Search

Let X t be a continuous local martingale satisfying X 0 =0 and K 1 q ( t )⩽〈 X 〉 t ⩽ K 2 q ( t ) a.s. for a nondecreasing function q with constants K 1 and K 2 . Define for a Borel function f, M t = ∫ 0 t f(X s ) d X s and M t ∗ = sup 0⩽s⩽t |M s | . If f is in L 2 and f ≠0 then for any slowly increasing function φ there exist two positive constants c and C such that for all stopping times T cEφ(M T ∗2 )⩽Eφ( q(T) )⩽C(Eφ(M T ∗2 )+1). Suppose that f 2 is even and ψ(x)= ∫ 0 x ( ∫ 0 y f 2 (t) d t) d y is moderate. If φ satisfies one of the 3 conditions: (i) φ is slowly increasing, (ii) φ is concave if f ∉ L 2 , and (iii) φ is moderate if ψ(x) is convex, then there exist two positive constants c and C such that for all stopping times T cEφ(M T ∗2 )⩽Eφ∘ψ( q(T) )⩽CEφ(M T ∗2 ). Define T r = inf {t>0; |M t |=r}, r>0 . The growth rate function of ET r γ can be found for appropriate γ , as an application of the above inequalities. The method of proving the main result also yields a similar type of two-sided inequality for the integrable Brownian continuous additive functional over all stopping times.

Different kinds of variations associated with a continuous two-parameter martingale bounded in L 4 are studied. As an application a “compact” Ito formula and a version of a two-parameter Tanaka formula are proved.

In this paper we prove the existence of the quadratic covariation [(∂F/∂x k )(X), X k ] for all 1⩽k⩽d, where F belongs locally to the Sobolev space W 1,p ( R d ) for some p>d and X is a d-dimensional smooth nondegenerate martingale adapted to a d-dimensional Brownian motion. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. As a consequence we obtain an extension of Ito's formula where the complementary term is one-half the sum of the quadratic covariations above.

Under the framework of G -expectation and G -Brownian motion, we introduce Ito’s integral for stochastic processes without assuming quasi-continuity. Then we can obtain Ito’s integral on stopping time interval. This new formulation permits us to obtain Ito’s formula for a general C 1 , 2 -function, which essentially generalizes the previous results of Peng (2006, 2008, 2009, 2010, 2010) [21] , [22] , [23] , [24] , [25] as well as those of Gao (2009) [8] and Zhang et al. (2010) [27] .

We trace Ito’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Ito’s formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Ito’s formula in mathematical finance in the 1970s. Throughout the paper, we treat Ito’s jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.

We develop a notion of nonlinear expectation --G-expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G-Brownian motion and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equation under our G-expectation., 27 pages

Motivated by applications in option pricing theory ( Peskir, 1997b ), (Research Report No. 386, Dept. Theoret. Statist. Aarhus, 19 pp.) we formulate and solve the following problem. Given a standard Brownian motion B=(Bt)t⩾0 and a centered probability measure μ on R having the distribution function F with a strictly positive density F′ satisfying ∫ 0 ∞ x log x μ( d x) there exists a cost function x↦c(x) in the optimal stopping problem sup τ E max 0⩽t⩽τ B t − ∫ 0 τ c(B t ) d t such that for the optimal stopping time τ ∗ we have B τ ∗ ∼μ. The cost function is explicitly given by the formula: c(x)= 1 2 F′(x) (1−F(x)) , where one incidentally recognizes x↦F′(x)/(1−F(x)) as the Hazard function of μ. There is also a simple explicit formula for the optimal stopping time τ ∗ , but the main emphasis of the result is on the existence of the underlying functional in the optimal stopping problem. The integrability condition on μ is natural and cannot be improved. The condition on the existence of a strictly positive density is imposed for simplicity, and more general cases could be treated similarly. The method of proof combines ideas and facts on optimal stopping of the maximum process ( Peskir, 1997a ), (Research Report No. 377, Dept. Theoret. Statist. Aarhas, 30 pp.) and the Azema–Yor solution of the Skorokhod-embedding problem ( Azema and Yor (1979a) , Azema and Yor (1979b) ), (Sem. Probab. XIII, Lecture Notes in Math., vol. 721, Springer, Berlin, pp. 90–115; 625–633). A natural connection between these two theories is established, and new facts of interest for both are displayed. The result extends in a similar form to stochastic integrals with respect to B, as well as to more general diffusions driven by B.

In this paper we establish some new theorems on pathwise uniqueness of solutions to the stochastic differential equations of the form of Xz=Z(s,0)+Z(0,t)−Z(0,0)+∫Rza(ξ,Xξ)dMξ+∫Rzb(ξ,Xξ)dAξ for z=(s,t)∈R+2 with non-Lipschitz coefficients, where M={Mz,z∈R+2} is a continuous square integrable martingale and A={Az,z∈R+2} is a continuous increasing process, Z is a continuous stochastic process on boundary ∂R+2 of R+2. We have proved existence theorem for the equation in Liang (1996a).

The objective of this paper is to present the principal results of a large part of stochastic calculus in a manner that should be comprehensible to readers having only the general notions of stochastic processes. Not all the theorems are proved in detail, but all the fundamental theorems are explained with clarity and precision, and with special attention to the motivations behind them.Given two real valued stochastic processes X and Y, the basic problem is to give a meaning to Z = ∫ Y dX in such a way that the integral sign is not misused. If X is a process whose paths are of bounded variation, then Z should coincide with the ordinary Lebesgue-Stieltjes integral taken path by path. If Y is a left continuous step function, then Z should coincide with the obvious choice: if Y is constant on ]t, u], then Zu-Zt is that constant times Xu-Xt. And finally, the Lebesgue dominated convergence theorem should hold: if the processes Yn converge to Y and all the Yn are dominated by a process Y' for which ∝ Y' dX is well defined, then Zn = ∫ Yn dX should converge to Z = ∫ Y dX in some sense.Starting with these requirements, it is shown that, if ∫ Y dX is defined for all predictable Y, then X must be a semimartingale. Conversely, the integral is well defined for all predictable Y and all semimartingales X.With the integrals defined, a number of their important properties are discussed. In particular, the integral Z is a semimartingale, and a change of variable formula (Ito's formula) holds for ⨍(Z). Finally, stochastic integral equations are introduced, and a general theorem is given on the existence and uniqueness of solutions.A bibliography with commentaries supplements the text for the benefit of those who would like to go deeper into the subject.