General Diffusion Processes Research Articles

Localized upper bounds of heat kernels for diffusions via a multiple Dynkin-Hunt formula

We prove that for a general diffusion process, certain assumptions on its behavior only within a fixed open subset of the state space imply the existence and sub- Gaussian type off-diagonal upper bounds of the global heat kernel on the fixed open set. The proof is mostly probabilistic and is based on a seemingly new formula, which we call a multiple Dynkin-Hunt formula, expressing the transition function of a Hunt process in terms of that of the part process on a given open subset. This result has an application to heat kernel analysis for the Liouville Brownian motion, the canonical diffusion in a certain random geometry of the plane induced by a (massive) Gaussian free field.

Exact Computation of Velocity Field and Probability Flux of Time-Evolving Probability Landscape of Stochastic Networks

Stochastic networks play important roles in cellular processes such as gene regulation, maintenance of epigenetic states, and decision of cellular fate. The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in cellular mesoscopic systems. The time-evolving probabilistic landscape governed by dCME gives a comprehensive picture of the stochastic behavior of the network. Probability flux and entropy production of stochastic network have been previously studied using Fokker-Plank equation based on Gaussian processes. Here we formulate the time-dependent evolution of the probability landscape as a general diffusion process [1], and construct the velocity field and the probability flux based on exactly computed time-evolving probabilistic landscape. We dispense with Gaussian approximations and the truncation of the jump operator necessary when using the Fokker-Plank equation. We computed the dynamics of probability fluxes and vector fields over the full state space across different time scale. We showed details of the time-evolving probability fluxes and vector fields for the ubiquitous birth-death process, the bistable Schlogl process, the oscillatory Schnakenberg model, and the toggle switch model at different time points using the exact ACME algorithm [2]. Our results identify important cellular states, characterize high probability paths and trajectories along which the probability mass of the network state evolve. Furthermore, we provide exact assessment of dissipative entropy production associated with the complex diffusion process.[1] Eisenberg, Bob et al. The Journal of Chemical Physics 133.10 (2010): 104104.[2] Cao, Youfang et al. BMC Systems Biology 2.1 (2008): 30.

Thermodynamics of the general diffusion process: Equilibrium supercurrent and nonequilibrium driven circulation with dissipation

Unbalanced probability circulation, which yields cyclic motions in phase space, is the defining characteristics of a stationary diffusion process without detailed balance. In over-damped soft matter systems, such behavior is a hallmark of the presence of a sustained external driving force accompanied with dissipations. In an under-damped and strongly correlated system, however, cyclic motions are often the consequences of a conservative dynamics. In the present paper, we give a novel interpretation of a class of diffusion processes with stationary circulation in terms of a Maxwell-Boltzmann equilibrium in which cyclic motions are on the level set of stationary probability density function thus non-dissipative, e.g., a supercurrent. This implies an orthogonality between stationary circulation $J^{ss}(x)$ and the gradient of stationary probability density $f^{ss}(x)>0$. A sufficient and necessary condition for the orthogonality is a decomposition of the drift $b(x)=j(x)+ D(x)\nabla\varphi(x)$ where $\nabla\cdot j(x)=0$ and $j(x)$ $\cdot\nabla\varphi(x)=0$. Stationary processes with Maxwell-Boltzmann equilibrium has an underlying conservative dynamics $\dot{x}= j(x)\equiv$ $\big(f^{ss}(x)\big)^{-1}J^{ss}(x)$, and a first integral $\varphi(x)\equiv-\ln f^{ss}(x)=$ const, akin to a Hamiltonian system. At all time, an instantaneous free energy balance equation exists for a given diffusion system; and an extended energy conservation law among a family of diffusion processes with different parameter $\alpha$ can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven $\varphi$-ascending steps and spontaneous $\varphi$-descending movements, alternated with iso-$\varphi$ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics.

Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity

We study generalized anomalous diffusion processes whose diffusion coefficient D(x, t) ∼ D0|x|αtβ depends on both the position x of the test particle and the process time t. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalized diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalized diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.

