The Representation of Functionals of Brownian Motion by Stochastic Integrals
01 Aug 1970-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 41, Iss: 4, pp 1282-1295
TL;DR: In this paper, it was shown that any finite functional of Brownian motion can be represented as a stochastic integral, where the integrand has the form of conditional expectations of the differential.
Abstract: 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.
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TL;DR: In this paper, a general stochastic model of a frictionless security market with continuous trading is developed, where the vector price process is given by a semimartingale of a certain class, and the general Stochastic integral is used to represent capital gains.
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Book•
01 Jan 1979TL;DR: In this paper, the second volume follows on from the first, concentrating on stochastic integrals, stochy differential equations, excursion theory and the general theory of processes.
Abstract: This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.
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TL;DR: In this article, a stochastic differential formulation of recursive utility is given sufficient conditions for existence, uniqueness, time consistency, monotonicity, continuity, risk aversion, concavity, and other properties.
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References
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TL;DR: In this paper, Kaczmarz and Steinhaus [I, pp. 143-144] showed that the equality W 1~~~~~~~~~~~~ |G a, ot(t) dx(t), *,Iap(t)-dx(t)] dwx (2.5) c 00 -p/2 L G(ui, *, up)euhu du,... du.
Abstract: (see, for example, Kaczmarz and Steinhaus [I, pp. 143-144]). Let (2.4) tap(t)} p = 1, 2, 3, be any orthonormal set of real functions, each belonging to L2(0, 1). Paley and 1 Wiener [II] have shown for each index p = 1, 2, that f ap(t) dx(t) exist as a generalized Stieltjes integral for almost all functions x(&) of C and that the equality W 1~~~~~~~~~~~~ |G a, ot(t) dx(t), * ,Iap(t) dx(t)] dwx (2.5) c 00 -p/2 L G(ui, * , up)euhu du, ... du.
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TL;DR: Theory of real and time continuous martingales has been developed recently by P. Meyer as discussed by the authors, who showed that there exists an increasing process "X" such that the number of martingale instances is increasing with time.
Abstract: Theory of real and time continuous martingales has been developed recently by P. Meyer [8, 9]. Let be a square integrable martingale on a probability space P. He showed that there exists an increasing process ‹X›t such that
556 citations