General Theory Of Stochastic Processes Research Articles

On the value of non-Markovian Dynkin games with partial and asymmetric information

We prove that zero-sum Dynkin games in continuous time with partial and asymmetric information admit a value in randomised stopping times when the stopping payoffs of the players are general càdlàg measurable processes. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The main novelties are that we do not assume a Markovian nature of the game nor a particular structure of the information available to the players. This allows us to go beyond the variational methods (based on PDEs) developed in the literature on Dynkin games in continuous time with partial/asymmetric information. Instead, we focus on a probabilistic and functional analytic approach based on the general theory of stochastic processes and Sion’s min-max theorem (Pacific J. Math. 8 (1958) 171–176). Our framework encompasses examples found in the literature on continuous time Dynkin games with asymmetric information and we provide counterexamples to show that our assumptions cannot be further relaxed.

A characterisation of isometries with respect to the Lévy-Prokhorov metric

According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L{\'e}vy--Prokhorov distance. Motivated by these facts, the famous Banach--Stone theorem, and some recent works related to characterisations of onto isometries of spaces of Borel probability measures, here we give a complete description of surjective isometries with respect to the L{\'e}vy--Prokhorov metric in case when the underlying metric space is a separable Banach space. Our result can be considered as a generalisation of L. Moln\'ar's earlier Banach--Stone-type result which characterises onto isometries of the space of all probability distribution functions on the real line wit respect to the L{\'e}vy distance. However, the present more general setting requires the development of an essentially new technique.

Stochastic description of electrochemical discharge using formalism of Kramers–Moyal expansion

The formalism of Kramers–Moyal expansion is used for a stochastic description of one-stage electrochemical discharge occurring via a single route. In the stochastic model under consideration, an electrochemical reaction is represented as two independent random series of anodic and cathodic elementary acts of charge transfer. Each of two random series obeys its generalized Poisson’s distribution. Three Kramers–Moyal expansions are determined. The first Kramers–Moyal expansion works near the equilibrium potential. It takes into consideration both (anodic and cathodic) series of elementary acts. The second expansion determines the stochastic behavior of electrochemical reaction at high anodic potentials. The third expansion controls the stochastic behavior of electrochemical reaction in the range of high cathodic overpotentials. The dependence of coefficients of the Kramers–Moyal expansion on the macroscopic parameters of electrochemical discharge is determined. In view of interdisciplinary character of the theory of noise and fluctuations, a stochastic description of electrochemical discharge within the framework of Kramers–Moyal expansion is of interest also for the general theory of stochastic processes.

Optional and Predictable Projections of Normal Integrands and Convex-Valued Processes

This article studies optional and predictable projections of integrands and convex-valued stochastic processes. The existence and uniqueness are shown under general conditions that are analogous to those for conditional expectations of integrands and random sets. In the convex case, duality correspondences between the projections and projections of epigraphs are given. These results are used to study projections of set-valued integrands. Consistently with the general theory of stochastic processes, projections are not constructed using reference measures on the optional and predictable sigma-algebras.

Andô–Douglas type characterization of optional projections and predictable projections

Optional projections and predictable projections of stochastic processes play important roles in the general theory of stochastic processes, semimartingale theory and stochastic calculus. They share some important properties with ordinary conditional expectations and generalized conditional expectations. While the characterization of ordinary conditional expectations and generalized conditional expectations has been studied by several authors, no similar work has been done for optional projections and predictable projections. This paper aims at undertaking this task by giving Andô–Douglas type characterization theorem for both.

