Forward Kolmogorov Research Articles

Bertrand & Cournot Mean Field Games

We study how continuous time Bertrand and Cournot competitions, in which firms producing similar goods compete with one another by setting prices or quantities respectively, can be analyzed as continuum dynamic mean field games. Interactions are of mean field type in the sense that the demand faced by a producer is affected by the others through their average price or quantity. Motivated by energy or consumer goods markets, we consider the setting of a dynamic game with uncertain market demand, and under the constraint of finite supplies (or exhaustible resources). The continuum game is characterized by a coupled system of partial differential equations: a backward HJB PDE for the value function, and a forward Kolmogorov PDE for the density of players. Asymptotic approximation enables us to deduce certain qualitative features of the game in the limit of small competition. The equilibrium of the game is further studied using numerical solutions, which become very tractable by considering the tail distribution function instead of the density itself. This also allows us to consider Dirac delta distributions to use the continuum game to mimic finite N-player nonzero-sum differential games, the advantage being having to deal with two coupled PDEs instead of N. We find that, in accordance with the two-player game, a large degree of competitive interaction causes firms to slow down production. The continuum system can therefore be used qualitative as an approximation to even small player dynamic games.

Local polynomial estimations of time-varying coefficients for local stationary diffusion models

This paper is dedicated to the study of local polynomial estimations of time-varying coefficients for a local stationary diffusion model. Based on local polynomial fitting, the estimations of drift parametric functions are obtained by using local weighted least squares method. By applying the forward Kolmogorov equation, the estimation of the diffusion coefficient is proposed. The consistency, asymptotic normality and uniform convergence of the estimations that we proposed are established.

Linear processes in stochastic population dynamics: theory and application to insect development.

We consider stochastic population processes (Markov jump processes) that develop as a consequence of the occurrence of random events at random time intervals. The population is divided into subpopulations or compartments. The events occur at rates that depend linearly on the number of individuals in the different described compartments. The dynamics is presented in terms of Kolmogorov Forward Equation in the space of events and projected onto the space of populations when needed. The general properties of the problem are discussed. Solutions are obtained using a revised version of the Method of Characteristics. After a few examples of exact solutions we systematically develop short-time approximations to the problem. While the lowest order approximation matches the Poisson and multinomial heuristics previously proposed, higher-order approximations are completely new. Further, we analyse a model for insect development as a sequence of E developmental stages regulated by rates that are linear in the implied subpopulations. Transition to the next stage competes with death at all times. The process ends at a predetermined stage, for example, pupation or adult emergence. In its simpler version all the stages are distributed with the same characteristic time.

Hermite Spectral Method to 1-D Forward Kolmogorov Equation and Its Application to Nonlinear Filtering Problems

In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problem with a real-time algorithm developed in [17]. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filter as a real-time solver to NLF.

On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Fellerʼs seminal paper. In particular, this paper extends Fellerʼs results for continuous Q-functions to measurable Q-functions and provides additional results.

Hitting Times, Occupation Times, Trivariate Laws and the Forward Kolmogorov Equation for a One-Dimensional Diffusion with Memory

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form dXt=σ(Xt,Xt)dWt, whereXt=m∧ inf0 ≤s≤tXs. In particular, we compute the expected time forXto leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula forX. We also show that (Xt,Xt) admits a joint density, which can be characterized in terms of two independent tied-down Brownian meanders (or, equivalently, two independent Bessel-3 bridges). Finally, we show that the joint density satisfies a generalized forward Kolmogorov equation in a weak sense, and we derive a new forward equation for down-and-out call options.

Hitting Times, Occupation Times, Trivariate Laws and the Forward Kolmogorov Equation for a One-Dimensional Diffusion with Memory

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form d Xt=σ(Xt,Xt)dWt, where Xt= m ∧ inf0 ≤s≤tXs. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at 0, and derive a new occupation time formula for X. We also show that (Xt,Xt) admits a joint density, which can be characterized in terms of two independent tied-down Brownian meanders (or, equivalently, two independent Bessel-3 bridges). Finally, we show that the joint density satisfies a generalized forward Kolmogorov equation in a weak sense, and we derive a new forward equation for down-and-out call options.

