Pure Diffusion Process Research Articles

Estimating Market Leverage: Maximum Likelihood and Kalman Filter

Market leverage is one of the main determinants for defining the optimal capital structure of the firm, and has a significant impact on several variables that affect the design of the firm's business strategy. This paper is focussed on the potential methodologies exploitable to obtain reliable estimates of the firm's market leverage. This is a relevant issue as many factors make obscure the actual dynamics of the firm's leverage. We estimate the unknown parameters that determine the stochastic properties of the unobservable variable, by using maximum likelihood estimation and the Kalman filter approach. The starting point of our analysis is the seminal Merton model (1974), which describes the firm's asset value dynamics with a pure diffusion process, improved with the early default risk as proposed by Duan et al. (2003). We report the simulation results for different possible estimation methods, and then perform an empirical analysis on real data of 25 European companies CDS premia.

Minimizing the risk of absolute ruin under a diffusion approximation model with reinsurance and investment

This paper studies the optimization problem with both investment and proportional reinsurance control under the assumption that the surplus process of an insurance entity is represented by a pure diffusion process. The company can buy proportional reinsurance and invest its surplus into a Black-Scholes risky asset and a risk free asset without restrictions. The authors define absolute ruin as that the liminf of the surplus process is negative infinity and propose absolute ruin minimization as the optimization scenario. Applying the HJB method the authors obtain explicit expressions for the minimal absolute ruin function and the associated optimal investment strategy. The authors find that the minimal absolute ruin function here is convex, but not S-shaped investigated by Luo and Taksar (2011). And finally, from behavioral finance point of view, the authors come to the conclusion: It is the restrictions on investment that results in the kink of minimal absolute ruin function.

International Stochastic Discount Factors and Stochastic Correlation

We consider an analytical framework that aims at investigating the impact of stochastic covariance and thus stochastic correlation between international economies. Using a Wishart pure diffusion process, we model the covariance matrix of country-specific stochastic discount factors (SDF) to incorporate stochastic volatility and unspanned stochastic covariance. We infer the dynamic structure of the SDFs and the time series of latent state variables by embedding a nonlinear filter in a quasi maximum likelihood estimation approach. Our findings suggest that at the outbreak of the financial crisis in the second half of the year 2008 stochastic discount factors face a sharp decrease in correlation associated with high levels of exchange rate volatility. After a short adaption period, local correlations stabilize at high positive levels. This indicates vanishing foreign currency risk premia which is in line with the breakdown of the carry trade phenomenon.

Slip velocity of rigid fibers in turbulent channel flow

In this study, the slip velocity between rigid fibers and a viscous carrier fluid is investigated for the reference case of turbulent channel flow. The statistical moments of the slip velocity are evaluated modelling fibers as prolate spheroids with Stokes number, St, ranging from 1 to 100 and aspect ratio, λ, ranging from 3 to 50. Statistics are compared one-to-one with those obtained for spherical particles (λ = 1) to highlight effects due to fiber elongation. Comparison is also made at different Reynolds numbers (Reτ =150, 180, and 300 based on the fluid shear velocity) to discuss effects due to an increase of turbulent fluctuations. Results show that elongation has a quantitative effect on slip velocity statistics, particularly evident for fibers with small St. As St increases, differences due to the aspect ratio tend to vanish and the relative translational motion between individual fibers and surrounding fluid is controlled by fiber inertia through preferential concentration. A clear manifestation of inertial effects is the different distribution of slip velocities for fibers trapped in sweep/ejection events and for fibers segregated in near-wall fluid streaks. The corresponding conditional probability distribution functions, shown here for the streamwise and wall-normal slip velocity components, are found to be non-Gaussian, thus suggesting that fiber motion relative to the fluid in high-shear flow regions may not be modelled as a pure diffusion process with constant diffusion coefficient. For the range of simulation parameters investigated, no significant Reynolds number effects are observed, indicating that fiber dynamics exhibit a scaling behavior with respect to the shear velocity up to Reτ =300.

