Square-integrable Martingales Research Articles

Convoluted smoothed kernel estimation for drift coefficients in jump-diffusion models

The occurrence of economic policies and other sudden and large shocks often bring out jumps in financial data, which can be characterized through continuous-time jump-diffusion model. In this paper, we will adopt convoluted smoothed approach to estimate unknown drift function of the potentially nonstationary diffusion models with jumps under high frequency sampling data. With Gaussian approximation of locally square-integrable martingales, we will establish large sample properties for the underlying nonparametric estimators. Furthermore, we construct Monte Carlo simulation study through three examples for the better finite-sample properties such as reduction of mean-squared error compared with the existing estimators. Finally, our estimator is verified through the actual data of Shibor in China for better performance.

An Alternative Definition of the Itô Integral for the Hilbert-Schmidt-Valued Stochastic Process

In this paper, using generalized Riemann approach, we give an alternative definition of the Itô integral of a Hilbert--Schmidt-valued stochastic process with respect to a Hilbert space-valued $Q$-Wiener process. We also show that this integral belongs to the space of all continuous square-integrable martingales. <br><br> Використовуючи узагальнений підхід Рімана, наведено альтернативне визначення інтеграла Іто для стохастичного процесу зі значеннями в просторі операторів Гільберта--Шмідта відносно $Q$-вінерівського процесу, що приймає значення у гілбертовому просторі. Також показано, що цей інтеграл належить до простору всіх неперервних квадратично інтегрованих мартингалів.

Useful Martingales for Stochastic Storage Processes with Lévy-Type Input

In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that the (local) martingales obtained are in fact square-integrable martingales which upon dividing by the time index converge to zero almost surely and in L2. The reflected Lévy-type process is considered as an example.

Useful Martingales for Stochastic Storage Processes with Lévy-Type Input

In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that the (local) martingales obtained are in fact square-integrable martingales which upon dividing by the time index converge to zero almost surely and in L 2. The reflected Lévy-type process is considered as an example.

Functional Itô calculus and stochastic integral representation of martingales

We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a nonanticipative "lifting" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Ito processes. By contrast with the Clark-Haussmann-Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.

Strong approximation of locally square-integrable martingales

In this paper, we consider the strong approximation for locally square-integrable martingales. In our results, the limit process may be a process with jumps. This is an extension of the former results.

Existence and continuity with respect to parameter of solutions to stochastic Volterra equations in a plane

In this paper we study stochastic Volterra equations in a plane. These equations contain integrals with respect to fields of locally bounded variation and square-integrable strong martingales. We prove the existence and the uniqueness of solutions of such equations with locally integrable (in some measure) trajectories, assuming that the coefficients of equations possess the Lipschitz property with respect to the functional argument. We prove that a solution of a stochastic Volterra integral equation in a plane is continuous with respect to parameter.

On the Necessity of the Lindeberg Condition for the Normal Convergence of Martingales

We give conditions on square-integrable martingales whose consequence is the “conditional” Lindeberg condition.

On the Convergence of Moments in a Martingale Central Limit Theorem

The paper investigates a limit theorem for randomly stopped, square-integrable martingales in the case when a limiting distribution is a mixture of normal distributions. Together with the weak convergence, the convergence of absolute moments is established.

A recurrence theorem for square-integrable martingales

A recurrence theorem for square-integrable martingales

On a martingale representation theorem of m. h. a. davis-a markovian approach

In this paper, we give a Markovian proof of a well-known result of M. H. A. Davis on representation of square-integrable martingales associated with arbitrary pure-jump processes. The main tool used to replace the original (non-Markovian) set up by a Markovian one is an interesting idea originally due to F. B. Knight. The main analysis then follows the classical line of investigating structures of martingale additive functionals of a Markov process, wherein a Levy system of the Markov process (in the classical sense-due to S. Watanabe) plays a vital role

Invariance principle for integral type functionals of square-integrable martingales

Let X n = { X n ( t): 0 ⩽ t ⩽1}, n ⩾ 0, be a sequence of square-integrable martingales. The main aim of this paper is to give sufficient conditions under which ∫ · 0 f n ( A n ( t), X n ( t)) d X n ( t) converges weakly in D[0, 1] to ∫ · 0 f 0( A 0( t), X 0( t)) d X 0 ( t) as n → ∞, where { A n , n ⩾ 0} is some sequence of increasing processes corresponding to the sequence { X n , n ⩾ 0}.

