共查询到20条相似文献,搜索用时 15 毫秒
1.
S. S. Sinelnikov 《Moscow University Mathematics Bulletin》2011,66(4):158-162
For a Lévy process X = (X
t
)0≤t<∞ we consider the time θ = inf{t ≥ 0: sup
s≤t
X
s
= sup
s≥0
X
s
}. We study an optimal approximation of the time θ using the information available at the current instant. A Lévy process being a combination of a Brownian motion with a drift
and a Poisson process is considered as an example. 相似文献
2.
For a wide class of local martingales (M
t
) there is a default function, which is not identically zero only when (M
t
) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability
of functions of sup
s≤t
M
s
in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales,
and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting
and last exit times.
Received: 6 August 1996 / Revised version: 27 July 1998 相似文献
3.
Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H
n
, n > 2. For ν > 0, the Brownian bridge B
(ν) of length ν on H is the process B
t
, 0 ≤t≤ν, conditioned by B
0 = B
ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge
on ℝ3). The same result holds for the simple random walk on an homogeneous tree.
Received: 4 December 1998 / Revised version: 22 January 1999 相似文献
4.
Let (M
t
) be any martingale with M
0≡ 0, an intermediate law M
1∼μ1, and terminal law M
2∼μ2, and let Mˉ
2≡ sup0≤
t
≤2
M
t
. In this paper we prove that there exists an upper bound, with respect to stochastic ordering of probability measures, on
the law of Mˉ
2. We construct, using excursion theory, a martingale which attains this maximum. Finally we apply this result to the robust
hedging of a lookback option.
Received: 26 December 1998 / Revised version: 20 April 2000 /?Published online: 15 February 2001 相似文献
5.
Jean-François Delmas 《Probability Theory and Related Fields》1999,114(4):505-547
We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting
behavior of the volume of the ɛ-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous
result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove
the support of X
t
is capacity-equivalent to [0, 1]2 in ℝd, d≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]4 in ℝd, d≥ 5.
Received: 7 April 1998 / Revised version: 2 October 1998 相似文献
6.
Summary LetX
t be a Brownian motion and letS(c) be the set of realsr0 such that üX
r+t
–X
r
üct, 0th, for someh=h(r)>0. It is known thatS(c) is empty ifc<1 and nonempty ifc>1, a.s. In this paper we prove thatS(1) is empty a.s.This research was partially supported by NSF Grant 9322689. 相似文献
7.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
8.
Aurel Spătaru 《Probability Theory and Related Fields》2006,136(1):1-18
Let X1, X2, . . . be i.i.d. random variables, and set Sn=X1+ . . . +Xn. Several authors proved convergence of series of the type f(ɛ)=∑ncnP(|Sn|>ɛan),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑ncnP(|Sn|>ɛan),ɛ>α, is equivalent to the convergence of the double sum ∑k∑ncnP(|Sn|>kan). Two exceptional series required deriving necessary and sufficient conditions for E[supn|Sn|(logn)η/n]<∞,0≤η≤1. 相似文献
9.
Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form
0
t
q(B
s
)ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324 相似文献
10.
S. Fourati 《Probability Theory and Related Fields》1998,110(1):13-49
Summary. We prove a conjecture of J. Bertoin: a Lévy process has increase times if and only if the integral is finite, where G and H are the distribution functions of the minimum and the maximum of the Lévy process killed at an independent exponential time.
The “if” part of the statement had been obtained before by R. Doney. Our proof uses different techniques, from potential theory
and the general theory of processes, and is self-contained. Our results also show that if P(X
t
<0)≤1/2 for all t small enough, then the process does not have increase times.
Received: 4 May 1995/In revised form: 6 May 1997 相似文献
11.
K. K. J. Kinateder Patrick McDonald David Miller 《Probability Theory and Related Fields》1998,111(4):469-487
Let X
t
be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using
the moments of the exit time for particles driven by X
t
, as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional
in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary
value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all
critical points are maxima.
Received: 23 January 1997 / Revised version: 21 January 1998 相似文献
12.
