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Let (V t) t ∈ [ 0 , L ] be a stochastic process with quadratic variation on a probability space (Ω , F , P) and Q (∋ 0) a dense subset of [ 0 , L ] , where [ 0 , L ] is regarded as the infinite interval [ 0 , ∞) when L = ∞. First, we introduce the L 0 (Ω) -module D Q (V) of V-differentiable noncausal processes on Q and V-derivative operator D V , Q = d Q d Q V defined on D Q (V) , which enjoys the modularity: D V , Q (α X + β Y) = α D V , Q X + β D V , Q Y for any X , Y ∈ D Q (V) and α , β ∈ L 0 (Ω). Second, we show that the class Q V , Q = { X ∈ D Q (V) | d [ X ] Q , t d [ V ] Q , t = | d Q X t d Q V t | 2 } forms an L 0 (Ω) -module, where [ ] Q , t stands for the quadratic variation on Q. As a result, we have the isometry: 〈 X , Y 〉 Q , t = 〈 D V , Q X , D V , Q Y 〉 L 2 ([ 0 , t ] , [ V ]) for any X , Y ∈ Q V , Q , where 〈 , 〉 Q , t stands for the quadratic covariation on Q. Finally, we present universal properties and examples of the stochastic integral I with D V , Q ∘ I = i d D (I) . This result is essentially used for solving the identification problem from the stochastic Fourier coefficients. [ABSTRACT FROM AUTHOR]

We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark–Hausmann–Bismut–Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation at the first place. We demonstrate the hedging strategy as explicitly as possible with some examples of the payoff function such as those used in exotic options, whose value at maturity depends on the prices over the entire time horizon. [ABSTRACT FROM AUTHOR]

In this paper, we prove a functional central limit theorem (FCLT) for a stochastic epidemic model with varying infectivity and general infectious periods recently introduced in R. Forien et al. [Epidemic models with varying infectivity, SIAM J. Appl. Math. 81 (2021), pp. 1893–1930]. The infectivity process (total force of infection at each time) is composed of the independent infectivity random functions of each infectious individual, which starts at the time of infection. These infectivity random functions induce the infectious periods (as well as exposed, recovered or immune periods in full generality), whose probability distributions can be very general. The epidemic model includes the generalized non–Markovian SIR, SEIR, SIS, SIRS models with infection-age dependent infectivity. In the FCLTs for the generalized SIR and SEIR models, the limits of the diffusion-scaled fluctuations of the infectivity and susceptible processes are a unique solution to a two-dimensional Gaussian-driven stochastic Volterra integral equations, and then given these solutions, the limits for the infected (exposed/infectious) and recovered processes are Gaussian processes expressed in terms of the solutions to those stochastic Volterra integral equations. We also present the FCLTs for the generalized SIS and SIRS models. [ABSTRACT FROM AUTHOR]

We discuss several properties of the generalized Hermite process, which is a non-Gaussian self-similar processes with self-similarity index belonging to the whole interval (0 , 1). We also define Wiener integral with respect to this process and as an application, we define and study its associated Ornstein–Uhlenbeck process. [ABSTRACT FROM AUTHOR]

We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces. [ABSTRACT FROM AUTHOR]

In this article, we study multiplicative SDE with fractional noise in a suitable sense, which can be regarded as a fractional Gruschin type process. Using the transfer principle and fractional integral and derivative operators, two kinds of Bismut type derivative formulae are established in this non-Markovian context under incompatible conditions. As an application, an explicit gradient estimate is derived. [ABSTRACT FROM AUTHOR]

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1 2 . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu and Nualart [Differential equations driven by Hölder continuous functions of order greater than 1/2, Stoch. Anal. Appl. Abel Symp. 2 (2007), pp. 399–413] and Nualart and Saussereau [Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391–409] where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Lévy process. We propose a new approach applying the theory of compensated-covariation stable families of martingales. Our main tool is a representation formula for products of elements of a compensated-covariation stable family, which enables us to consider Lévy processes, with both jumps and Gaussian part. [ABSTRACT FROM AUTHOR]

Higher moment a priori estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral. [ABSTRACT FROM AUTHOR]