Dynamic Programming for General Linear Quadratic Optimal Stochastic Control with Random Coefficients

We are concerned with the linear-quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable (but essentially bounded) processes and the terminal state-weighting matrix in the cost functional is allowed to be random. Under suitable conditions, we prove that the value field $\mathbb{V}(t,x,\omega), (t,x,\omega)\in [0,T]\times R^n\times \Omega$, is quadratic in $x$ and has the following form: $\mathbb{V}(t,x)=\langle \mathbb{K}_tx, x\rangle$, where $\mathbb{K}$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $\mathbb{K}$ is a continuous semimartingale of the form $\mathbb{K}_t=\mathbb{K}_0-\int_0^t \, dk_s+\int_0^t\sum_{i=1}^d L_s^i\, dW_s^i, \ t\in [0,T]$, with $k$ being a continuous process of bounded variation and the stochastic integral $\int_0^\cdot\sum_{i=1}^d L^i_s\, dW^i_s$ being a bounded mean oscillation martingale, and that $(\mathbb{K}, L)$ with $L:=(L^1, \ldots, L^d)$ is a solution to the associated backward stochastic Riccati equation (BSRE), whose generator is highly nonlinear in the unknown pair of processes. The uniqueness is also proved via a localized completion of squares in a self-contained manner for a general BSRE. The existence and uniqueness of an adapted solution to a general BSRE was initially proposed by the French mathematician J. M. Bismut [SIAM J. Control Optim., 14 (1976), pp. 419--444; Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978, pp. 180--264] and subsequently listed by S. Peng [Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), S. Chen et al., eds., Kluwer, Boston, 1999, pp. 265--273] as an open problem for backward stochastic differential equations. It had remained open until a general solution by the author [SIAM J. Control Optim., 42 (2003), pp. 53--75] via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. The present paper is its companion and gives the second but more comprehensive (seemingly much simpler, but appealing to the advanced tool of Doob--Meyer's decomposition theorem and the general theory of stochastic processes, in addition to the DPP) adapted solution to a general BSRE via the DPP. Further extensions to the jump-diffusion control system and to the general nonlinear control system are possible.

Continuous-Time Public Good Contribution Under Uncertainty

We study a continuous-time problem of optimal public good contribution under uncertainty for an economy with an finite number of agents. Each agent can allocate his wealth between private consumption and repeated but irreversible contributions to increase the stock of some public good. We study the corresponding social planner problem and the case of strategic interaction between the agents and we characterize the optimal investment policies by a set of necessary and sufficient stochastic Kuhn-Tucker conditions. Suitably combining arguments from Duality Theory and the General Theory of Stochastic Processes, we prove an abstract existence result for a Nash equilibrium of our public good contribution game. Also, we show that our model exhibits a dynamic free rider effect. We explicitly evaluate it in a symmetric Black-Scholes setting with Cobb-Douglas utilities and we show that uncertainty and irreversibility of public good provisions do not affect free-riding.

Steady-state parameter sensitivity in stochastic modeling via trajectory reweighting

Parameter sensitivity analysis is a powerful tool in the building and analysis of biochemical network models. For stochastic simulations, parameter sensitivity analysis can be computationally expensive, requiring multiple simulations for perturbed values of the parameters. Here, we use trajectory reweighting to derive a method for computing sensitivity coefficients in stochastic simulations without explicitly perturbing the parameter values, avoiding the need for repeated simulations. The method allows the simultaneous computation of multiple sensitivity coefficients. Our approach recovers results originally obtained by application of the Girsanov measure transform in the general theory of stochastic processes [A. Plyasunov and A. P. Arkin, J. Comput. Phys. 221, 724 (2007)]. We build on these results to show how the method can be used to compute steady-state sensitivity coefficients from a single simulation run, and we present various efficiency improvements. For models of biochemical signaling networks, the method has a particularly simple implementation. We demonstrate its application to a signaling network showing stochastic focussing and to a bistable genetic switch, and present exact results for models with linear propensity functions.

An axiomatic approach to the valuation of cash flows

We model a stream of cash flows as an optional stochastic process, and value the cash flows by using a continuous and strictly positive linear functional. By applying a representation theorem from the general theory of stochastic processes we are able to study this valuation principle, as well as properties of the stochastic discount factor it implies. This approach to valuation is useful in the non-presence of a financial market, as is often the case when valuing cash flows arising from insurance contracts and in the application of real options.

