dependent parameters governing the dynamics of the firm values and interest rate. We develop some numerical approximation of the analytical valuation, which is given implicitly through Voltera integral equation related to the density of the first-passage- time that a firm reaches some specified default barrier. For some appropriate default barrier arising from financial considerations we obtain a closed-form solution, which is more flexible for numerical calculation.

Nucleation is the first step in the formation of a new phase in a thermodynamic system. The Classical Nucleation Theory (CNT) is the traditional theory used to describe this phenomenon. The object of this thesis is to investigate nucleation beyond one of the most significant limitations of the CNT: the assumption that the surface tension of a nucleating cluster of the new phase is independent of the cluster’s size and has the same value that it would have in the bulk of the new phase. In order to accomplish this, we consider a microscopic, two-dimensional Lattice Gas Automata (LGA) model of precipitate nucleation in a supersaturated system, with model input parameters Ess (solid particle-to-solid particle bonding energy), Esw (solid particle-to-water particle bonding energy), η (next-to-nearest neighbour bonding coeffiicent in solid phase), and Cin (initial solute concentration). The LGA method was chosen for its advantages of easy implementation, low memory requirements, and fast computation speed. Analytical results for the system’s concentration and the crystal radius as functions of time are derived and the former is fit to the simulation data in order to determine the system’s equilibrium concentration. A mean first-passage time (MFPT) technique is used to obtain the nucleation rate and critical nucleus size from the simulation data. The nucleation rate and supersaturation are evaluated using a modification to the CNT that incorporates a two-dimensional, radius-dependent surface tension term. The Tolman parameter, δ, which controls the radius-dependence of the surface tension, decreases (increases) as a function of the magnitude of Ess (Esw), at fixed values of η and Esw (Ess). On the other hand, δ increases as η increases while Ess and Esw are held constant. The constant surface tension term of the CNT, Σ0, increases (decreases) with increasing magnitudes of Ess (Esw) fixed values of Esw (Ess), and increases as η is increased. Together, these results indicate an increase in the radius-dependent surface tension, Σ, with respect to increasing magnitude of Ess relative to the magnitude of Esw. Σ0 increases linearly as a function of the change in energy during an attachment or detachment reaction, |ΔE|, however with a slope less than that predicted for a crystal that is uniformly packed at maximum density.

Nucleation is the first step in the formation of a new phase in a thermodynamic system. The Classical Nucleation Theory (CNT) is the traditional theory used to describe this phenomenon. The object of this thesis is to investigate nucleation beyond one of the most significant limitations of the CNT: the assumption that the surface tension of a nucleating cluster of the new phase is independent of the cluster’s size and has the same value that it would have in the bulk of the new phase. In order to accomplish this, we consider a microscopic, two-dimensional Lattice Gas Automata (LGA) model of precipitate nucleation in a supersaturated system, with model input parameters Ess (solid particle-to-solid particle bonding energy), Esw (solid particle-to-water particle bonding energy), η (next-to-nearest neighbour bonding coeffiicent in solid phase), and Cin (initial solute concentration). The LGA method was chosen for its advantages of easy implementation, low memory requirements, and fast computation speed. Analytical results for the system’s concentration and the crystal radius as functions of time are derived and the former is fit to the simulation data in order to determine the system’s equilibrium concentration. A mean first-passage time (MFPT) technique is used to obtain the nucleation rate and critical nucleus size from the simulation data. The nucleation rate and supersaturation are evaluated using a modification to the CNT that incorporates a two-dimensional, radius-dependent surface tension term. The Tolman parameter, δ, which controls the radius-dependence of the surface tension, decreases (increases) as a function of the magnitude of Ess (Esw), at fixed values of η and Esw (Ess). On the other hand, δ increases as η increases while Ess and Esw are held constant. The constant surface tension term of the CNT, Σ0, increases (decreases) with increasing magnitudes of Ess (Esw) fixed values of Esw (Ess), and increases as η is increased. Together, these results indicate an increase in the radius-dependent surface tension, Σ, with respect to increasing magnitude of Ess relative to the magnitude of Esw. Σ0 increases linearly as a function of the change in energy during an attachment or detachment reaction, |ΔE|, however with a slope less than that predicted for a crystal that is uniformly packed at maximum density.

This thesis consists of two novel contributions to the computation of first passage time distribution for Brownian motion. First, we extend the known formula for boundary crossing probabilities for Brownian motion to the discontinuous piecewise linear boundary. Second, we derive explicit formula for the first passage time density of Brownian motion crossing piecewise linear boundary. Further, we demonstrate how to approximate the boundary crossing probabilities and density for general nonlinear boundaries. Moreover, we use Monte Carlo simulation method and develop algorithms for the numerical computation. This method allows one to assess the accuracy of the numerical approximation. Our approach can be further extended to compute two-sided boundary crossing probabilities.