Simeon M. Berman 《Annals of the Institute of Statistical Mathematics》1984,36(1):301-321
Summary Let {X
n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate
mixing conditions, it is shown thatS
n=Xn,1+…+X
n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f
n(x)∼ defined on a stationary sequence {X
j∼, whereX
n.f=fn(Xj). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of
a general type.
This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship
of the National Sciences Foundation, Grant MCS 82-01119. 相似文献
13.
B. K. Driver 《Applied Mathematics and Optimization》1999,39(2):179-210
Let W(M) be the based (at o∈ M) path space of a compact Riemannian manifold M equipped with Wiener measure ν . This paper is devoted to considering vector fields on W(M) of the form X
s
h
(
σ
)=P
s
(
σ
)h
s
(
σ ) where P
s
(
σ ) denotes stochastic parallel translation up to time s along a Wiener path σ
∈ W(M) and {h
s
}
s∈ [0,1]
is an adapted T
o
M -valued process on W(M). It is shown that there is a large class of processes h (called adapted vector fields) for which we may view X
h
as first-order differential operators acting on functions on W(M) . Moreover, if h and k are two such processes, then the commutator of X
h
with X
k
is again a vector field on W(M) of the same form.
Accepted 5 May 1997 相似文献
14.
Saharon Shelah 《Israel Journal of Mathematics》2012,191(2):507-543
We force 2 λ to be large, and for many pairs in the interval (λ, 2 λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)+3, hL(X) ≤ s(X)+3 and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ 0 we get hd(X), hL(X) ≤ N 2. 相似文献
15.
Hirofumi Osada 《Probability Theory and Related Fields》2001,119(2):275-310
We construct a family of diffusions P
α = {P
x} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d
H
< α < ∞, where d
H
= log(3
d
− 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions P
α are reversible with invariant measures μ = μ[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3α·μ(A) for all A∈ℬ(F^). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the
Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the
diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet.
Received: 30 September 1999 / Revised version: 15 June 2000 / Published online: 24 January 2000 相似文献
16.
In this paper we study C0-semigroups on X × Lp( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on
the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞).
Dedicated to the memory of H. H. Schaefer 相似文献
17.
Consider a d-dimensional Brownian motion X = (X
1,…,X
d
) and a function F which belongs locally to the Sobolev space W
1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f
k
(X), X
k
] involving the weak first partial derivatives f
k
of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X
k
] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation
results for forward and backward stochastic integrals.
Received: 16 March 1998 / Revised version: 4 April 1999 相似文献
18.
Marc Arnaudon 《Probability Theory and Related Fields》1997,108(2):219-257
Summary. We prove that the derivative of a differentiable family X
t
(a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ
t
)0≤
t
≤1, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x
0∈M, there exists an analytic family a↦X(a) of continuous martingales such that X
1(a)=L(a). For this, we investigate the convexity of the tangent spaces T
(
n
)
M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to
a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that
every ℱ1-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C
1 connection is reachable by a V-valued martingale.
Received: 14 March 1996/In revised form: 12 November 1996 相似文献
19.
Marius Junge 《Positivity》2006,10(2):201-230
For n independent random variables f1, . . . ,fn and a symmetric norm || ||X on ℝn, we show that for 1≤ p < ∞
Here
is the disjoint sum of the fi's and h* is the non-increasing rearrangement. Similar results (where Lp is replaced by a more general rearrangement invariant function space) were obtained first by Litvak, Gordon, Schütt and Werner
for Orlicz spaces X and independently by S. Montgomery-Smith [22] for general X but without an explicit analysis of the order of growth for the constant in the upper estimate. The order is optimal and obtained from combinatorial estimates for doubly stochastic matrices. The result extends to Lorentz-norms
lf, q on ℝn under mild assumptions on f. We give applications to the theory of noncommutative Lp spaces. 相似文献
20.
Suppose K is a compact convex set in ℝ2 and X
i
, 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X
i
we study the asymptotic properties of certain statistics of the convex hull of the sample.
Received: 24 July 1996/Revised version: 24 February 1998 相似文献