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In this paper we study the stochastic control problem of a partially observed (multidimensional) stochastic system driven by both Brownian motions and fractional Brownian motions. In the absence of the powerful tool of Girsanov transformation, we introduce and study new stochastic processes which are used to transform the original problem to a "classical one"". The adjoint backward stochastic differential equations and the necessary condition satisfied by the optimal control (maximum principle) are obtained. [ABSTRACT FROM AUTHOR]

A variant of the optimal control problem is considered which is nonstandard in that the performance index contains 'stochastic' integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be 'fitted' with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes. [ABSTRACT FROM AUTHOR]

Consider a periodic, mean-reverting Ornstein–Uhlenbeck process X = { X t , t ≥ 0 } of the form d X t = L (t) + α X t d t + d B t H , t ≥ 0 , where L (t) = ∑ i = 1 p μ i ϕ i (t) is a periodic parametric function, and { B t H , t ≥ 0 } is a fractional Brownian motion of Hurst parameter 1 2 ≤ H < 1 . In the "ergodic" case α < 0 , the parametric estimation of (μ 1 , ... , μ p , α) based on continuous-time observation of X has been considered in Dehling et al. (Stat Inference Stoch Process 13:175–192, 2010; Stat Inference Stoch Process 20:1–14, 2016) for H = 1 2 , and 1 2 < H < 1 , respectively. In this paper we consider the "non-ergodic" case α > 0 , and for all 1 2 ≤ H < 1 . We analyze the strong consistency and the asymptotic distribution for the estimator of (μ 1 , ... , μ p , α) when the whole trajectory of X is observed. [ABSTRACT FROM AUTHOR]

We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by α -stable processes for α ∈ (0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is P -a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations. [ABSTRACT FROM AUTHOR]

A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as $d{X_{t}}=\theta (\mu +{X_{t}})dt+d{B_{t}^{a,b}}$, $t\ge 0$, with unknown parameters $\theta >0$, $\mu \in \mathbb{R}$ and $\alpha :=\theta \mu $, whereas ${B^{a,b}}:=\{{B_{t}^{a,b}},t\ge 0\}$ is a weighted fractional Brownian motion with parameters $a>-1$, $|b|

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that the Lipschitz coefficients of the remaining parts are small. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with H > 1 / 2 , and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model. [ABSTRACT FROM AUTHOR]

We define in this work a notion of Young differential inclusion dzt ? F(zt) dxt, for an a-H?older control x, with a > 1/2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, ?-H?older continuous set-valued map on the interval [0, 1] has a selection with finite p-variation, for p > 1/?. We also give a notion of solution to the rough differential inclusion dzt ? F(zt) dt + G(zt) dXt, for an a-H?older rough path X with a ? (1/3, 1/2], a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values. [ABSTRACT FROM AUTHOR]

In this paper we explain the relationship between Stieltjes type integrals of Young, Dushnik and Kurzweil for functions with values in Banach spaces. To this aim also several new convergence theorems will be stated and proved.

A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as ..., t ≥ 0, with unknown parameters θ > 0, μ ∈ ℝ and α := θμ, whereas ... is a weighted fractional Brownian motion with parameters a > -1, b < 1, b < a + 1. Least square-type estimators (...) and (...) are provided, respectively, for (θ,μ) and (θ, α) based on a continuous-time observation of {Xt, t ∈ [0, T ]} as T → ∞. The strong consistency and the joint asymptotic distribution of (...) and (...) are studied. Moreover, it is obtained that the limit distribution of ... is a Cauchy-type distribution, and ... and ... are asymptotically normal. [ABSTRACT FROM AUTHOR]

We study a problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as d X t = θ (μ + X t) d t + d G t , t ≥ 0 with unknown parameters θ > 0 , μ ∈ R and α : = θ μ , where G is a Gaussian process. We provide least square-type estimators (θ ~ T , μ ~ T) and (θ ~ T , α ~ T) , respectively, for (θ , μ) and (θ , α) based a continuous-time observation of { X t , t ∈ [ 0 , T ] } as T → ∞ . Our aim is to derive some sufficient conditions on the driving Gaussian process G in order to ensure the strongly consistency and the joint asymptotic distribution of (θ ~ T , μ ~ T) and (θ ~ T , α ~ T) . Moreover, we obtain that the limit distribution of θ ~ T is a Cauchy-type distribution, and μ ~ T and α ~ T are asymptotically normal. We apply our result to fractional Vasicek, subfractional Vasicek and bifractional Vasicek processes. This work extends the results of El Machkouri et al. (J Korean Stat Soc 45:329–341, 2016) studied in the case where μ = 0 . [ABSTRACT FROM AUTHOR]

