In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued Lévy process (which is not required to satisfy any moment condition), with respect to a cylindrical Lévy processes with (weak) second moments and with respect to a Lévy-valued random martingale measures with finite second moment. As an application of our theory of integration we prove existence and uniqueness of solutions for stochastic stochastic partial differential equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients.
We introduce and discuss Lévy-type cylindrical martingale problems on separable reflexive Banach spaces. Our main observations are the following: Cylindrical martingale problems have a one-to-one relation to weak solutions of stochastic partial differential equations. Moreover, well-posed problems possess the strong Markov property and a Cameron-Martin-Girsanov-type formula holds. As applications, we derive existence and uniqueness results.
We prove limit theorems for cylindrical martingale problems associated to Lévy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by Lévy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential …
Martingale representation theorem for set-valued martingales was proposed by M. Kisielewicz [J. Math. Anal. Appl. 2014]. We shall prove that the result holds only for very special case: the set-valued martingale degenerates to the point-valued one. A revised representation theorem for a special kind of non-degenerate set-valued martingales is presented.
In this paper we give necessary and sufficient conditions for a cylindrical continuous local martingale to be the stochastic integral with respect to a cylindrical Brownian motion. In particular we consider the class of cylindrical martingales with closed operator-generated covariations. We also prove that for every cylindrical continuous local martingale $M$ there exists a time change $τ$ such that $M\circ τ$ is Brownian representable.
In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $γ$-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gund…
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of normalized maxima of partial sums.
We prove maximal inequalities for $L_q$-valued martingales obtained by stochastic integration with respect to compensated random measures. A version of these estimates for integrals with respect to compensated Poisson random measures were first obtained by Dirksen (arXiv:1208:3885) using arguments based on inequalities for sums of independent Banach-space-valued random variables, geometric properties of Banach spaces, and decoupling inequalities. Our proofs are completely different and rely almost exclusively on classical stochastic analysis for real semimartingales.
A cylindrical Levy process does not enjoy a cylindrical version of the semi-martingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Levy processes in Hilbert spaces. The space of admissible integrands consists of adapted stochastic processes with values in the space of Hilbert-Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as …
We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs approximates the original solution. We also consider the hedging problem in bond markets and prove that, for an approximately attainable contingent claim, the sequence of locally risk-minimizing strategies based on small markets converges to the generalized hedging strategy.
In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double …
Let (f_n) be a mean zero vector valued martingale sequence. Then there exist vector valued functions (d_n) from [0,1]^n such that int_0^1 d_n(x_1,...,x_n) dx_n = 0 for almost all x_1,...,x_{n-1}, and such that the law of (f_n) is the same as the law of (sum_{k=1}^n d_k(x_1,...,x_k)) . Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales. This paper may be found at http://math.missouri.edu/~stephen/preprints
We study stochastic Korteweg - de Vries equation driven by Lévy noise consisting of the compensated time homogeneous Poisson random measure and a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. In proof of the existence theorem we use the Galerkin approximation and several auxiliary results suitable for the problem considered.
This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $ϕ$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapień, McConnell, and Woyczyński. The vast majority of the assertions presented…
We introduce a stochastic integral with respect to cylindrical Lévy processes with finite $p$-th weak moment for $p\in [1,2]$. The space of integrands consists of $p$-summing operators between Banach spaces of martingale type $p$. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process.
This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of $S=\sum_{i=1}^n X_i$ given some $σ$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0 $ as the vector $S$. Assuming that the conditional variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, an…
In this paper non-asymptotic exponential and moment estimates are derived for tail of distribution for discrete time martingale and martingale transform by means of martingale differences in the terms of moments and tails of distributions of summands and multipliers. We show also the exactness of obtained estimations and consider some applications in the theory of limit theorem for Banach space valued martingales.
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (2001) and Fan et al. (2012). In the multidimensional ca…
We introduce the local martingale problem associated to semilinear stochastic evolution equations driven by a cylindrical Wiener process and establish a one-to-one correspondence between solutions of the martingale problem and (analytically) weak solutions of the stochastic equation. We also prove that the solutions of well-posed equations are strong Markov processes. We apply our results to semilinear stochastic equations with additive noise where the semilinear term is merely measurable and to stochastic reaction-diffusion equations with Hölder continuous multiplicative noise.
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale …
In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given.
In this paper we consider (probability-)measure valued processes, which we call MVMs, which have a natural martingale structure. Following previous work of Eldan and Cox-Källblad, these processes are known to have a close connection to the solutions to the Skorokhod Embedding Problem. In this paper, we consider properties of these processes, and in particular, we are able to show that the MVMs connected to the Bass and Root embeddings have natural measure-valued analogues which also possess natural optimality properties. We also introduce a new MVM which is a generalisation of both the Bass an…
Let $X$ be a $\mathbb{C}$-valued random variable with the property that $$X \ \text{ has the same law as }\ \sum_{j\ge1} T_j X_j$$ where $X_j$ are i.i.d.\ copies of $X$, which are independent of the (given) $\mathbb{C}$-valued random variables $ (T_j)_{j\ge1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins' martingale with complex parameter.
