Skorohod Integral Research Articles

Local linear estimator for fractional diffusions

In this paper, we investigate the nonparametric local linear estimator for the drift function of stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter [Formula: see text]. The drift function is one-sided dissipative Lipschitz that ensures the ergodic property of the SDE. We derive the strong consistency of the proposed estimator with proper bandwidth selectors associated with the determined Hurst parameter [Formula: see text]. The main tools are the ergodic theorem, Malliavin calculus, and a maximum inequality for It[Formula: see text]–Skorohod integrals.

Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈ (0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM)∂u(t,x)∂t=Lu(t,x)+u(t,x)dBH,λ(t)dt,(t,x)∈R+×Rd, where L is the infinitesimal generator of some time-homogeneous Markov process {Xt}t≥0 and {BH,λ(t)}t∈R is a tempered fractional Brownian motion with Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0 which is independent of {Xt}t≥0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formulau(t,x)=Ex[f(Xt)]eBH,λ(t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that limt→∞⁡ln⁡‖u(t,⋅)‖pg(t)=0 in the almost surely sense where p∈R and the function g:R+→R+ grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals.

Normal Approximation of Kabanov–Skorohod Integrals on Poisson Spaces

AbstractWe consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process.

Order estimate of functionals related to fractional Brownian motion

Nualart and Yoshida (2019) presented a general scheme for asymptotic expansion of Skorohod integrals. However, when applying the general theory to a variation of a process related to a fractional Brownian motion, one repeatedly needs estimates of the order of Lp-norms and Sobolev norms of functionals that are complicated as a randomly weighted sum of products of multiple integrals of the fractional Brownian motion. To resolve the difficulties, in this paper, we construct a theory of exponents based on a graphical representation of the structure of the functionals, which enables us to tell an upper bound of the order of functionals by a simple rule. We also show how the exponents change by the actions of the Malliavin derivative and its projection. The exponents are useful in various studies of limit theorems as well as in applications to asymptotic expansion.

A Malliavin Calculus Computation of the Greeks Theta and Vega of Asian Option and Best of Asset Option

We determine the Theta and the Vega sensitivity of Asian Option (AO) and Best of Asset option (BAO) via the properties of Malliavin calculus. These sensitivities which are represented by the Greeks are obtained with skorohod integral and the integration by part technique for stochastic variation of the Malliavin calculus. The weight functions of the Greeks for Asian Option (AO) and the Best of Asset option (BAO) were derived and this was used to determine expressions for the Greeks.

Asymptotic expansion and estimates of Wiener functionals

Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and other random symbols. To specify these random symbols, it is necessary to classify the level of the effect of each term appearing in the stochastic expansion of the variable in question. To solve this problem, we consider a class L of certain sequences (In)n∈N of Wiener functionals and give a systematic way of estimation of the order of (In)n∈N. Based on this method, we introduce a notion of exponent of the sequence (In)n∈N, and investigate the stability and contraction effect of the operators Dun and D on L, where un is the integrand of a Skorohod integral. After constructed these machineries, we derive asymptotic expansion of the variation having anticipative weights. An application to robust volatility estimation is mentioned.

Stochastic integrals and Gelfand integration in Fréchet spaces

We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida–Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space.

Exponential tightness of a family of Skorohod integrals

Exponential tightness of a family of Skorohod integrals is studied in this paper. We first provide a counterexample to illustrate that in general the exponential tightness with speed ε similar to Itô integral does not hold, even for any speed εα with α>0. Then, some characterizations of this subject are given. Application is also provided to illustrate our results.

Sensitivity of option prices via fuzzy Malliavin calculus

In this paper, we define the Malliavin derivative and the Skorohod integral for fuzzy stochastic processes for the sensitivity analysis of the option prices. The price dynamics could be modelled by equations involving uncertainties that lead to modelling with fuzzy processes in equations. We introduce a fuzzy stochastic differential equation for financial pricing models under an asset price that follows a fuzzy stochastic process. If there is no explicit solution to the equations, the sensitivity analysis, and the estimates are computationally expensive, and the result will be inaccurate due to the estimates and the derivative of the payoff function. The Malliavin calculus and the integration by parts formula in fuzzy space can be exploited to bypass the derivative of the payoff function.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

Limit theorems for singular Skorohod integrals

In this paper we prove the convergence in distribution of sequences of It\^o and Skorohod integrals with integrands having singular asymptotic behavior. These sequences include stochastic convolutions and generalize the example $\sqrt n\int _0^1 t^n B_tdB_t$ first studied by Peccati and Yor in 2004.