Explicit form of approximate transition probability density functions of diffusion processes

A continuous-time diffusion process is very popular in modeling and provides useful tools to analyze particularly, but not restricted to, a variety of economic and financial variables. The transition probability density function (TPDF) of a diffusion process plays an important role in understanding and explaining the dynamics of the process. A new way to find closed-form approximate TPDFs for multivariate diffusions is proposed in this paper. This method can be applied to general multivariate time-inhomogeneous diffusion processes, as long as, roughly speaking, they have smooth drift and volatility functions. A diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix. We have established how to obtain the approximate TPDF of a reducible diffusion explicitly. When a diffusion process is not reducible, an explicit form of approximate TPDF can be obtained by using the results in Aït-Sahalia (2008) and Choi (2013). The TPDF expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate TPDF for a multivariate jump diffusion. Monte Carlo simulation studies of conducting maximum likelihood estimation (MLE) using our approximations provide convincing evidence that our TPDF expansion can be used for the MLE when the true TPDF is unavailable. We also applied our approximate TPDFs to option pricing. The differences between our option prices and those from the Extended Black–Scholes formula are shown to be quite small. This implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models.

Multi-Market, Multi-Product New Product Diffusion: Decomposing Local, Foreign, and Indirect (Cross-Product) Effects

In this study, we address an important issue largely ignored in existing diffusion research—the simultaneous diffusion of related (here, complementary) products across multiple interacting countries. In doing so, we demonstrate that incorporating prior diffusion of complementary products in an international framework leads to an enhanced substantive understanding of the evolution of cross-country diffusion. The limited prior research on cross-product interactions has focused exclusively on a single country. We extend this research by building a more complete view of the role that prior diffusion of two interacting technologies play both within as well as across countries. Specifically, we decompose—on a country-by-country basis—the impact of three factors on the diffusion of any product: (a) prior diffusion of the product within each country, (b) prior diffusion of the same product in other counties, and (c) prior diffusion of a related product. This decomposition leads to a number of important strategic insights. We estimate and graph the three effects over time for each product and country using a comprehensive data set that covers the diffusion of PCs and the Internet over two decades—from 1981 to 2009—and across 19 countries. There are a number of interesting findings. First, we find that home PC diffusion was driven predominantly by local effects—the more individuals saw the penetration of home PCs grow locally, the greater the likelihood of adoption. Second, we find very different effects for the Internet—Internet adoption was driven by a combination of influences: (a) local network effects, (b) foreign network effects, and (c) cross-product effects. These results suggest that diffusion of one product can facilitate the diffusion of another product and that the impact can be asymmetric across products. When taken in aggregate, these results highlight the importance of incorporating and estimating cross-product effects in a multi-market new product diffusion context—one is able to obtain a more complete view of the impact of strategic decisions within a general diffusion process in markets that develop and evolve dynamically over time.

Theoretical study of diffusion processes around a non-rotating neutron star

The general relativistic diffusion process on curved space-time manifold around a non-rotating neutron star has been analyzed. The general relativistic diffusion equation of diffusive particles around non-rotating neutron star is derived by constructing phase space in the parametrization of observer time in the hyperbolic coordinate system. This diffusion equation describes the stochastic dynamic of particles around non-rotating neutron stars. In this work we also have studied the diffusion processes around a non-rotating neutron star for asymptotic case.

Dynamic preferences for popular investment strategies in pension funds

In this paper, we characterize dynamic investment strategies that are consistent with the expected utility setting and more generally with the forward utility setting. Two popular dynamic strategies in the pension funds industry are used to illustrate our results: a constant proportion portfolio insurance (CPPI) strategy and a life-cycle strategy. For the CPPI strategy, we are able to infer preferences of the pension fund’s manager from her investment strategy, and to exhibit the specific expected utility maximization that makes this strategy optimal at any given time horizon. In the Black–Scholes market with deterministic parameters, we are able to show that traditional life-cycle funds are not optimal to any expected utility maximizers. We also prove that a CPPI strategy is optimal for a fund manager with HARA utility function, while an investor with a SAHARA utility function will choose a time-decreasing allocation to risky assets in the same spirit as the life-cycle funds strategy. Finally, we suggest how to modify these strategies if the financial market follows a more general diffusion process than in the Black–Scholes market.

Liquidation risk in the presence of Chapters 7 and 11 of the US bankruptcy code

We aim at quantitatively measuring the liquidation risk of a firm subject to both Chapters 7 and 11 of the US bankruptcy code. The firm value is modeled by a general time-homogeneous diffusion process in which the drift and volatility are level dependent and can be easily adjusted to reflect the state changes of the firm. An explicit formula for the probability of liquidation is established, based on which we gain a quantitative understanding of how the capital structures before and during bankruptcy affect the probability of liquidation.