Backward Solution of Markov Chains and Markov Regenerative Processes: Formalization and Applications

In this paper we investigate the computation, and the stochastic interpretation, of backward probabilities of Markov chains (transient and steady-state probabilities derived from backward Kolmogorov equations) and its extension to the case of Markov Regenerative Processes (MRP). The study is then extended to the case of non-ergodic settings, which enlights a substantial difference between the forward solution process (based on forward Kolmogorov equations) and the backward one.We shall clarify the role that backward solutions play in computing absorption probabilities and in the model-checking of stochastic logics as CSL and CSLTA, which typically require the steady state solution of a non-ergodic CTMC and MRP respectively. Moreover we show that the algorithm for the computation of the whole set of states that satisfy a CSL formula, which is standard practice in CSL model-checkers, can be seen as a case of computation of backward probabilities of Continuous Time Markov Chains (CTMCs). The backward computation of MRP is then inserted in the context of matrix-free solution technique, which allows to deal with MRP of much bigger size than the standard approach based on the computation and solution of the embedded Markov chain.

Analytic Solution of Convection-Diffusion Partial Differential Equations Using Girsanov Theorem, Feynman-Kac Formula and Brownian Bridge

In this brief research note, we try to find solution of a large class of convection-diffusion forward Kolmogorov equations of the type that typically appear in theory of derivatives pricing and stochastic volatility modeling. Our technique is based on a change of coordinates that makes the diffusion part of associated SDE linear. We use Girsanov theorem to add drift to this scaled Brownian motion and Feynman-Kac Formula is used to take into account killing term in the PDE. We solve the PDE at points of interest by analytically calculating conditional value of Girsanov exponential and Feynman Exponential using Brownian Bridging.

Translation invariant diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σij, bi and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σij ★ \(\tilde y\), bi ★ \(\tilde y\) are assumed to be locally Lipshitz.Here ★ denotes convolution and \(\tilde y\) is the distribution which on functions, is realised by the formula \(\tilde y\left( r \right): = y\left( { - r} \right)\). The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.

Reconstructing Macroeconomics, A Perspective from Statistical Physics and Combinatorial Stochastic Processes: Book Review

Professors Aoki and Yoshikawa adopted a variety of concepts from statistical physics and combinatorial stochastic processes to various problems in economics (such as labor markets, real growth, consumption, unemployment, financial market, productivity difference, etc.). In order to analyze these phenomena, they depart from the standard methods of model construction and analysis in mainstream economics and use methods that generally fall into two broad categories. One deals with stochastic dynamics, and the other with the formation of clusters and random combinatorial analysis. The authors build their new macroeconomics on the observation that the huge number of heterogeneous agents act stochastically different based on individual insights, tastes, goals, etc. In this approach the properties of economic systems are described by macroscopic dynamical equations of motion that are nonlinear partial differential equations, such as backward and forward Kolmogorov equations (known as a Chapman-Kolmogorov equation and Fokker-Planck equations).

Solution of Convection-Diffusion Partial Differential Equations Using Girsanov Theorem and Conditional Brownian Local Time

In this brief research note, we try to find solution of a large class of convection-diffusion backward or forward Kolmogorov equations of the type that typically appear in theory of derivatives pricing and stochastic volatility modeling. Our technique is based on a change of coordinates that makes the diffusion part of associated SDE linear. Using Brownian Bridging technique, we find conditional local time of variance scaled Brownian motion. We use Girsanov theorem to add drift to this scaled Brownian motion and solve the PDE at points of interest by integrating conditional value of exponential martingale using conditional Brownian local time. In the Backward Kolmogorov equation, the final data is modeled as a sum of Dirac Deltas of finite width and total solution is a superposition of sum of individual solutions from Dirac Deltas.

The complexity in sociological theory: an axiomatic description

This article puts forth an axiomatic description of the complexity of an object of sociological investigation. The proposed axioms allow us to determine complexity within the framework of mathematical sociology such as the variational principle, which is formed relative to the state of the object of sociological investigation. On the basis of this principle we can conclude the equation of state, which coincides with the stationary forward Kolmogorov equation.

Applications and Properties of Different Kind of Wavelet

We consider dynamical systems in which a (typically vectorvalued) dependentvariable evolves according to autonomousdynamics switching randomly according to Markovian laws that change with the value of the dependent variable. Such systems are known as “random evolutions” or, in electrical engineering contexts, as “switching systems”. Systems of this type are encountered in applications from electrical engineering to cell biology (our paper was inspired by a recent model for genetic oscillators). We review the derivation of the forward Kolmogorov master equations for the probabilities to find the system in a certain state at some time. In the limit of an infinite switching rate solutions of our system converge almost surely to solutions of an averaged problem. The classical tool of a ChapmanEnskog expansion, wellknown from kinetic theory, provides diffusion approximations to first order in the scale parameter. Our work focusses on a few typical examples. The analysis is formal, but we expect that the results hold rigorously in analogy to earlier results from kinetic theory (see [11]).