We show that geometric integrals of the type ∫ Ω f d g 1 ∧ d g 2 can be defined over a two-dimensional domain Ω when the functions f , g 1 , g 2 : R 2 → R are just Hölder continuous with sufficiently large Hölder exponents and the boundary of Ω has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R. Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map g = (g 1 , g 2) when f (x) = F (x , g (x)) with F sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper. [ABSTRACT FROM AUTHOR]

We provide a draft of a theory of geometric integration of "rough differential forms" which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by Züst, and provides also an alternative (and more geometric) view on the standard construction of rough paths. To simplify the exposition, we limit ourselves to integration of rough k-forms with k ≤ 2 . [ABSTRACT FROM AUTHOR]

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a γ -Hölder continuous function 𝜃 with γ ∈ (2 3 , 1). Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to 𝜃 is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374]. [ABSTRACT FROM AUTHOR]

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1 / 2 < H < 1. Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., H = 1/2), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind { X t , t ≥ 0 } defined as d X t = 𝜃 (μ + X t) d t + d Y t , G (1) , t ≥ 0 , with unknown parameters 𝜃 > 0 and μ ∈ ℝ , where Y t , G (1) : = ∫ 0 t e − s d G a s with a t = He t H , and { G t , t ≥ 0 } is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of 𝜃 and μ based on the continuous-time observations { X t , t ∈ [ 0 , T ] } as T → ∞ , we impose some technical conditions on the process G , which are satisfied, for instance, if G is a fractional Brownian motion with Hurst parameter H ∈ (0 , 1) , G is a subfractional Brownian motion with Hurst parameter H ∈ (0 , 1) or G is a bifractional Brownian motion with Hurst parameters (H , K) ∈ (0 , 1) × (0 , 1 ]. Our method is based on pathwise properties of { X t , t ≥ 0 } and { Y t , G (1) , t ≥ 0 } proved in the sequel. [ABSTRACT FROM AUTHOR]

We prove the exponential stability of the zero solution of a stochastic differential equation with a Hölder noise, under the strong dissipativity assumption. As a result, we also prove that there exists a random pullback attractor for a stochastic system under a multiplicative fractional Brownian noise. [ABSTRACT FROM AUTHOR]

In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leão, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t. Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite p -variation (p ≥ 2) regularity is also discussed. Under stronger regularity conditions in the sense of finite (p , q) -variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established. [ABSTRACT FROM AUTHOR]

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion withH> 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that the local time associated with symmetric α -stable processes is of bounded p -variation for any p > 2 α − 1 partly based on Barlow’s estimation of the modulus of the local time of such processes. The fact that the local time is of bounded p -variation for any p > 2 α − 1 enables us to define the integral of the local time ∫ − ∞ ∞ ▿ − α − 1 f ( x ) d x L t x as a Young integral for less smooth functions being of bounded q -variation with 1 ≤ q < 2 3 − α . When q ≥ 2 3 − α , Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric α -stable processes for 2 3 − α ≤ q < 4 . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of a mild solution for neutral impulsive fractional stochastic delay differential equations driven by Brownian motion, and fractional Brownian motion with the Hurst parameter H ∈ (1/2,1), by using Banach fixed point theorem in a Hilbert space [ABSTRACT FROM AUTHOR]

We investigate the process of eigenvalues of a fractional Wishart process defined by N = B ⁎ B , where B is the matrix fractional Brownian motion recently studied in [18] . Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter H ∈ ( 1 / 2 , 1 ) , we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process , which constitutes the family of fractional dilations of the free Poisson distribution. [ABSTRACT FROM AUTHOR]