Fast Greeks by simulation: The block adjoint method with memory reduction

The pathwise method is one of the approaches to calculate the option price sensitivities by Monte Carlo simulation. Under the general diffusion process, we usually use the Euler scheme to simulate the path of the underlying asset, which requires small time spaces to assure the convergence. The adjoint method can be used to accelerate the calculation of the Greeks in such case. However, it needs to store the intermediate information along the path. In this paper, we propose to use the block simulation to further accelerate the calculation. Block simulation can be seen as an extension of the common Monte Carlo simulation. It is simple to implement, without any extra work and loss in accuracy. It also has the flexibility on the block division, fitting to the computation environment. Moreover, we use the extended forward-path method along with the real time calculation strategy to do the memory reduction so that it can be combined with the adjoint method better. The numerical tests show our method can accelerate the adjoint method by several times. Our method is relatively even more efficient in the high-dimensional case.

OPTIMAL TIME-CONSISTENT PORTFOLIO AND CONTRIBUTION SELECTION FOR DEFINED BENEFIT PENSION SCHEMES UNDER MEAN–VARIANCE CRITERION

AbstractWe investigate two mean–variance optimization problems for a single cohort of workers in an accumulation phase of a defined benefit pension scheme. Since the mortality intensity evolves as a general Markov diffusion process, the liability is random. The fund manager aims to cover this uncertain liability via controlling the asset allocation strategy and the contribution rate. In order to have a more realistic model, we study the case when the risk aversion depends dynamically on current wealth. By solving an extended Hamilton–Jacobi–Bellman system, we obtain analytical solutions for the equilibrium strategies and value function which depend on both current wealth and mortality intensity. Moreover, results for the constant risk aversion are presented as special cases of our models.

Lifetime Asset Allocation with Idiosyncratic and Systematic Mortality Risks

This paper considers the lifetime asset allocation problem with both idiosyncratic and systematic longevity risks, in which the stochastic mortality model is given by a general diffusion process. A wage earner can invest in a zero-coupon bond, a stock and a longevity bond, consume part of his wealth and purchase life insurance or annuity so as to maximize the expected utility from consumption, terminal wealth and bequest. The problem is solved via the dynamic programming principle and the Hamilton-Jacobi-Bellman equation. General solutions and special solutions are derived for the general diffusion mortality model and the square-root mortality model, respectively. To illustrate our results, numerical examples based on special solutions are provided. It is shown that idiosyncratic mortality risk has significant impacts on the wage earner’s investment, consumption, life insurance purchase and bequest decisions regardless of the length of the decision-making horizon, while systematic mortality risk only has significant impacts on the wage earner’s investment in the zero-coupon bond and the longevity bond. Since systematic mortality risk can be hedged by trading the longevity bond, its impacts on consumption, purchase of life insurance and bequest are not significant, especially when the decision-making horizon is short.

Survey on the role of accelerator modes for anomalous diffusion: the case of the standard map.

We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected ("dragged") by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the Lévy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter K, up to K = 70. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large K ≥ 2π, but also of higher periods (2,3,4,...) at smaller values of K ≤ 2π.

Water Absorption Properties of Heat-Treated Bamboo Fiber and High Density Polyethylene Composites

To modify water absorption properties of bamboo fiber (BF) and high density polyethylene (HDPE) composites, heat treatment of BFs was performed prior to compounding them with HDPE to form the composites. The moisture sorption property of the composites was measured and their diffusion coefficients (Dm) were evaluated using a one-dimensional diffusion model. Moisture diffusion coefficient values of all composites were in the range of 0.115x10-8 to 1.267x10-8 cm2/s. The values of Dm decreased with increasing BF heat-treatment temperature, and increased with increasing BF loading level. The Dm value of 40 wt% bamboo fiber/HDPE composites with BFs treated with 100 oC was the greatest (i.e., 1.267x10-8cm2/s). Morphology analysis showed increased fiber-matrix interfacial bonding damage due to fiber swelling and shrinking from water uptaking and drying. The mechanism of water absorption of the composite, indicated a general Fickian diffusion process.