Understanding the evaporation of spherical drops in quiescent environment

Evaporation of suspended micro-liter drops in room temperature, atmospheric conditions and in quiescent air is not controlled by liquid transport across the drop interface (phase change) but by vapor transport from the drop surface to the surrounding. The vapor transport can be decomposed into convection and diffusion. Literature assumes the convection is negligible, also the evaporation process is assumed as quasi-steady (Maxwell assumptions). Although, the Maxwellian models are relatively accurate, the validity of the assumptions was never investigated independently. In this paper, it was shown that relaxing the steady-state assumption alone results in 60% error in predicting the evaporation rate. Also, using the Péclet number concept, it was found the evaporation is not purely diffusive. This means that evaporation is not a steady-state nor pure-diffusive process. Relaxing both Maxwell assumptions together, a transient and diffusive-convective (TDC) model was developed. It was found that only for liquids with small vapor concentration values (e.g. water), the TDC and Maxwell based models provide similar results (likely due to fortuitous cancelling of transient and convective effects for such liquids). For liquids with large vapor concentration values the TDC should be used.

JOVIAN STRATOSPHERE AS A CHEMICAL TRANSPORT SYSTEM: BENCHMARK ANALYTICAL SOLUTIONS

We systematically investigated the solvable analytical benchmark cases in both one- and two-dimensional (1D and 2D) chemical-advective-diffusive systems. We use the stratosphere of Jupiter as an example but the results can be applied to other planetary atmospheres and exoplanetary atmospheres. In the 1D system, we show that CH_4 and C_2H_6 are mainly in diffusive equilibrium, and the C_2H_2 profile can be approximated by modified Bessel functions. In the 2D system in the meridional plane, analytical solutions for two typical circulation patterns are derived. Simple tracer transport modeling demonstrates that the distribution of a short-lived species (such as C_2H_2) is dominated by the local chemical sources and sinks, while that of a long-lived species (such as C_2H_6) is significantly influenced by the circulation pattern. We find that an equator-to-pole circulation could qualitatively explain the Cassini observations, but a pure diffusive transport process could not. For slowly rotating planets like the close-in extrasolar planets, the interaction between the advection by the zonal wind and chemistry might cause a phase lag between the final tracer distribution and the original source distribution. The numerical simulation results from the 2D Caltech/JPL chemistry-transport model agree well with the analytical solutions for various cases.

A Family of Density Expansions for Lévy-Type Processes with Default

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asymptotic series. Additionally, for pure diffusion processes, we derive an asymptotic expansion for the implied volatility induced by European calls/puts. Our method is numerically efficient; approximate transition densities and European option prices are computed via Fourier transforms; approximate bond prices are computed as finite series. Additionally, as in Pagliarani et al. (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Numerical examples confirming the effectiveness and versatility of our method are provided, as is sample Mathematica code.

Direct Numerical Simulation of the turbulent wake behind a heated cylinder

Direct Numerical Simulations were conducted to describe a well-known and widely studied configuration, i.e. flow field development downstream from a cylinder under the mixed convection regime, which has too rarely been considered. The Richardson number studied was equal to 2.77 and the Reynolds number equal to 1000; under such conditions, thermal instability development along the cylinder was found to interact with and pronouncedly disturb upper shear-layer development. Whole flow behavior in the back of the cylinder is consequently asymmetric. The resulting flow corresponds to complex features ranging from Kelvin–Helmholtz instabilities to pure buoyant diffusion process and Von Karman alley, the latter being significantly deviated upward.

Conditional Jumps in Volatility and Their Economic Determinants

The volatility of financial returns is affected by rapid and large increments. Such movements can be hardly generated by a pure diffusive process for stochastic volatility. On the contrary jumps in volatility are important because they allow for rapid increases, like those observed during stock market crashes. We propose an extension of HAR model for estimating the presence of jumps in volatility, using the realized-range measure as a volatility proxy. By focusing on a set of 36 NYSE stocks, we show that, once that squared jumps in prices are disentangled from integrated variance, then there is a positive probability of jumps in volatility, conditional on the past information set. We then focus on the contribution of jumps during periods of financial turmoil. We analyze the dependence between the first principal component of the volatility jumps with a set of financial covariates including VIX, S&P500 volume, CDS, and Federal Fund rates. We observe that CDS captures large part of the expected jumps moves, verifying the common interpretation that large and sudden increases in volatility in stock markets over some days in the recent financial crisis have been caused by credit deterioration of US bank sector. Finally, we extend the model incorporating the credit-default swap in the dynamics of the jump size and intensity. The estimates confirm the significant contribution of the credit-default swap to the dynamics of the volatility jump size.