Inference for systems of stochastic differential equations from discretely sampled data: a numerical maximum likelihood approach

Maximum likelihood estimation of discretely observed diffusion processes is mostly hampered by the lack of a closed form solution of the transient density. It has recently been argued that a most generic remedy to this problem is the numerical solution of the pertinent Fokker–Planck (FP) or forward Kolmogorov equation. Here we expand existing work on univariate diffusions to higher dimensions. We find that in the bivariate and trivariate cases, a numerical solution of the FP equation via alternating direction finite difference schemes yields results surprisingly close to exact maximum likelihood in a number of test cases. After providing evidence for the efficiency of such a numerical approach, we illustrate its application for the estimation of a joint system of short-run and medium-run investor sentiment and asset price dynamics using German stock market data.

Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

We consider morphological and linear scale spaces on the space ℝ3⋊S 2 of 3D positions and orientations naturally embedded in the group SE(3) of 3D rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The strength of these enhancements is that they are expressed in a moving frame of reference attached to fiber fragments, allowing us to diffuse along the fibers and to erode orthogonal to them. The linear scale spaces are described by forward Kolmogorov equations of Brownian motions on ℝ3⋊S 2 and can be solved by convolution with the corresponding Green’s functions. The morphological scale spaces are Bellman equations of cost processes on ℝ3⋊S 2 and we show that their viscosity solutions are given by a morphological convolution with the corresponding morphological Green’s function. For theoretical underpinning of our scale spaces on ℝ3⋊S 2 we introduce Lagrangians and Hamiltonians on ℝ3⋊S 2 indexed by a parameter η∈[1,∞). The Hamiltonian induces a Hamilton-Jacobi-Bellman system that coincides with our morphological scale spaces on ℝ3⋊S 2. By means of the logarithm on SE(3) we provide tangible estimates for both the linear- and the morphological Green’s functions. We also discuss numerical finite difference upwind schemes for morphological scale spaces (erosions) of Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI), which allow extensions to data-adaptive erosions of DW-MRI. We apply our theory to the enhancement of (crossing) fibres in DW-MRI for imaging water diffusion processes in brain white matter.

Solution of Stochastic Volatility Models Using Variance Transition Probabilities and Path Integrals

In this paper, we solve the problem of solution of stochastic volatility models in which the volatility diffusion can be solved by a one dimensional Fokker-planck equation. We use one dimensional transition probabilities for the evolution of PDE of variance. We also find dynamics of evolution of expected value of any path dependent function of stochastic volatility variable along the PDE grid. Using this technique, we find the conditional expected values of moments of log of terminal asset price along every node of one dimensional forward Kolmogorov PDE. We use the conditional distribution of moments of above path integrals along the variance grid and use Edgeworth expansions to calculate the density of log of asset price. Main result of the paper gives dynamics of evolution of conditional expected value of a path dependent function of volatility (or any other SDE) at any node on the PDE grid using just one dimensional PDE if we can describe its one step conditional evolution between different nodes of the PDE.

European Option Under Jump-Diffusion and Stochastic Interest Rate

A one-dimensional partial differential-difference equation (pdde) under forward measure is developed to value European option under jump-diffusion, stochastic interest rate and local volatility. The corresponding forward Kolmogorov partial differential-difference equation for transition probability density is a also developed to value the options for various strikes at a given maturity time.The mathematical formulation of those equations is verified numerically by comparing their finite difference computation results with those of the Monte Carlo simulations. For the Kolmogorov equation, an alternate numerical method called the redistribution method is also developed. The redistribution method is based on the moments of the transition probability density and avoids some of the difficulties of a finite difference method.

European Option Under Cox-Ingersoll-Ross Model for Stochastic Interest Rate

European option under local volatility and Cox-Ingersoll-Ross model of short rate is computed from one-dimensional partial differential equations: the Black-Scholes equation for option price and the forward Kolmogorov equation for probability transition density. Both the computations are verified by Monte Carlo simulations.

Linear Programming and Constrained Average Optimality for General Continuous-Time Markov Decision Processes in History-Dependent Policies

This paper attempts to study the constrained average optimality for continuous-time Markov decision processes in the class of randomized history-dependent policies. The states and actions are in general Polish spaces, and the transition rates are allowed to be unbounded. The optimality criterion to be optimized is expected average costs, multiple constraints are imposed on similar expected average costs, and all costs may be unbounded from above and from below. Under suitable conditions, we first show the existence of a constrained optimal policy by improving the concept of a stable policy in the previous literature and using the analogue of the forward Kolmogorov equation. Then, we develop a linear program (LP), which is equivalent to the constrained optimality problem and is used to obtain a constrained optimal policy. By introducing suitable operators and conditions, we further establish the dual program (DP) of the LP, show that the LP and DP are solvable, and show that there is no duality gap between them. Finally, we use a cash flow model and a controlled birth and death system to illustrate the applications of our main results.