Skorohod and rough integration for stochastic differential equations driven by Volterra processes

Given a solution Y to a rough differential equation (RDE), a recent result (Ann. Probab. 47 (2019) 1–60) extends the classical Itô-Stratonovich formula and provides a closed-form expression for ∫Y∘dX−∫YdX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X, where X is a Gaussian process with finite p-variation less than 3. In this paper, we extend this result to Gaussian processes with finite p-variation such that 3≤p<4. The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kernels satisfying a natural condition, which however still allows the result to encompass many standard examples, including fractional Brownian motion with Hurst parameter H>14. As an application we recover Itô formulas in the case where the vector fields of the RDE governing Y are commutative.

Backward stochastic differential equations driven by fractional noise with non-Lipschitz coefficients

In this paper, we study the backward stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1∕2, where the coefficient is non-Lipschitz continuous and the stochastic integral is the Skorohod integral.

Cauchy Formula for Affine SDE with Skorohod Integral

The Cauchy representation formula enables to obtain a solution to a nonhomogeneous equation with the help of the linear homogeneous part solution and nonhomogeneities. In case of known asymptotics of the linear homogeneous part solution, we can establish some properties of behavior of a solution to nonhomogeneous equation. For diffusion equations the Cauchy formula was ascertained and successfully applied for different cases. In this paper, the Cauchy representation formula for a solution to a multidimentional affine SDE with the Skorohod integral is established. Conditions for inclusion of the solution into generalized Wiener functional spaces are given.

Non-Lipschitz anticipated backward stochastic differential equations driven by fractional Brownian motion

In this paper, we obtain the existence and uniqueness of the solutions of anticipated backward stochastic differential equations driven by fractional Brownian motion under the non-Lipschitz condition, where Hurst index H is greater than 1∕2 and the associated stochastic integral is the Skorohod integral.

Asymptotics for discrete time hedging errors under fractional Black–Scholes models

In this paper, we analyze the hedging errors due to discrete time trading, the exact growth rates of discrete time hedging errors under fractional Black–Scholes (BS) models with the fractional pathwise integral and the fractional Wick–Itô–Skorohod integral are investigated respectively.

Drift parameter estimation for nonlinear stochastic differential equations driven by fractional Brownian motion

ABSTRACTWe derive the strong consistency of the least squares estimator (LSE) for the drift coefficient of a fractional stochastic differential system. The drift coefficient is one-sided dissipative Lipschitz and the driving noise is additive and fractional with Hurst parameter . We assume that continuous observation is possible. The main tools are ergodic theorem and Malliavin calculus. As a by-product, we derive a maximum inequality for Skorohod integrals, which plays an important role to obtain the strong consistency of the LSE.

A Stratonovich–Skorohod integral formula for Gaussian rough paths

Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t}=V(Y_{t})\circ\mathrm{d}\mathbf{X}_{t}$, we present a closed-form correction formula for $\int Y\circ\mathrm{d}\mathbf{X}-\int Y\,\mathrm{d}X$, that is, the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-Itô conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p<3$, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with $H>\frac{1}{3}$. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in $L^{2}(\Omega)$ by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional Young–Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a rebalancing of terms.

Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance

This paper develops mixed-normal approximations for probabilities that vectors of multiple Skorohod integrals belong to random convex polytopes when the dimensions of the vectors possibly diverge to infinity. We apply the developed theory to establish the asymptotic mixed normality of the realized covariance matrix of a high-dimensional continuous semimartingale observed at a high-frequency, where the dimension can be much larger than the sample size. We also present an application of this result to testing the residual sparsity of a high-dimensional continuous-time factor model.

Asymptotic analysis for hedging errors in models with respect to geometric fractional Brownian motion

ABSTRACTIn this paper, we investigate asymptotic behaviour for rates of discrete time hedging errors in models with respect to geometric fractional Brownian motion with Hurst parameter . We analyse the rates of hedging errors due to discrete time trading when the true strategy is known, and the exact rates of convergence and limit distributions of hedging errors for models with respect to the fractional pathwise integral and the fractional Wick–Itô–Skorohod integral are investigated.