Two-dimensional Diffusion Processes Research Articles

Two-curve Green’s function for 2-SLE: the boundary case

We prove that for $\kappa\in(0,8)$, if $(\eta_1,\eta_2)$ is a $2$-SLE$_\kappa$ pair in a simply connected domain $D$ with an analytic boundary point $z_0$, then $\lim_{r\to 0^+}r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\eta_j)<r,j=1,2]$ converges to a positive number for some $\alpha>0$, which is called the two-curve Green's function. The exponent $\alpha$ equals $\frac{12}{\kappa}-1$ or $2(\frac{12}{\kappa}-1)$ depending on whether $z_0$ is one of the endpoints of $\eta_1$ and $\eta_2$. We also find the convergence rate and the exact formula of the Green's function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.

Markov chain approximation and measure change for time-inhomogeneous stochastic processes

In this paper, we propose a general time-inhomogeneous continuous-time Markov chain (CTMC) framework for the approximation of the general one-dimensional and two-dimensional time-inhomogeneous diffusion processes. For the approximating CTMC, we can perform a change of measure, choose the minimal relative entropy measure to determine the measure uniquely, and finally establish the convergence. Therefore, the proposed methodology covers the stochastic processes that are hard to perform a change of measure, and is applicable to valuation problems driven by models not only under the risk-neutral probability measure but also under the physical probability measure.

Density symmetries for a class of 2-D diffusions with applications to finance

We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models.

Large and moderate deviations of realized covolatility

In this note, we consider the large and moderate deviation principle of the estimators of the integrated covariance of two-dimensional diffusion processes when they are observed only at discrete times in a synchronous manner. The proof is extremely simple. It is essentially an application of the contraction principle for the results given in the case of the volatility.

Two-dimensional diffusion processes as models in lifetime studies

Let X(t) denote the remaining useful lifetime of a machine, and Y(t) be a standard Brownian motion. Assume that the derivative ρ[X(t), Y(t)] of X(t) is a deterministic function of (at least) Y(t). We consider the two-dimensional degenerate diffusion process (X(t), Y(t)). We obtain explicit expressions for the expected value of the random variable T(x, y) denoting the first time the machine must be replaced, or repaired, for various functions ρ[X(t), Y(t)].

Similarity Solutions of Partial Differential Equations in Probability

Two‐dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.

Mean First-Passage Time to Zero for Wear Processes

To model the remaining lifetime X(t) of machines, two-dimensional degenerate diffusion processes (X(t), Y(t)) are considered. The process Y(t) is assumed to be a geometric Brownian motion, while the derivative of X(t) is a negative deterministic function of X(t) and Y(t). Hence, the process X(t) is decreasing. We calculate explicitly for a class of processes the expected value of the random variable that denotes the time it takes X(t) to reach c, given that X(0) = x > c and Y(0) = y > 0. The method of similarity solutions is used to solve the appropriate Kolmogorov backward equation.

Dehydration kinetics of muscovite by in situ infrared microspectroscopy

Dehydration behavior of muscovite flake was investigated at 760–860°C by using in situ high-temperature IR microspectroscopy for the OH absorption band around 3,620 cm−1. Isothermal kinetic heating experiments at each temperature gave detailed decrease curves of the OH band area with time. These curves have been simulated by the first and second order reactions or mono- and two-dimensional diffusion processes. The mono-dimensional diffusion was found to give the best fit to the experimental data and apparent diffusion coefficients D were determined at 760–860°C with the activation energy of 290 ± 20 kJ/mol. The apparent diffusion coefficients D varied with the sample thickness L. This variation can be explained by an m layers model with a unit length of L′ with a constant diffusion coefficient D′. Therefore, the dehydration process might be rate-limited by mono-dimensional diffusion through tetrahedral silicate sheet perpendicular to (001) planes of muscovite with a unit length of L′.

Number of arm selection in two-dimensional diffusion processes

We introduce an algorithm to generate two-dimensional diffusion-limited star-branched aggregates (DLSA) attaching bi-functional monomers successively to a central colloidal particle with any desired number of reactive sites. The proposed algorithm produces star-shaped aggregates that grow forever and show a power law polydispersity in the chemical length of the arms near the central colloid. More interestingly, it gives rise to a number of arm selection consisting in that only a small number of arms (around five) define the final structure at relatively large distances from the central colloid, independently of the initial number of reactive sites and the size of the central colloid. We characterize the structure of the aggregates by means of the particle-particle correlation function, analyze its scaling properties and obtain the fractal dimension.

Bounded perturbations of two-dimensional diffusion processes with nonlocal conditions near the boundary

We study the existence of Feller semigroups arising in the theory of multidimensional diffusion processes. We study bounded perturbations of elliptic operators with boundary conditions containing an integral over the closure of the domain with respect to a nonnegative Borel measure without assuming that the measure is small. We state sufficient conditions on the measure guaranteeing that the corresponding nonlocal operator is the generator of a Feller semigroup.

Unbounded perturbations of two-dimensional diffusion processes with nonlocal boundary conditions

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. Unbounded perturbations of elliptic operators (in particular, integro-differential operators) are considered in plane bounded regions. Their domain of definition is given by a nonlocal boundary condition involving the integral over the closure of the region with respect to a nonnegative Borel measure. The support of the measure may intersect with the boundary, and the measure need not be small. We formulate sufficient conditions on unbounded perturbations of elliptic operator and on the Borel measure in the nonlocal boundary condition which guarantee that the corresponding nonlocal operator is a generator of a Feller semigroup.