Dynamic nuclear polarization assisted spin diffusion for the solid effect case

The dynamic nuclear polarization (DNP) process in solids depends on the magnitudes of hyperfine interactions between unpaired electrons and their neighboring (core) nuclei, and on the dipole-dipole interactions between all nuclei in the sample. The polarization enhancement of the bulk nuclei has been typically described in terms of a hyperfine-assisted polarization of a core nucleus by microwave irradiation followed by a dipolar-assisted spin diffusion process in the core-bulk nuclear system. This work presents a theoretical approach for the study of this combined process using a density matrix formalism. In particular, solid effect DNP on a single electron coupled to a nuclear spin system is considered, taking into account the interactions between the spins as well as the main relaxation mechanisms introduced via the electron, nuclear, and cross-relaxation rates. The basic principles of the DNP-assisted spin diffusion mechanism, polarizing the bulk nuclei, are presented, and it is shown that the polarization of the core nuclei and the spin diffusion process should not be treated separately. To emphasize this observation the coherent mechanism driving the pure spin diffusion process is also discussed. In order to demonstrate the effects of the interactions and relaxation mechanisms on the enhancement of the nuclear polarization, model systems of up to ten spins are considered and polarization buildup curves are simulated. A linear chain of spins consisting of a single electron coupled to a core nucleus, which in turn is dipolar coupled to a chain of bulk nuclei, is considered. The interaction and relaxation parameters of this model system were chosen in a way to enable a critical analysis of the polarization enhancement of all nuclei, and are not far from the values of (13)C nuclei in frozen (glassy) organic solutions containing radicals, typically used in DNP at high fields. Results from the simulations are shown, demonstrating the complex dependences of the DNP-assisted spin diffusion process on variations of the relevant parameters. In particular, the effect of the spin lattice relaxation times on the polarization buildup times and the resulting end polarization are discussed, and the quenching of the polarizations by the hyperfine interaction is demonstrated.

Are there contagion effects in information technology and business process outsourcing?

We model the diffusion of IT outsourcing using announcements about IT outsourcing deals. We estimate a lognormal diffusion curve to test whether IT outsourcing follows a pure diffusion process or there are contagion effects involved. The methodology permits us to study the consequences of outsourcing events, especially mega-deals with IT contract amounts that exceed US$1 billion. Mega-deals act, we theorize, as precipitating events that create a strong basis for contagion effects and are likely to affect decision-making by other firms in an industry. Then, we evaluate the role of different communication channels in the diffusion process of IT outsourcing by testing for the fit of the mixed influence model at the industry level. This helps us to evaluate the consistency of evidence at two different levels of analysis. We also evaluate two flexible diffusion models: the Gompertz and Weibull models. Our results show that the diffusion patterns of IT outsourcing do not appear to be lognormal, suggesting that IT outsourcing does not follow a pure diffusion process. Instead, we find the presence of contagion effects in the diffusion of IT outsourcing. During periods of the most rapid outsourcing growth – the contagion periods – the actions of the large and more visible firms may provide exemplars for smaller firms, reducing their inhibitions about committing to IT outsourcing. We also find that the results of the mixed influence and the Weibull models, which provide the best fit for overall IT outsourcing diffusion patterns, are potentially indicative of the existence of spillovers that might drive the observed contagion effects at the industry level.

Simulations of Metal Oxidation in Lead Bismuth Eutectic at a Mesoscopic Level

The corrosiveness of lead bismuth eutectic (LBE), as an ideal coolant candidate in reactors and accelerator driven systems (ADSs), presents a critical challenge for safe applications. One of the effective ways to protect the materials is to form and maintain a protective oxide film along the structural material surfaces by active oxygen control technology. The oxidation of metals in LBE environment has been investigated numerically at a mesoscopic scale. A novel stochastic cellular automaton (CA) model has been proposed considering the transport of oxygen along the grain boundaries. The proposed mesoscopic CA model has been mapped with the experimental data. A parametric study was conducted in order to check the importance of the main explicit parameters of the mesoscopic model. The boundary condition at the far end of the specimen has been investigated for the CA model. The model has benchmarked with the analytical solution and with the previous work of a pure diffusion process, and significant agreement has been reached. The developed CA model can be used to solve diffusion problem.

