Doob-Meyer Decomposition Research Articles

A Meyer-Itô formula for stable processes via fractional calculus

The infinitesimal generator of a one-dimensional strictly alpha -stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order alpha and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if alpha in (0,1) and the recurrent potential if alpha in (1,2). The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order alpha . One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada [1]. This result is used to find a Doob-Meyer (or semimartingale) decomposition for |X_t - x|^{gamma } with X a recurrent strictly stable process of index alpha and gamma in (alpha -1,alpha ), generalizing the work of Engelbert and Kurenok [2] to the asymmetric case.

G-Expectation for Conformable Backward Stochastic Differential Equations

In this paper, we study the applications of conformable backward stochastic differential equations driven by Brownian motion and compensated random measure in nonlinear expectation. From the comparison theorem, we introduce the concept of g-expectation and give related properties of g-expectation. In addition, we find that the properties of conformable backward stochastic differential equations can be deduced from the properties of the generator g. Finally, we extend the nonlinear Doob–Meyer decomposition theorem to more general cases.

The Itô integral and near-martingales in Riesz spaces

A new class of the Itô integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is a motivitation to construct near-martingales in Riesz spaces. Furthermore, we state Doob–Meyer decomposition theorem for near-submartingales in Riesz spaces.

A Doob-Meyer decomposition under model ambiguity: the case of compactness

A Doob-Meyer decomposition under model ambiguity: the case of compactness

Stopped processes and Doob's optional sampling theorem

Using the spectral measure μS of the stopping time S, we define the stopping element XS as a Daniell integral ∫XtdμS for an adapted stochastic process (Xt)t∈J that is a Daniell summable vector-valued function. This is an extension of the definition previously given for right-order-continuous sub martingales with the Doob-Meyer decomposition property. The more general definition of XS necessitates a new proof of Doob's optional sampling theorem, because the definition given earlier for sub martingales implicitly used Doob's theorem applied to martingales. We provide such a proof, thus removing the heretofore necessary assumption of the Doob-Meyer decomposition property in the result. Another advancement presented in this paper is our use of unbounded order continuity of a stochastic process, which properly characterizes the notion of continuity of sample paths almost everywhere, found in the classical theory.

Formula omitted] control for Poisson-driven stochastic systems

The paper aims to investigate the H∞ control problem for systems perturbed by jump random noise, i.e., Poisson-driven stochastic systems (SSs). Firstly, this paper uses the Doob-Meyer decomposition and measure theory to give a model transformation method, and Poisson-driven SSs, which are SSs driven by semi-martingale, are transformed into SSs driven by compensated Poisson process. Since SSs driven by compensated Poisson process are SSs driven by martingale, we can use effective properties and tools in martingale theory to investigate the H∞ control problem. Secondly, this paper utilizes martingale theory to deal with the jump item and the sum of stochastic integrals (SIs) with respect to (w.r.t.) the continuous part of states in Itô formula, and then gives an equivalent Itô formula for SDEs driven by compensated Poisson process. Thirdly, on the basis of these, this paper presents a simple H∞ controller design method. Furthermore, the design criterion contains information about the average number of jump random events in a unit time, and one can utilize convex optimization algorithm to estimate the maximum average number of jump random events in a unit time, which the closed-loop system can tolerate to achieve the stability and H∞ performance. Finally, the usefulness of the design result is verified by two numerical examples.

Strong differential subordinates for noncommutative submartingales

We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case (Ann. Probab. 22 (1994) 995–1025). Then we establish the maximal weak-type $(1,1)$ inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty $ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-$\lambda$ inequalities.

The monotone case approach for the solution of certain multidimensional optimal stopping problems

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob–Meyer decomposition in continuous time, also in its multiplicative versions. The approach via these decompositions leads to explicit solutions for a variety of examples, including multidimensional versions of the house-selling and burglar’s problem, the Poisson disorder problem, and an optimal investment problem.

Projections of scaled Bessel processs

Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.

Filtration Shrinkage, the Structure of Deflators, and Failure of Market Completeness

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are them- selves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.

Optional decomposition of optional supermartingales and applications to filtering and finance

ABSTRACTThe classical Doob–Meyer decomposition and its uniform version the optional decomposition are stated on probability spaces with filtrations satisfying the usual conditions. However, the comprehensive needs of filtering theory and mathematical finance call for their generalizations to more abstract spaces without such technical restrictions. The main result of this paper states that there exists a uniform Doob–Meyer decomposition of optional supermartingales on unusual probability spaces. This paper also demonstrates how this decomposition works in the construction of optimal filters in the very general setting of the filtering problem for optional semimartingales. Finally, the application of these optimal filters of optional semimartingales to mathematical finance is presented.

Mutual Information, Relative Entropy and Estimation Error in Semi-Martingale Channels

Fundamental relations between information and estimation have been established in the literature for the continuous time Gaussian and Poisson channels. In this paper, we demonstrate that such relations hold for a much larger family of continuous-time channels. We introduce the family of semi-martingale channels where the channel output is a semi-martingale stochastic process, and the channel input modulates the characteristics of the semi-martingale. For these channels, which includes as a special case the continuous time Gaussian and Poisson models, we establish new representations relating the mutual information between the channel input and output to an optimal causal filtering loss, thereby unifying and considerably extending results from the Gaussian and Poisson settings. Extensions to the setting of mismatched estimation are also presented where the relative entropy between the laws governing the output of the channel under two different input distributions is equal to the cumulative difference between the estimation loss incurred by using the mismatched and optimal causal filters, respectively. The main tool underlying these results is the Doob–Meyer decomposition of a class of sub-martingales. The results in this paper can be viewed as the continuous-time analogues of recent generalizations for relations between information and estimation for discrete-time Levy channels.