INITIAL ENLARGEMENT IN A MARKOV CHAIN MARKET MODEL

Enlargement of filtrations is a classical topic in the general theory of stochastic processes. This theory has been applied to stochastic finance in order to analyze models with insider information. In this paper we study initial enlargement in a Markov chain market model, introduced by Norberg. In the enlarged filtration, several things can happen: some of the jumps times can be accessible or predictable, but in the original filtration all the jumps times are totally inaccessible. But even if the jumps times change to accessible or predictable, the insider does not necessarily have arbitrage possibilities.

Enlargements of filtrations and path decompositions at non stopping times

Azema associated with an honest time L the supermartingale Open image in new window and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermartingales, which in turn will naturally provide many examples of enlargements of filtrations. We combine this characterization with some arguments from both initial and progressive enlargements of filtrations to establish some path decomposition results, closely related to or reminiscent of Williams' path decomposition results. In particular, some of the fragments of the paths in our decompositions end or start with a new family of random times which are not stopping times, nor honest times.

An essay on the general theory of stochastic processes

This text is a survey of the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school and which are usually written in French. The material presented in the last three chapters is less standard and takes into account some recent developments.

Some random times and martingales associated with BES0(δ) processes (0

In this paper, we study Bessel processes of dimension $\delta\equiv2(1-\mu)$, with $0<\delta<2$, and some related martingales and random times. Our approach is based on martingale techniques and the general theory of stochastic processes (unlike the usual approach based on excursion theory), although for $0<\delta<1$, these processes are even not semimartingales. The last time before 1 when a Bessel process hits 0, called $g_{\mu}$, plays a key role in our study: we characterize its conditional distribution and extend Paul L\'{e}vy's arc sine law and a related result of Jeulin about the standard Brownian Motion. We also introduce some remarkable families of martingales related to the Bessel process, thus obtaining in some cases a one parameter extension of some results of Az\'{e}ma and Yor in the Brownian setting: martingales which have the same set of zeros as the Bessel process and which satisfy the stopping theorem for $g_{\mu}$, a one parameter extension of Az\'{e}ma's second martingale, etc. Throughout our study, the local time of the Bessel process also plays a central role and we shall establish some of its elementary properties.

$\sigma$-Localization and $\sigma$-Martingales

This paper introduces the concept of {\em $\sigma$-localization}, which is a generalization of localization in the general theory of stochastic processes. The $\sigma$-localized class derived from the set of martingales is the class of {\em $\sigma$-martingales}, which plays an important role in mathematical finance. These processes and the corresponding {\em $\sigma$-martingale measures} are considered in detail. By extending the stochastic integral with respect to compensated random measures, a canonical representation of $\sigma$-martingales as for local martingales is derived.

Nonlocal stochastic mixing-length theory and the velocity profile in the turbulent boundary layer

Turbulence mixing by finite size eddies will be treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a stochastic closure hypothesis, which implies a well defined recipe for the calculation of sampling and transition rates. The connection with the general theory of stochastic processes will be established. The relation with other nonlocal turbulence models (e.g. transilience and spectral diffusivity theory) is also discussed. Using an analytical sampling rate model (satisfying exchange) the theory is applied to the boundary layer (using a scaling hypothesis), which maps boundary layer turbulence mixing of scalar densities onto a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process. The resulting transpport equation for longitudinal momentum P x ≡ ϱ U is solved for a unified description of both the inertial and the viscous sublayer including the crossover. With a scaling exponent ε ≈ 0.58 (while local turbulence would amount to ε → ∞) the velocity profile U + = ƒ(y +) is found to be in excellent agreement with the experimental data. Inter alia (i) the significance of ε as a turbulence Cantor set dimension, (ii) the value of the integration constant in the logarithmic region (i.e. if y + → ∞), (iii) linear timescaling, and (iv) finite Reynolds number effects will be investigated. The (analytical) predictions of the theory for near-wall behaviour (i.e. if y + → 0) of fluctuating quantities also perfectly agree with recent direct numerical simulations.

Strong approximations of semimartingales by processes with independent increments

Strong approximation theorems for continuous time semimartingales are obtained by combining some techniques of the general theory of stochastic processes with some of the direct approximation of dependent random variables by independent ones. Continuous processes with independent increments whose variance functions increase polynomially or exponentially are considered as approximating processes. The basic assumptions of the main results only contain rates of convergence for certain probabilities. In particular, moment assumptions are not required. Some almost sure invariance principles for partial sum processes with nonlinear growth of variance and for functionals of Markov processes are derived by applying the main results.