In this article, we investigate the L 2 -convergence rate for the discretization error of the Clark-Ocone representation of a random variable ϕ (L T) , where (L t ) 0 ⩽ t ⩽ T is a pure-jump Lévy process. As in the Brownian case, we find that the exact rate is subject to the Sobolev-differentiability index (in the sense of classical Malliavin calculus) of the space to which ϕ (L T) belongs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the representation of a class of non-Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular, the underlying process can be seen as a non-Gaussian extension of the Ornstein–Uhlenbeck process, hence generalizing the representation results of Muravlev, Russian Math. Surveys 66 (2), 2011 as well as Harms and Stefanovits, Stochastic Process. Appl. 129, 2019 to the non-Gaussian case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the question of existence and uniqueness of strong solutions of stochastic differential equations with respect to optional semimartingales. Optional semimartingales have right and left limits (ladlag) but are not necessarily continuous, therefore, defined on un usual probability spaces. Integration theory with respect to optional semimartingales is well developed. However, not much attention is given to stochastic equations of optional semimartingales. An existence and uniqueness result for strong solutions of a very general class of optional stochastic equations with non-Lipschitz coefficients is given. Moreover, potential applications to physics and finance are given. [ABSTRACT FROM AUTHOR]

Equation with the symmetric integral with respect to stochastic measure is considered. For the integrator, we assume only σ-additivity in probability and continuity of the paths. It is proved that the averaging principle holds for this case, the rate of convergence to the solution of the averaged equation is estimated. [ABSTRACT FROM AUTHOR]

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus. We give conditions under which there exist unique solutions of such equations. Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus. As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE. [ABSTRACT FROM AUTHOR]

We study the optimal discretization error of stochastic integrals driven by a multidimensional continuous Brownian semimartingale. In the previous works a pathwise lower bound for the renormalized quadratic variation of the error was provided together with an asymptotically optimal discretization strategy, i.e. for which the lower bound is attained. However the construction of the optimal strategy involved the knowledge about the diffusion coefficient of the semimartingaleunder study. In this work we provide a model-adaptive asymptotically optimal discretization strategy that does not require any prior knowledge about the model. We prove the optimality for quite general class of discretization strategies based on kernel techniques for adaptive estimation and previously obtained optimal strategies that use random ellipsoid hitting times. [ABSTRACT FROM AUTHOR]

In this paper, we prove some random fixed point theorems for Hardy-Rogers self-random operators in separable Banach spaces and, as some applications, we show the existence of a solution for random nonlinear integral equations in Banach spaces. Some stochastic versions of deterministic fixed point theorems for Hardy-Rogers self mappings and stochastic integral equations are obtained. [ABSTRACT FROM AUTHOR]

In this paper we study the properties of a new multidimensional continuous-time stochastic covariance process, the Stochastic Volatility Factor model. Two helpful conditional characteristic functions, one needed for estimation and the other used in the pricing of financial derivatives are provided, together with the properties of the instantaneous dependence structure. Conditions for stationarity, ergodicity and mixing properties of the increments are studied. The estimation of the model is performed using the Continuum-Generalized Method of Moments (CGMM). A simulation exercise is included showing parameters are recovered, confirming identifiability. The model is also calibrated to exemplary financial data. [ABSTRACT FROM AUTHOR]

We study a general impulse control problem with execution delay, which means that there is a time lag Δ > 0 between a decision and the timing of its execution. Since the classical approach is not applicable when the underlying dynamics of the controlled system is a general adapted stochastic process, we suggest general probabilistic tools, namely the Snell envelope and reflected backward stochastic differential equations to model the profitability of the controller, and find the related optimal strategy. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness for stochastic functional differential equations driven by a one-dimensional fractional Brownian motion with Hurst parameter H > 1/2. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral. [ABSTRACT FROM AUTHOR]

We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals. [ABSTRACT FROM PUBLISHER]

In this article, we introduce a Lévy analogue of the spatially homogeneous Gaussian noise of [5], and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered measure μ on, whose density is given byfor a symmetric complex-valued functionh. Without assuming that the Fourier transform of μ is a non-negative function, we identify a large class of integrands with respect to this noise. As an application, we examine the linear stochastic heat and wave equations driven by this type of noise. [ABSTRACT FROM PUBLISHER]

We construct an iterated stochastic integral with respect to fractional Brownian motion (fBm) withH>1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment Theorem of Nualart and Peccati [10], we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart [2]. [ABSTRACT FROM AUTHOR]

Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is an extension of fBm enabling to control the local regularity of the process. It is obtained by replacing the constant Hurst parameterHof fBm by a functionh(t), thus allowing for a finer modelling of various phenomena. In this work we extend to mBm the construction of the Wick–Itô stochastic integral with respect to fBm, as originally proposed in Bender (Stoch. Process. Appl. 104 (2003), pp. 81–106), Bender (Bernouilli 9(6) (2003), pp. 955–983), Biagini et al. (Proceedings of Royal Society, special issue on stochastic analysis and applications, 2004, pp. 347–372) and Elliott and Van der Hoek (Math. Finance 13(2) (2003), pp. 301–330). In that view, a multifractional white noise is defined and used to integrate with respect to mBm a large class of stochastic processes using Wick products. Itô formulas (both for tempered distributions and for functions with sub-exponential growth) are obtained, as well as a Tanaka Formula. [ABSTRACT FROM AUTHOR]

We derive convex comparison inequalities for stochastic integrals of the formand, whereare adapted processes with respect to the filtration generated by a standard Brownian motion, andis an independent Brownian motion. Our method uses forward–backward stochastic integration and the Malliavin calculus, and is also applied to jump–diffusion processes. [ABSTRACT FROM AUTHOR]

Given a random functionand an orthonormal basisin the, we are concerned with the basic properties of its stochastic Fourier coefficient (SFC) which is defined by the stochastic integral with respect to the Brownian motion,. More precisely we are concerned in this note with the problem of reconstructing the functionfrom its SFCs. We will show as our main result the affirmative answer when the random function belongs to a certain restricted but wide enough subclass, which is the class of causal Wiener functionals. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system ofksecond-order nonlinear stochastic partial differential equations with spatial dimension, driven by aq-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and Hölder's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of stochastic partial differential equations (SPDEs). In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span. We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix spanwhich contains this point has a non-void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension. [ABSTRACT FROM AUTHOR]

In this paper, we present an elementary and self-contained proof of the stochastic Fubini theorem, which states that one can interchange a Lebesgue integral and a stochastic integral. The integrability conditions we use are weaker and more natural than the usual conditions in the literature. In particular, we do not need integrability in , and we use -integrability instead of -integrability in the additional parameter. [ABSTRACT FROM AUTHOR]

We consider nonlinear stochastic integrals of Itô-type w.r.t. a family of semimartingales which depend on a spatial parameter. These integrals were introduced by Carmona/Nualart, Kunita, and Le Jan. The extension of the elementary nonlinear integral is based on the condition that the semimartingale kernel has nice continuity properties in the spatial parameter. We investigate the case that continuity is not available and suggest different directions of generalization. This brings us beyond the case that any integral can be approximated by integrals with integrands taking only finitely many values. [ABSTRACT FROM PUBLISHER]

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given. [ABSTRACT FROM AUTHOR]

An (unbounded) operator Ξ on Boson Fock space over L2(R+) is called regular if it is an admissible white noise operator such that the conditional expectations [image omitted] give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation. [ABSTRACT FROM AUTHOR]

In this paper, we consider complex-valued Brownian motion with p-adic time index and the associated abstract Wiener space. We define symmetric stochastic integrals with respect to p-adic Brownian motion. We also provide a sufficient condition for the existence of symmetric stochastic integrals and present a relation to the adjoint of the Malliavin derivatives. [ABSTRACT FROM AUTHOR]

We study several properties of the sub-fractional Brownian motion (fBm) introduced by Bojdecki et al. related to those of the fBm. This process is a self-similar Gaussian process depending on a parameter H ∈ (0, 2) with non stationary increments and is a generalization of the Brownian motion (Bm). The strong variation of the indefinite stochastic integral with respect to sub-fBm is also discussed. [ABSTRACT FROM AUTHOR]

The well-known results on the American put option problem obtained for the classical Black–Scholes model are generalized to the case of a diffusion model with level-dependent volatility. An early exercise premium representation of the value function of the American put option is established. The proof of this result is based on the properties of a stochastic integral with respect to an arbitrary continuous semimartingale over the predictable subsets of its zeros. A nonlinear integral equation is derived for the optimal exercise boundary of the American put option. The uniqueness of the solution of this integral equation is established by introducing the corresponding Cauchy problem with discontinuous right-hand side and with a unique solution in the second order Sobolev space. [ABSTRACT FROM AUTHOR]

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242. [ABSTRACT FROM AUTHOR]

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero. we show that in the one-parameter case the solution is an exponential--thus positive--function while in the two-parameter setting the solution is negative on a non negligible set. [ABSTRACT FROM AUTHOR]