Markov processes with spatial delay: Path space characterization, occupation time and properties

In this paper, we study one-dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one-dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator’s domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the generator has restrictions involving second derivatives. In this paper we provide a pathwise characterization for processes with delay in terms of an SDE and an occupation time formula involving the symmetric local time. This characterization provides an explicit Doob–Meyer decomposition, demonstrating that such processes are semi-martingales and that all of stochastic calculus including Itô formula and Girsanov formula applies. We also establish an occupation time formula linking the time that the process spends at a delay point with its symmetric local time there. A physical example of a stochastic dynamical system with delay is lastly presented and analyzed.

From the decompositions of a stopping time to risk premium decompositions

The occurrence of some events can impact asset prices and produce losses. The amplitude of these losses are partly determined by the degree of predictability of those events by the market investors, as risk premiums build up in an asset price as a compensation of the anticipated losses. The aim of this paper is to propose a general framework where these phenomena can be properly defined and quantified.Our focus are the default events and the defaultable assets, but the framework could apply to any event whose occurrence impacts some asset prices.We provide the general construction of a default time under the so called (H) hypothesis, which reveals a useful way in which default models can be built, using both market factors and idiosyncratic factors. All the relevant characteristics of a default time (i.e. the Azema supermartingale and its Doob-Meyer decomposition) are explicitly computed given the information about these factors.We then define the default event risk premiums and the default adjusted probability measure. These concepts are useful for pricing defaultable claims in a framework that includes possible economic shocks, such as jumps of the recovery process or of some default-free assets at the default time. These formulas are not classic and we point out that the knowledge of the default compensator (or the intensity process when the default time is totally inaccessible) is not a sufficient quantity for finding explicit prices; the Azema supermartingale and its Doob-Meyer decomposition are needed. The progressive enlargement of a filtration framework is the right tool for pricing defaultable claims in non standard frameworks where non defaultable assets or recovery processes may react at the default event.

Rebalancing static super-replications

This paper proposes a trading strategy that dynamically rebalances static super-replicating portfolios, which is very useful for both investment and hedging strategies. In order to investigate general properties of the strategy, we derive the Doob–Meyer decomposition for the value process without any specifications of models under the continuous processes of the underlying variables. In particular, we find that the increasing part of the decomposition characterizes the performance of the strategy. Also, we obtain more concrete features for cross-currency and one-touch options based on our general framework. Moreover, numerical examples for cross-currency options demonstrate the effectiveness of our strategy for investment and hedging.

The Itô integral for martingales in vector lattices

In the setting of a Dedekind complete Riesz space on which a conditional expectation is defined, we study the Itô integral. This integral obtains the integration of a stochastic process of a continuous parameter relative to a martingale. A considerable part of the paper is devoted to aspects of integration of a vector-valued function with respect to a vector measure. Here we use the Dobrakov integral that is, in our case, a generalization of the well known Bartle integral. We discuss natural and predictable processes to prove a general version of the Doob–Meyer decomposition of a submartingale. This decomposition provides an essential tool used in the definition of the stochastic integral. We derive the properties of the stochastic integral that are useful in developing the theory of stochastic processes in Riesz spaces.

Quadratic BSDEs with jumps: Related nonlinear expectations

In this paper, we continue the study of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables [Formula: see text], started in our accompanying paper [12]. Relying on the existence and uniqueness result of [12], we define the corresponding [Formula: see text]-expectations and study some of their properties. We obtain in particular a nonlinear Doob–Meyer decomposition for [Formula: see text]-submartingales and a downcrossing inequality which implies their regularity in time. As a consequence of these results, we also obtain a converse comparison theorem for our class of BSDEs. Finally, we provide a dual representation for the corresponding dynamic risk measures, and study the properties of their inf-convolution, giving several explicit examples.

Fukushima type decomposition for semi-Dirichlet forms

We present a Fukushima type decomposition in the setting of general quasi-regular semi-Dirichlet forms. The decomposition is then employed to give a transformation formula for martingale additive functionals. Applications of the results to some concrete examples of semi-Dirichlet forms are given at the end of the paper. We discuss also the uniqueness question about the Doob-Meyer decomposition on optional sets of interval type.

On &lt;i&gt;g&lt;/i&gt;&gt;Evaluations with L&lt;sup&gt;&lt;i&gt;p&lt;/i&gt;&lt;/sup&gt; Domains under Jump Filtration

Given p ∈ (1, 2), the unique Lp solutions of backward stochastic differential equations with jumps (BSDEJs) allow us to extend the notion of g−evaluations, in particular g−expectations, to the jump case with Lp domains. We explore many important properties of the extended g−evaluations including optional sampling, upcrossing inequality, Doob-Meyer decomposition, generator representation and Jensen’s inequality. Most of these results are important for the further development of jump-filtration consistent nonlinear expectations with Lp domains.

A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using It\^o's calculus and ideas of Arnold, Carlen and Ju \cite{Arnoldcarlenju}, we obtain moreover a new Bakry Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is non-intrisic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide examples where the classic Bakry Emery criterion fails, but our non-intrisic criterion applies without modifying the law of the diffusion process.