Volatility: A hidden Markov process in financial time series

Volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, volatility is unobservable and only the price is known. Diffusion theory has many common points with the research on volatility, the key of the analogy being that volatility is a time-dependent diffusion coefficient of the random walk for the price return. We present a formal procedure to extract volatility from price data by assuming that it is described by a hidden Markov process which together with the price forms a two-dimensional diffusion process. We derive a maximum-likelihood estimate of the volatility path valid for a wide class of two-dimensional diffusion processes. The choice of the exponential Ornstein-Uhlenbeck (expOU) stochastic volatility model performs remarkably well in inferring the hidden state of volatility. The formalism is applied to the Dow Jones index. The main results are that (i) the distribution of estimated volatility is lognormal, which is consistent with the expOU model, (ii) the estimated volatility is related to trading volume by a power law of the form sigma proportional, variant V0.55, and (iii) future returns are proportional to the current volatility, which suggests some degree of predictability for the size of future returns.

Solvable local and stochastic volatility models: supersymmetric methods in option pricing

In this paper we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black–Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in a paper by Albanese et al. (Albanese, C., Campolieti, G., Carr, P. and Lipton, A., Black–Scholes goes hypergeometric. Risk Mag., 2001, 14, 99–103). For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3 / 2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.

Volatility: A Hidden Markov Process in Financial Time Series

The volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, the volatility is unobservable and only the price is known. Diffusion theory has many common points with the research on volatility, the key of the analogy being that volatility is the time-dependent diffusion coefficient of the random walk for the price return. We present a formal procedure to extract volatility from price data, by assuming that it is described by a hidden Markov process which together with the price form a two-dimensional diffusion process. We derive a maximum likelihood estimate valid for a wide class of two-dimensional diffusion processes. The choice of the exponential Ornstein-Uhlenbeck (expOU) stochastic volatility model performs remarkably well in inferring the hidden state of volatility. The formalism is applied to the Dow Jones index. The main results are: (i) the distribution of estimated volatility is lognormal, which is consistent with the expOU model; (ii) the estimated volatility is related to trading volume by a power law; and (iii) future returns are proportional to the current volatility which suggests some degree of predictability for the size of future returns.

Modeling the Role of Incisures in Vertebrate Phototransduction

Phototransduction is mediated by a G-protein-coupled receptor-mediated cascade, activated by light and localized to rod outer segment (ROS) disk membranes, which, in turn, drives a diffusion process of the second messengers cGMP and Ca 2+ in the ROS cytosol. This process is hindered by disks—which, however, bear physical cracks, known as incisures, believed to favor the longitudinal diffusion of cGMP and Ca 2+. This article is aimed at highlighting the biophysical functional role and significance of incisures, and their effect on the local and global response of the photocurrent. Previous work on this topic regarded the ROS as well stirred in the radial variables, lumped the diffusion mechanism on the longitudinal axis of the ROS, and replaced the cytosolic diffusion coefficients by effective ones, accounting for incisures through their total patent area only. The fully spatially resolved model recently published by our group is a natural tool to take into account other significant details of incisures, including their geometry and distribution. Using mathematical theories of homogenization and concentrated capacity, it is shown here that the complex diffusion process undergone by the second messengers cGMP and Ca 2+ in the ROS bearing incisures can be modeled by a family of two-dimensional diffusion processes on the ROS cross sections, glued together by other two-dimensional diffusion processes, accounting for diffusion in the ROS outer shell and in the bladelike regions comprised by the stack of incisures. Based on this mathematical model, a code has been written, capable of incorporating an arbitrary number of incisures and activation sites, with any given arbitrary distribution within the ROS. The code is aimed at being an operational tool to perform numerical experiments of phototransduction, in rods with incisures of different geometry and structure, under a wide spectrum of operating conditions. The simulation results show that incisures have a dual biophysical function. On the one hand, since incisures line up from disk to disk, they create vertical cytoplasmic channels crossing the disks, thus facilitating diffusion of second messengers; on the other hand, at least in those species bearing multiple incisures, they divide the disks into lobes like the petals of a flower, thus confining the diffusion of activated phosphodiesterase and localizing the photon response. Accordingly, not only the total area of incisures, but their geometrical shape and distribution as well, significantly influence the global photoresponse.

Solvable Local and Stochastic Volatility Models: Supersymmetric Methods in Option Pricing

In this paper, we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3/2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.

First-passage problems for degenerate two-dimensional diffusion processes

LetY(t) be a one-dimensional diffusion process anddX(t)=Y(t)dt. The process (X(t), Y(t)) is considered in the second quadrant. First, the probability that (X(t), Y(t)) will hit thex-axis before they-axis is computed explicitly whenY(t) is a standard Brownian motion or a particular case of the Bessel process. Next, an exact expression is obtained for the average time it takes (X(t), Y(t)) to exit the regionC={(x, y) ∈ ℝ2 :x < 0,0 <d 1 <y <d 2 whenY(t) is a standard Brownian motion. The solution is expressed as a generalized Fourier series.

Exact solutions to two-dimensional homing problems

Two-dimensional controlled Wiener and Bessel diffusion processes are considered in circles. Both the infinitesimal means and variances of the controlled processes depend on the control variables. The processes are controlled until they hit the circles for the first time. The objective is to minimize the expected value of the time spent by the controlled processes inside the circles, while taking the control costs into account. Explicit and exact expressions are obtained for the optimal values of the control variables as well as for the value functions. The method of similarity solutions is used to solve the appropriate dynamic programming equation.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.