In 1962 Lamperti introduced a transformation that associates to every non-trivial self-similar process a strictly stationary process. This transform has been widely studied for Gaussian processes and in particular for fractional Brownian motion. Our aim is to analyze various properties of the Lamperti transform of the fractional Brownian sheet. We give the stochastic differential equation satisfied by this transform and we represent it as a series of independent Ornstein-Uhlenbeck sheets. [ABSTRACT FROM AUTHOR]

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu (J Theor Probab 27:220-248, 2014). Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used. [ABSTRACT FROM AUTHOR]

We consider the ordinary differential equation (ODE) d x t = b ( t , x t ) d t + d w t where w is a continuous driving function and b is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path w on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of ρ - irregularity and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H ∈ ( 0 , 1 ) , we prove that almost surely the ODE admits a solution for all b in the Besov–Hölder space B ∞ , ∞ α + 1 with α > − 1 / 2 H . If α > 1 − 1 / 2 H then the solution is unique among a natural set of continuous solutions. If H > 1 / 3 and α > 3 / 2 − 1 / 2 H or if α > 2 − 1 / 2 H then the equation admits a unique Lipschitz flow. Note that when α < 0 the vector field b is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply. [ABSTRACT FROM AUTHOR]

In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-valued continuous semimartingales. Under finite (p; q)- variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition ... Here, Xt = {X(s); 0 ≤ s ≤ t} is the continuous semimartingale path up to time t ∊ [0; T], ▽h is the horizontal derivative, (▽xw Fs)(xXs) is a weak derivative of F with respect to the terminal value x of the modified path xXs and ▽wFs(Xs) = (▽xw Fs)(xXs)| x=X(s). The double integral is interpreted as a spacetime 2D-Young integral with differential d(s;x)ℓx(s), where ℓ is the local-time of X. Under less restrictive joint variation assumptions on (▽xw Ft)(xXt), functional Itô formulas are established when X is a stable symmetric process. Singular cases when x → (▽xwFt)(xXt) is smooth off random bounded variation curves are also discussed. The results of this paper extend previous change of variable formulas in Cont and Fournié (2013) and also Peskir (2005), Feng and Zhao (2006) and Elworthy et al. (2007) in the context of path-dependent functionals. In particular, we provide a pathwise path-dependent version of the classical Föllmer-Protter-Shiryaev formula for continuous semimartingales given by Follmer et al. (1995). [ABSTRACT FROM AUTHOR]

We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional Hölder continuous Gaussian processes of order γ ∈ ( 1 / 2 , 1 ) . Using the stochastic calculus with respect to the Young integral we show that these eigenvalues do not collide at any time with probability one. When the matrix process has entries that are fractional Brownian motions with Hurst parameter H ∈ ( 1 / 2 , 1 ) , we find a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding matrix fractional Brownian motion. A new generalized version of the Itô formula for the multidimensional fractional Brownian motion is first established. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first study an ordinary integral equation, where the integral is defined in the Young sense, and we prove an existence result and the boundedness of the solutions. Then, we apply this result pathwise to solve the stochastic problem. [ABSTRACT FROM AUTHOR]

Abstract: The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with , in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory – existence, uniqueness, convergence of the Euler scheme, flow property … – which are spread out among several articles. [ABSTRACT FROM AUTHOR]

In this note, we consider an optimal control problem associated to a differential equation driven by a Hölder continuous function g of index β > 1/2. We split our study into two cases. If the coefficient of dgt does not depend on the control process, we prove an existence theorem for a slightly generalized control problem, that is we obtain a literal extension of the corresponding situation for ordinary differential equations. If the coefficient of dgt depends on the control process, we also prove an existence theorem but here we are obliged to restrict the set of controls to sufficiently regular functions. [ABSTRACT FROM AUTHOR]

In 1936, L. C. Young proved that the Riemann-Stieltjes integral ∫ba ƒ dg exists, if ƒ ∈ BVp, g ∈ BVq, 1/p + 1/p > 1 and ƒ ,g do not have common discontinuous points. In this note, using Henstock's approach, we prove that ∫ba ƒ dg still exists without assuming the condition on discontinuous points. Some convergence theorems axe also proved. [ABSTRACT FROM AUTHOR]