A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + σ(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $σ:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and $\mathcal{L}$ the $L^2$-generator of a Lévy process. Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large $t$) at most a precise exponential rate for the $\mathcal{L}$; and grows in time at most a precise…
Let $Φ$ be a nuclear space and let $Φ'_β$ denote its strong dual. In this paper we introduce sufficient conditions for a cylindrical process in $Φ'$ to have a version that is a $Φ'_β$-valued continuous or cádlág process. We also establish sufficient conditions for the existence of such a version taking values and having finite moments in a Hilbert space continuously embedded in $Φ'_β$. Finally, we apply our results to the study of properties of cylindrical martingales in $Φ'$.
We show that the canonical decomposition (comprising both the Meyer-Yoeurp and the Yoeurp decompositions) of a general $X$-valued local martingale is possible if and only if $X$ has the UMD property. More precisely, $X$ is a UMD Banach space if and only if for any $X$-valued local martingale $M$ there exist a continuous local martingale $M^c$, a purely discontinuous quasi-left continuous local martingale $M^q$, and a purely discontinuous local martingale $M^a$ with accessible jumps such that $M = M^c + M^q + M^a$. The corresponding weak $L^1$-estimates are provided. Important tools used in the…
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustra…
Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [21]. We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck, Jourdain, Margheriti, and Pammer [5], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w…
In this paper, we study the connections between three concepts - the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo (bounded mean oscillation) coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these two equivalent conditions are satisfied. Finally, we apply our results on l…
We prove sharp maximal inequalities for $L^q$-valued stochastic integrals with respect to any Hilbert space-valued local martingale. Our proof relies on new Burkholder-Rosenthal type inequalities for martingales taking values in an $L^q$-space.
In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of Hukuhara difference between sets, in order to compensate the lack of "inverse" operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental i…
Consider a real-valued branching random walk in the boundary case. Using the techniques developed by Aïdékon and Shi [5], we give two integral tests which describe respectively the lower limits for the minimal position and the upper limits for the associated additive martingale.
For a Hilbert space valued martingale $(f_n)$ and an adapted sequence of positive random variables $(w_n)$, we show the weighted Davis type inequality \[ \mathbb{E} \Bigl( |f_0| w_0 + \frac{1}{4} \sum_{n=1}^{N} \frac{|df_n|^2}{f^*_n} w_n \Bigr) \leq \mathbb{E} ( f^*_N w^*_N). \] This inequality is sharp and implies several results about the martingale square function. We also obtain a variant of this inequality for martingales with values in uniformly convex Banach spaces.
In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach spa…
Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $σ$-field of the invariant events by all reflec…
In this work, we introduce a theory of stochastic integration with respect to symmetric $α$-stable cylindrical Lévy processes. Since $α$-stable cylindrical Lévy processes do not enjoy a semi-martingale decomposition, our approach is based on a decoupling inequality for the tangent sequence of the Radonified increments. This approach enables us to characterise the largest space of predictable Hilbert-Schmidt operator-valued processes which are integrable with respect to an $α$-stable cylindrical Lévy process as the collection of all predictable processes with paths in the Bochner space $L^α$. W…
We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.
In this work stochastic integration with respect to cylindrical Levy processes with weak second moments is introduced. It is well known that a deterministic Hilbert-Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Levy process by random Hilbert-Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic int…
In this paper we consider Meyer-Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$, any $X$-valued $L^p$-martingale $M$ has a unique decomposition $M = M^d + M^c$ such that $M^d$ is a purely discontinuous martingale, $M^c$ is a continuous martingale, $M^c_0=0$ and \[ \mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_{p,X} \mathbb E \|M_{\infty}\|^p. \] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-l…
Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given.
The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with t…
We consider the problem of finding a real valued martingale fitting specified marginal distributions. For this to be possible, the marginals must be increasing in the convex order and have constant mean. We show that, under the extra condition that they are weakly continuous, the marginals can always be fitted in a unique way by a martingale which lies in a particular class of strong Markov processes. It is also shown that the map that this gives from the sets of marginal distributions to the martingale measures is continuous. Furthermore, we prove that it is the unique continuous method of fi…
Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration of measure on product spaces for separately Lipschitz functions are presented, including ones concerning the central moments of the norm of the sums of independent random vectors in any separable Banach space.
Given the univariate marginals of a real-valued, continuous-time martingale, (respectively, a family of measures parameterised by $t \in [0,T]$ which is increasing in convex order, or a double continuum of call prices) we construct a family of pure-jump martingales which mimic that martingale (respectively, are consistent with the family of measures, or call prices). As an example, we construct a fake Brownian motion. Then, under a further `dispersion' assumption, we construct the martingale which (within the family of martingales which are consistent with a given set of measures) has the smal…
A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in the 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of the sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on the infinite-dimensional martingales, most of the results seem to be new even for the one-dimensional one…
Positive $T$-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval $T=[0,1]$ and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of us…