Stochastic integration in Fock space

In this paper, using purely Hubert space-theoretic methods, an analogue of the Itό integral is constructed in the symmetric Fock space of a direct integral § of Hilbert spaces over the real line. The classical Itό integral is the special case when §=L 2 [0, ∞). An explicit formula is obtained for the projection onto the space of 'non-anticipating functionals', which is then used to prove that simple non-anticipating functionals are dense in the space of all non-anticipating functionate. After defining the analogue of the Itό integral, its isometric nature is established. Finally, the range of this 'integral' is identified; this last result is essentially the Kunita-Watanabe theorem on square-integrable martingales.

The Itô-Clifford integral

We present a theory of non-commutative stochastic integration analogous to the Itô-theory. It is shown that Wick products of Fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted (operatorvalued) square-integrable integrands. For square-integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined, and a Doob-Meyer decomposition for supermartingales obtained.

Orthogonal functionals of independent-increment processes

In analogy with the Wiener-Itô theory of multiple integrals and orthogonal polynominals, a set of functionals of general square-integrable martingales is presented which, in the case of independent-increments processes, is orthogonal and complete in the sense that every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L^{2}</tex> -functional of the independent-increment process can be represented as an infinite sum of these elementary functionals. The functionals are iterated integrals of the basic martingales, similar to the multiple iterated integrals of Itô and can be also thought of as being the analogs of the powers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,x,x^{2}, \cdots</tex> of the usual calculus. The analogy is made even clearer by observing that expanding the Doleans-Dade formula for the exponential of the process in a Taylor-like series leads again to the above elementary functionals. A recursive formula for these functionals in terms of the basic martingale and of lower order functionals is given, and several connections with the theory of reproducing kernel Hilbert spaces associated with independent-increment processes are obtained.

The Multiplicity of an Increasing Family of $\Sigma$-Fields

Let $(\Omega, \mathscr{F}, P)$ be a probability space and let $\mathscr{F}_t, t \in R_+$, be an increasing family of sub-$\sigma$-fields of $\mathscr{F}$ such that $\mathscr{F}_0$ is trivial and $\mathscr{F} = V_t\mathscr{F}_t$. Let $\mathscr{M}^2$ be the family of all square-integrable martingales $m_t$ with $m_0 = 0$. Suppose that $L^2(\Omega, \mathscr{F}, P)$ is separable. Then there exists a finite or countable sequence in $\mathscr{M}^2, m_t^1, m_t^2, \cdots$, such that (i) the stable subspaces generated by $m_t^i, m_t^j$ are orthogonal for $i \neq j$; (ii) $\langle m^1\rangle \succ \langle m^2\rangle \succ\cdots$ where $\langle m^i\rangle$ is the nonnegative measure on the predictable $\sigma$-field on $\Omega \times R_+$ induced by the quadratic variation process $\langle m^i\rangle$ of $m^i$, and (iii) every $m$ in $\mathscr{M}^2$ has a representation $m_t = \sum_i \int^t_0 \phi_i(s) dm_s^i$ a.s. for some predictable integrands $\phi_i$. Furthermore, if $n_t^1, n_t^2, \cdots$ is another such sequence, then $\langle n^i\rangle \sim \langle m^i\rangle$ for all $i$.

Representation by stochastic integrals of square-integrable martingales

Representation by stochastic integrals of square-integrable martingales