Optimization Applications of Compensators of Poisson Random Measures

Compensator or dual predictable projection is one of the main concepts of the general theory of stochastic processes, established by a French probabilistic school. In general, it is a considerably abstract concept and difficult to grasp. But in the Poisson case, it affords very powerful tools to calculate the expectation of certain stochastic integrals with respect to Poisson processes and random measures. Naturally, compensators can be used to solve certain problems of applied probability, involved in Poisson processes and random measures. In this direction, Brémaud and Jacod [1] had used the compensators of Poisson processes to solve the optimal dispatching problem, investigated by Ross [4] originally. In this note, we use the compensators of Poisson random measures to solve two optimal stopping problems: house selling and the burglary problem. The solutions are simple and completely rigorous as well. Our objective is to attract more attention to this method of using compensators. To facilitate reading, we introduce some fundamental results on marked point processes and Poisson random measures and their compensators.

Previsible sets for hyperfinite filtrations

Previsible (or predictable) stochastic processes are defined for any “filtration” over a probability space (Dellacherie and Meyer (1978), IV. 61). This technical definition gives previsible processes certain “predictability properties” such as not being able to oscillate in unison with martingale differentials. Thus previsibility has become one essential ingredient in “The General Theory of Stochastic Processes”.

The martingale method: Introductory sketch and access to the literature

Since 1975 there has developed a powerful and sophisticated theory of statistical inference for point processes known as the ‘martingale method’. Because it is based on deep and in many cases technical results in the (French) general theory of stochastic processes, it has remained inaccessible to many who might usefully apply it. Yet the fundamental ideas (as opposed to the details, which can be formidable) are not hopelessly arcane; our purposes here are to describe the theory in a manner that reveals its underlying conceptual simplicity (although oversimplified, (6) below is the essential idea), to illustrate it with three examples, and to give some points of entry into the complex and now rather extensive literature. This is not a guide to application, much less even an incomplete development of the theory, but merely (so to speak) a silhoutte.

The Theory of Risk and Some Applications

Some of the classical assumptions and formulae of the collective risk theory are presented first and a model utilizing dynamic programming is constructed. Some classical applications of risk theory such as solvency tests, risk reserves, reinsurance, and rate making are briefly reviewed. The scope of the theory is extended to the field of dynamic system design. Finally, problems concerning the teaching of risk theory to actuaries and non-actuaries is discussed. 1. Theory or practice? Conventional insurance mathematics operates mainly by deterministic quantities, such as mortalities, loss ratios, and their expected or capitalized values, and so on. However, the deterministic models give a truncated picture of the insurance business, because random and other fluctuations are omitted. Attempts to complete the deterministic model by including the probabilistic variations have been made for a long time. The so-called individual risk theory was developed mainly as a supplementary field to life insurance mathematics. Subsequently, following an idea originally introduced by Filip Lundberg, the risk business of an insurer (or any other collective) as a whole has been interpreted as a stochastic process and collective risk theory is built mainly on that basis. The simultaneous development of the general theory of stochastic processes and its numerous applications has influenced fruitfully the development of risk theory, too. A well-developed theory already exists, even if many questions are still open. When speaking of risk theory we limit the consideration, as the European risk theorists conventionally do, to concern the risk process of an insurer as a whole. Theories dealing with individual risks, behaviour of policyowners or individual risk contracts are not dealt with even if many of the formulae are also applicable to them. By some simplifying one observes that research has had two main directions: the scientific school, which has successfully developed the theory on the basis of rigorous discipline, and the school, which has attempted to modify the existing theories and formulae to make them applicable in the environments of insurers. Often the starting point has been a practical problem, e.g. the making of premium rates (including the safety margin), defining the minimum need for reserve funds, choosing the level of reassurance, and so on. In practice, many rules of thumb have been applied and to turn for help to risk theory, which originally was