Functionals of diffusion processes as stochastic integrals

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, ) and by Øksendal (Stochastic Differential Equations. Springer, New York, ). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

Stochastic integration for inhomogeneous Wiener process in the dual of a nuclear space

In this thesis, stochastic processes closely related to a prominent process called Brownian motion is introduced. The processes are called local Brownian motions. Two rich families are constructed, which can be used to model stochastic dynamical systems in such fields as finance, biology, and physics. Stochastic calculus with respect to local Brownian motion is developed, and strong solutions are presented for some stochastic differential equations driven by the noise generated by local Brownian motions.

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

STOCHASTIC INTEGRATION

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let , , and be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of into . If is an integrable, -valued predictable process and is an -valued square integrable martingale, then there exists a -valued process called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.

In this chapter, we consider studies involving cadlag processes. We pay particular attention to semi-martingales with respect to a given filtration; these processes will always be taken with cadlag paths. We present the definition of stochastic integrals with respect to a square integrable martingale, and we extend the definition to stochastic integrals with respect to a local martingale. Then, we introduce semi-martingales, quadratic covariation processes for semi-martingales and some general versions of Ito’s formula and Girsanov’s theorem. We give necessary and sufficient conditions for the existence of an equivalent martingale measure. We end the chapter with a brief survey of valuation in an incomplete market.

This thesis consists of two parts, each part concentrating on a different problem from the theory of Stochastic Integration. Chapter 1 contains the introduction explaining the results in this dissertation in general terms. We use the infinite dimensional space S'R endowed with the gaussian measure mu. The Hilbert space ( L2) is defined as (L2) (L2) ≡ L2( S'R , mu) and our results are based on the Gel'fand triple ( S )beta ⊂ (L2) ⊂ S* b . The necessary preliminary background in white noise analysis are well elaborated in Chapter 2. In Chapter 3 we present a generalization of the Ito Formula to anticipating processes in the white noise framework. We first introduce an extension of the Ito integral to anticipating processes called the Hitsuda Skorokhod integral. We then use the anticipating Ito formula for processes of the type theta(X(t), B(c)), where X(t) is a Wiener integral and B(t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion for a ≤ c ≤ b, which was obtained in Professor H. H. Kuo's book, White Noise Distribution Theory, as a motivation to obtain our first main result. We generalize the formula in Prof. Kuo's book to processes of the form theta(X(t),F) where X(t) is a Wiener integral and F ∈ W 1/2, a special subspace of (L 2). Chapter 4 contains the second part of our work. We first state the Clark-Ocone (C-O) formula in the white noise framework. We then verify the formula for Brownian functionals using the very important white noise tool, the S-transform. As our second main result, we extend the formula to generalized Brownian functionals of the form F=Sinfinityn=0 :˙⊗n:,fn where fn ∈ L 2 R ⊗&d4;n . As examples we verify the formula for the Hermite Brownian functional: B(t)n:t using the Ito formula. We also compute the C-O formula for the Donsker's delta function using the result for the Hermite Brownian functional. Finally, the formula is generalized to compositions of tempered distributions with Brownian motion, i.e, to functionals of the form f( B(t)) where f ∈ S'R and B(t) = B( t) = 〈·, 1&sqbl0;0,t&parr0; 〉 is Brownian motion.

This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale $X$. The criterion is minimization of $\\mathbf{L}^2$–distance, or "least-squares". This problem has a straightforward and well-known solution when $X$ is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure $P$. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to $P$, under which $X$ behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it "has no gap" (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also "strong", in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of "variance-optimal" and "mean-variance efficient" portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Föllmer, D. Sondermann and most notably M. Schweizer.

IG-OU processes are a subclass of the non-Gaussian processes of Ornstein–Uhlenbeck type, which are important models appearing in financial mathematics and elsewhere. The simulation of these processes is of interest for its applications in statistical inference. In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables—one has an inverse Gaussian distribution and the other has a compound Poisson distribution. And in distribution, the compound Poisson random variable is equal to a sum of Poisson-distributed number positive random variables, which are independent identically distributed and have a common specified density function. The exact simulation of the IG-OU processes, proceeding from time 0 and going in steps of time interval Δ, is achieved via the representation of the stochastic integral. Comparing to the approximate method, which is based on Rosinski’s infinite series representation of the same stochastic integral, by the quantile–quantile plots, the advantage of the exact simulation method is obvious. In addition, as an application, we provide an estimator of the intensity parameter of the IG-OU processes and validate its superiority to another estimator by our exact simulation method.

The Wiener process (or the Brownian motion) is the starting point and the basis for all the following chapters. This is why we will consider this process more explicitly. It is a continuous-time process having as prominent a position in stochastic calculus as the Gaussian distribution in statistics. After introducing its defining properties intuitively, we will discuss important characteristics in the third section. Examples derived from the Wiener process will conclude the exposition.

In this paper we develop a stochastic calculus with respect to a Gaussian process of the form $B_t = \int^t_0 K(t, s)\, dW_s$, where $W$ is a Wiener process and $K(t, s)$ is a square integrable kernel, using the techniques of the stochastic calculus of variations. We deduce change-of-variable formulas for the indefinite integrals and we study the approximation by Riemann sums.The particular case of the fractional Brownian motion is discussed.

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index H≥1 2. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if H>3 4 and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a “controlled” process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals.

We give stopped Doob inequalities for p-th moment, 0<p<∞, of stochastic convolution integrals ∫(0,t] U(t,s)φ s − dM s in a Hilbert space, where M is a Hilbert space-valued cadlag square integrable martingale, φ is an operator-valued predictable process and U(t,s) is a contraction-type evolution operator. We also generalize the previous results and try to get smaller constants. 1This research was partially supported by Sharif University of Technology.