Nucleation is the first step in the formation of a new phase in a thermodynamic system. The Classical Nucleation Theory (CNT) is the traditional theory used to describe this phenomenon. The object of this thesis is to investigate nucleation beyond one of the most significant limitations of the CNT: the assumption that the surface tension of a nucleating cluster of the new phase is independent of the cluster’s size and has the same value that it would have in the bulk of the new phase. In order to accomplish this, we consider a microscopic, two-dimensional Lattice Gas Automata (LGA) model of precipitate nucleation in a supersaturated system, with model input parameters Ess (solid particle-to-solid particle bonding energy), Esw (solid particle-to-water particle bonding energy), η (next-to-nearest neighbour bonding coeffiicent in solid phase), and Cin (initial solute concentration). The LGA method was chosen for its advantages of easy implementation, low memory requirements, and fast computation speed. Analytical results for the system’s concentration and the crystal radius as functions of time are derived and the former is fit to the simulation data in order to determine the system’s equilibrium concentration. A mean first-passage time (MFPT) technique is used to obtain the nucleation rate and critical nucleus size from the simulation data. The nucleation rate and supersaturation are evaluated using a modification to the CNT that incorporates a two-dimensional, radius-dependent surface tension term. The Tolman parameter, δ, which controls the radius-dependence of the surface tension, decreases (increases) as a function of the magnitude of Ess (Esw), at fixed values of η and Esw (Ess). On the other hand, δ increases as η increases while Ess and Esw are held constant. The constant surface tension term of the CNT, Σ0, increases (decreases) with increasing magnitudes of Ess (Esw) fixed values of Esw (Ess), and increases as η is increased. Together, these results indicate an increase in the radius-dependent surface tension, Σ, with respect to increasing magnitude of Ess relative to the magnitude of Esw. Σ0 increases linearly as a function of the change in energy during an attachment or detachment reaction, |ΔE|, however with a slope less than that predicted for a crystal that is uniformly packed at maximum density.

Molecular cell biology has been an intensively studied interdisciplinary field with the rapid development of experimental techniques and fast upgrade of computational hardware and numerical tools. Recent technological developments have led to single-cell experiments which allow us to probe the role of stochasticity in cellular processes. Stochastic modeling of the corresponding processes is thus an essential ingredient for the understanding and interpretation of cellular systems of interest. In this thesis, we explore several nucleation-based stochastic cellular processes, i.e. Min protein oscillation in Escherichia coli, pausing phenomena in DNA transcription, and single-molecule enzyme kinetics. We focus on the key experimental results and build up stochastic models accordingly to provide quantitative insights to the underlying physical mechanisms for the corresponding biological processes. We utilize specific mathematical methods and computational algorithms to gain a better understanding and make predictions for further experimental explorations in the relevant fields. / Ph. D.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

The heterogeneous condensation of a vapour onto a substrate is a key step in a wide range of chemical and physical process that occur in both nature and technology. For example, dust and pollutant aerosol particles, ranging in size from several microns down to just a few nanometers, serve as cloud condensation nuclei in the atmosphere, and nanoscale structured surfaces provide templates for the controlled nucleation and growth of variety of complex materials. While much is known about the general features of heterogeneous nucleation onto macroscopic surfaces, much less is understood about both the dynamics and thermodynamics of nucleation involving nanoscale heterogeneities. The goal of this thesis is to understand the general features of condensation of vapours onto different types of nanoscale heterogeneity that range in degree of solubility from being insoluble, to partially miscible through to completely miscible. The heterogeneous condensation of the Lennard-Jones vapour onto an insoluble nanoscale seed particle is studied using a combination of molecular dynamics simulations and thermodynamic theory. The nucleation rate and free energy barrier are calculated from molecular dynamics using the mean first passage time method. These results show that the presence of a weakly interacting seed has no effect on the formation of small cluster embryos but accelerates the rate by lowering the free energy barrier of the larger clusters. A simple phenomenological model of film formation on a small seed is developed by extending the capillarity based liquid drop model. It captures the general features of heterogeneous nucleation, but a comparison with the simulation results show that the model significantly overestimates the height of the nucleation barrier while providing good estimates of the critical film size. A non-volatile liquid drop model that accounts for solution non-ideality is developed to describe the thermodynamics of partially miscible and fully miscible droplets in a solvent vapour. The model shows ideal solution drops dissolve always spontaneously, but partially miscible drops exhibit a free energy surface with two minima, associated with a partially dissolved drop and a fully dissolved drop, separated by a free energy barrier. The solubility transition between the two drops is shown to follow a hysteresis loop as a function of system volume similar to that observed in deliquescence. A simple lattice gas model describing the absorption of mono-layers of vapour onto the particle is also developed. Finally, molecular dynamics simulation of miscible and partially miscible binary Lennard-Jones mixtures are also used to study this system. For all cases studied, condensation onto the drop occurs spontaneously. Sub-monolayers of the solvent phase form when the system volume is large. At smaller system volumes, complete film formation is observed and the dynamics of film growth are dominated by cluster-cluster coalescence. Some degree of mixing into the core of the particle is observed for the miscible mixtures for all volumes. However, mixing of the solvent into the particle core only occurs below an onset volume for the partially miscible case, suggesting the presence of a solubility transition similar to the one described by the thermodynamic model.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.