Lévy area for Gaussian processes: A double Wiener-Itô integral approach

Let {X₁(t)}_0≤t≤1 and {X₂(t)}_0≤t≤1 be two independent continuous centered Gaussian processes with covariance functions R₁ and R₂. We show that if the covariance functions are of finite p-variation and q-variation respectively and such that p⁻¹+q⁻¹>1, then the Lévy area can be defined as a double Wiener-Itô integral with respect to an isonormal Gaussian process induced by X₁ and X₂. Moreover, some properties of the characteristic function of that generalised Lévy area are studied.     

Distribution and characteristic functions for correlated complex Wishart matrices

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t₁, t₂, where t₁<t₂. In particular, we derive the following results: the joint density of the elements of A(t₁), A(t₂), the joint density of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂), the characteristic function of the elements of A(t₁), A(t₂), the characteristic function of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂). In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Dynamic Markov bridges motivated by models of insider trading

Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X₁=Z₁. We call X a dynamic bridge, because its terminal value Z₁ is not known in advance. We compute its semimartingale decomposition explicitly under both its own filtration F^X and the filtration F^X,Z jointly generated by X and Z. Our construction is heavily based on parabolic partial differential equations and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading that can be viewed as a non-Gaussian generalization of the model of Back and Pedersen (1998) [3], where the insider’s additional information evolves over time.     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.     

The influence of a log-type small ball factor in the study of the lower classes

Let {BH₁,H₂(t₁,t₂),t₁?0,t₂?0} be a fractional Brownian sheet with indexes 0<H₁,H₂<1. When H₁=H₂:=H, there is a logarithmic factor in the small ball function of the sup-norm statistic of BH,H. First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of BH,H. Then we characterize the sufficiency part of the lower classes of these statistics by an integral test. Finally, when we consider the sup-norm statistic, the influence of the log-type small ball factor in the necessity part is measured by a second integral test.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

The influence of the nonrecent past in prediction for stochastic processes

Consider the stochastic processes X₁, X₂,… and Λ₁, Λ₂,… where the X process can be thought of as observations on the Λ process. We investigate the asymptotic behavior of the conditional distributions of X_t+v given X₁,…, X_t and Λ_t+v given X₁,…, X_t with regard to their dependency on the “early” part of the X process. These distributions arise in various time series and sequential decision theory problems. The results support the intuitively reasonable and often used (as a basic tenet of model building) assumption that only the more recent past is needed for near optimal prediction.     

Quasi sure quadratic variation of local times of smooth semimartingales

In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑_Δn(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

CHAOS EXPANSION OF LOCAL TIME OF FRACTIONAL BROWNIAN MOTIONS

We find the chaos expansion of local time ? _T ^(H)(x,·) of fractional Brownian motion with Hurst coefficient H∈(0,1) at a point x∈R ^d . As an application we show that when H ₀ d<1 then ? _T ^(H)(x,·)∈L ²(μ). Here μ denotes the probability law of B ^(H) and H ₀=max{H ₁,…,H _d }. In particular, we show that when d=1 then ? _T ^(H)(x,·)∈L ²(μ) for all H∈(0,1).     

Hamilton-Jacobi semi-groups in infinite dimensional spaces

Let (X,ρ) be a Polish space endowed with a probability measure μ. Assume that we can do Malliavin Calculus on (X,μ). Let be a pseudo-distance. Consider QtF(x)=infy∈X{F(y)+d²(x,y)/2t}. We shall prove that QtF satisfies the Hamilton-Jacobi inequality under suitable conditions. This result will be applied to establish transportation cost inequalities on path groups and loop groups in the spirit of Bobkov, Gentil and Ledoux.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Local independence of fractional Brownian motion

Let _σ(t,t′)${\sigma }_{(t,t^{\prime })}$ be the sigma-algebra generated by the differences _Xs−_Xs′$X_s-X_{s^{\prime }}$ with s,s^′∈(t,t^′)$s,s^{\prime }\in (t,t^{\prime })$, where _{(Xt)−∞<t<∞}${\left(X_t\right)}_{-\infty <t<\infty }$ is the fractional Brownian motion with Hurst index H∈(0,1)$H\in (0,1)$. We prove that for any two distinct timepoints _t1$t_1$ and _t2$t_2$ the sigma-algebras _{σ(t1−ε,t1+ε)}${\sigma }_{(t_1-\epsilon ,t_1+\epsilon )}$ and _{σ(t2−ε,t2+ε)}${\sigma }_{(t_2-\epsilon ,t_2+\epsilon )}$ are asymptotically independent as ε↘0$\epsilon \searrow 0$. We show the independence in the strong sense that Shannon’s mutual information between the two σ$\sigma$-algebras tends to zero as ε↘0$\epsilon \searrow 0$. Some generalizations and quantitative estimates are also provided.