Recovery of time dependent volatility coefficient by linearization

We study the problem of reconstruction of special time dependent local volatility from market prices of options with different strikes at two expiration times. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an operator $W$ which is linear in perturbation of volatility. We further simplify the linearized inverse problem and obtain unique solvability result in basic functional spaces. By using the Laplace transform in time we simplify the kernels of integral operators for $W$ and we obtain uniqueness and stability results for volatility under natural condition of smallness of the spacial interval where one prescribes the (market) data. We propose a numerical algorithm based on our analysis of the linearized problem.

The forward-path method for pricing multi-asset American-style options under general diffusion processes

The forward-path method is one of the effective memory reduction approaches in pricing multi-asset American-style options by the Monte Carlo simulation. However, when the underlying assets follow a general diffusion process, it would be difficult to apply this method. In this paper we propose a simple and effective method, which applies the Milstein–Taylor explicit and implicit schemes and makes it possible to apply the forward-path method under general diffusion processes. It preserves the advantage of high reduction with relatively low cost of the forward-path method. Moreover, it works as a plug-in and one can flexibly substitute the schemes with different ones for different problems. It is proven that our method can conveniently restore the simulated path with strong convergence order 1.0. The numerical results indicate our method has high pricing accuracy and efficiency.

Propagator for the Fokker-Planck equation with an arbitrary diffusion coefficient

We consider a general diffusion process that is force-free and the corresponding Fokker-Planck equation with an arbitrary diffusion coefficient. A propagator for the Fokker-Planck equation of the Stratonovich form is obtained based on random walks. The characteristics of the solution are analyzed.

Nonmyopic optimal portfolios in viable markets

We provide a representation for the nonmyopic optimal portfolio of an agent consuming only at the terminal horizon when the single state variable follows a general diffusion process and the market consists of one risky asset and a risk-free asset. The key term of our representation is a new object that we call the “rate of macroeconomic fluctuation” whose properties are fundamental for the portfolio dynamics. We show that, under natural cyclicality conditions, (i) the agent’s hedging demand is positive (negative) when the product of his prudence and risk tolerance is below (above) two and (ii) the portfolio weights decrease in risk aversion. We apply our results to study a general continuous-time capital asset pricing model and show that under the same cyclicality conditions, the market price of risk is countercyclical and the price of the risky asset exhibits excess volatility.

Dividend optimization for regime-switching general diffusions

We consider the optimal dividend distribution problem of a financial corporation whose surplus is modeled by a general diffusion process with both the drift and diffusion coefficients depending on the external economic regime as well as the surplus itself through general functions. The aim is to find a dividend payout scheme that maximizes the present value of the total dividends until ruin. We show that, depending on the configuration of the model parameters, there are two exclusive scenarios: (i)the optimal strategy uniquely exists and corresponds to paying out all surpluses in excess of a critical level (barrier) dependent on the economic regime and paying nothing when the surplus is below the critical level;(ii)there are no optimal strategies.

A decomposition of irreversible diffusion processes without detailed balance

As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of canonical conservative dynamics with respect to a positive, differentiable stationary density ρ(x) is introduced: \documentclass[12pt]{minimal}\begin{document}$\dot{x}=j(x)$\end{document}ẋ=j(x) in which ∇·(ρ(x)j(x)) = 0. Such systems have a conserved “generalized free energy function” F[u] = ∫u(x, t)ln (u(x, t)/ρ(x))dx in phase space with a density flow u(x, t) satisfying ∂ut = −∇·(ju). Any general stochastic diffusion process without detailed balance, in terms of its Fokker-Planck equation, can be decomposed into a reversible diffusion process with detailed balance and a canonical conservative dynamics. This decomposition can be rigorously established in a function space with inner product defined as ⟨ϕ, ψ⟩ = ∫ρ−1(x)ϕ(x)ψ(x)dx. Furthermore, a law for balancing F[u] can be obtained: The non-positive dF[u(x, t)]/dt = Ein(t) − ep(t) where the “source” Ein(t) ⩾ 0 and the “sink” ep(t) ⩾ 0 are known as house-keeping heat and entropy production, respectively. A reversible diffusion has Ein(t) = 0. For a linear (Ornstein-Uhlenbeck) diffusion process, our decomposition is equivalent to the previous approaches developed by Graham and Ao, as well as the theory of large deviations. In terms of two different formulations of time reversal for a same stochastic process, the meanings of dissipative and conservative stationary dynamics are discussed.