BARRIERS TO THE SPREAD OF NEUTRAL ALLELES IN THE CYTONUCLEAR SYSTEM

Neutral alleles can eventually pass a hybrid zone and their initial clines generated by a pure diffusion process dissipate with time, irrespective of the presence or absence of physical barriers. However, the transient neutral clines at the nuclear or organelle sites can be reinforced by the cytonuclear disequilibrium generated by diploid seed and haploid pollen dispersal. In this study, the spread of a neutral allele in an ecological zone of hermaphrodite plants is examined under three cytonuclear systems for genomes with contrasting modes of inheritance (paternal, maternal, and biparental inheritance). The results show that the transient neutral clines can exhibit the spatial pattern similar to the selective clines from separate genomes although discordance between them exists. The spread of a neutral allele is not only related to the vectors of seed and pollen dispersal but also to the mode of its inheritance. Pollen dispersal facilitates the direct effects of the selective organelle sites with paternal inheritance on the spread of a neutral nuclear allele. It also enhances its indirect effects on the spread of a neutral organelle allele with maternal inheritance via modifying the cytonuclear disequilibrium. A positive relationship exists between the barriers to the spread of selective nuclear (or organelle) and neutral organelle (or nuclear) alleles. An asymmetric barrier to the spread of the neutral alleles exists on the two sides of the physical barrier, given the presence of symmetric barrier to the spread of the selective alleles. These theoretical predictions highlight the effects of cytonuclear disequilibrium on the spread of a neutral allele and draw attention to our empirical cline analysis with neutral markers.

Term structure of sovereign spreads: a contingent claim model

This paper proposes a simple structural model to estimate the termstructure and the implied default probability of a selected group of emerging countries, which account for 54% of the JPMorgan EMBIG index on average for the period 2000-2005. The real exchange rate dynamic, modeled as a pure diffusion process, is assumed to trigger default. The calibrated model generates sovereign spread curves consistent to market data. The results suggest that the market is systematically overpricing spreads for Brazil in 100 basis points, whereas for Mexico, Russia and Turkey the model is able to reproduce the market behavior.

Stochastic pension fund control in the presence of Poisson jumps

We consider the problem of optimal funding and asset allocation for a defined benefit pension scheme by assuming that the pension fund can be invested in a risk-free asset and a risky asset whose return follows a jump diffusion process. We extend existing literature which mainly assumes that the risky asset’s return follows a pure diffusion process. In a stochastic analysis of the optimal policies we show that the optimal contribution and asset allocation policies have similar forms as in the pure diffusion approaches, but with a modification for the effect of jumps. These results hold under both constant pension scheme benefit outgo and stochastic pension scheme benefit outgo. Using a sensitivity analysis of the effect of the mean jump magnitude on the asset allocation policy, we show that increasing (in absolute terms) the mean jump magnitude reduces the allocation in the risky asset and increases the allocation in the risk-free asset.

FRACTIONAL CALCULUS AND THE SCHRÖDINGER EQUATION

In this paper, a derivation of the fractional Schrödinger equation is presented for the simple case of a pure diffusive process with dissipation. The Gaussian white noise is replaced by more general kinds of white noise and both the Markovian both the Markovian (β = 1) and non-Markovian case (0 < β < 1) are considered.

The Term Structure of Sovereign Spreads in Emerging Markets

A simple structural model to estimate the term structure of sovereign spreads and the implied default probability of a selected group of emerging countries can account for more than 50% of the J.P. Morgan EMBIG index. The real exchange rate dynamics, modeled as a pure diffusion process, are assumed to trigger default events. Relaxing the hypothesis of market completeness, the model generates sovereign spread curves consistent with market data. The robustness of the model indicates a need to reexamine the criticism that structural models underestimate market spreads. The results suggest that the market tends to overprice the spreads for Brazil by 120 basis points on average, but it reproduces the market behavior for Mexico, Russia, and Turkey. These findings are supported by credit rating agency upgrades of the ratings for Brazil in the second half of 2004, just after the study period.

A Lévy process-based framework for the fair valuation of participating life insurance contracts

In this communication, we develop suitable valuation techniques for a with-profit/unitized with profit life insurance policy providing interest rate guarantees, when a jump-diffusion process for the evolution of the underlying reference portfolio is used. Particular attention is given to the mispricing generated by the misspecification of a jump-diffusion process for the underlying asset as a pure diffusion process, and to which extent this mispricing affects the profitability and the solvency of the life insurance company issuing these contracts.

An approximate solution for a fractional diffusion-wave equation using the decomposition method

The partial differential equation of diffusion is generalized by replacing the first order time derivative by a fractional derivative of order α, 0 < α ⩽ 2. An approximate solution based on the decomposition method is given for the generalized fractional diffusion (diffusion-wave) equation. The fractional derivative is described in the Caputo sense. Numerical example is given to show the application of the present technique. Results show the transition from a pure diffusion process ( α = 1) to a pure wave process ( α = 2).

Option pricing with Weyl–Titchmarsh theory

In the Black–Merton–Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schrödinger spectral problem, which has been well studied in quantum mechanics. With Weyl–Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.