共查询到20条相似文献,搜索用时 31 毫秒
1.
For a one-parameter process of the form Xt=X0+∫t0φsdWs+∫t0ψsds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(Xt) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W2, z∈R2+} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let Xz be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(Xz) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(Xz) on increasing paths in . 相似文献
2.
3.
In this paper we prove the existence of the quadratic covariation [f(X),X], where f is a locally square integrable function and X
t =
t
0
u
s
dW
s is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. 相似文献
4.
This paper provides new necessary and sufficient conditions for a Gaussian random field to have a Gohberg-Krein representation in terms of an n-parameter Wiener process (n > 1). As an application, it demonstrates the nonexistence of a Gohberg-Krein representation of Ws,t ? stW1,1 in terms of the two-parameter Wiener process Ws,t with (s, t) ? [0, a] × [0, b] for 0 < a < 1, 0 < b < 1. 相似文献
5.
In this paper, the authors study a double random integral of the form ∫01∫01f(s,t) M(ds) M(dt), where M(0,t) is a stable process with independent increments. Basically, the Wiener approach is used, and the existence of the above integral is established for a wide class of functions f. 相似文献
6.
Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ
d
with d ≥ 4 and let |W
r
(t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for
| Wr (t) | - \mathbbE| Wr (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case. 相似文献
7.
A. Járai 《Aequationes Mathematicae》2001,61(3):205-211
Summary. We prove that a solution f of the functional equation¶¶f(t)=h(t,y,f(g1(t,y)),...,f(gn(t,y))) f(t)=h(t,y,f(g_1(t,y)),\dots,f(g_n(t,y))) ¶ having locally bounded variation is a C¥ {\cal C}^\infty -function. 相似文献
8.
W. J. Park 《Journal of multivariate analysis》1974,4(4):479-485
Strassen's version of the law of the iterated logarithm is extended to the two-parameter Gaussian process {X(s, t); ε(s, t) [0, ∞)2} with the covariance function R((s1,t1),(s2,t2)) = min(s1,s2)min(t1,t2). 相似文献
9.
Andreas Basse 《Journal of Theoretical Probability》2009,22(4):811-826
The aim of the present paper is to characterize the spectral representation of Gaussian semimartingales. That is, we provide
necessary and sufficient conditions on the kernel K for X
t
=∫
K
t
(s) dN
s
to be a semimartingale. Here, N denotes an independently scattered Gaussian random measure on a general space S. We study the semimartingale property of X in three different filtrations. First, the ℱ
X
-semimartingale property is considered, and afterwards the ℱ
X,∞-semimartingale property is treated in the case where X is a moving average process and ℱ
t
X,∞=σ(X
s
:s∈(−∞,t]). Finally, we study a generalization of Gaussian Volterra processes. In particular, we provide necessary and sufficient
conditions on K for the Gaussian Volterra process ∫
−∞
t
K
t
(s) dW
s
to be an ℱ
W,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case. 相似文献
10.
We consider a hypoelliptic two-parameter diffusion. We first prove a sharp upper bound in small time (s, t)[0, 1]2 for the Lp-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of22 log ps, t(x, y), as ↓0, where x is the initial condition of the diffusion, =
, and ps, t(x, y) is the density of the hypoelliptic two-parameter diffusion. 相似文献
11.
Let {W i (t), t ∈ ?+}, i = 1, 2, be two Wiener processes, and let W 3 = {W 3(t), t ∈ ? + 2 } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P f = P{W 1(t 1) + W 2(t 2) + W 3(t) + f(t) ≤ u(t), t ∈ ? + 2 }, where f, u : ? + 2 → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W 1(t 1) + W 2(t 2) + W 3(t). It turns out that our approach is also applicable for the additive Brownian pillow. 相似文献
12.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
13.
For ν(dθ), a σ-finite Borel measure on R
d
, we consider L
2(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫
t
0
e
−λ(θ)(
t
−
s
)
dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫
t
0
g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions
of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process
t
(τ)≗Y(τt∧t).
Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001 相似文献
14.
The existence of positive solutions of the Fredholm nonlinear equation y(t) = h(t) + ∫T0k(t, s)[f(y(s)) + g(y(s))] ds is discussed. It is assumed that f is a continuous, nondecreasing function and g is continuous, nonincreasing, and possibly singular. 相似文献
15.
Martin Becker 《Computational Management Science》2010,7(1):1-17
We introduce a method for generating (Wx,T(m,s),mx,T(m,s),Mx,T(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , where Wx,T(m,s)W_{x,T}^{(\mu,\sigma)} denotes the final value of a Brownian motion starting in x with drift μ and volatility σ at some final time T, mx,T(m,s) = inf0 £ t £ TWx,t(m,s)m_{x,T}^{(\mu,\sigma)} = {\rm inf}_{0\leq t \leq T}W_{x,t}^{(\mu,\sigma)} and Mx,T(m,s) = sup0 £ t £ T Wx,t(m,s)M_{x,T}^{(\mu,\sigma)} = {\rm sup}_{0\leq t \leq T} W_{x,t}^{(\mu,\sigma)} . By using the trivariate distribution of (Wx,T(m,s),mx,T(m,s),Mx,T(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , we obtain a fast method which is unaffected by the well-known random walk approximation errors. The method is extended
to jump-diffusion models. As sample applications we include Monte Carlo pricing methods for European double barrier knock-out
calls with continuous reset conditions under both models. The proposed methods feature simple importance sampling techniques
for variance reduction. 相似文献
16.
Wen heng Wang 《数学学报(英文版)》2002,18(4):727-736
Let {W(t); t≥ 0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables $ \sup _{{0 \leqslant t \leqslant T - \alpha _{T} }} \inf _{{f \in S}} \sup _{{0 \leqslant x \leqslant 1}} {\left| {Y_{{t,T}} {\left( x \right)} - f{\left( x \right)}} \right|} Let {W(t); t≥ 0} be a standard Wiener process and S be the Strassen set of functions. We investigate the exact rates of convergence to zero (as T→∞) of the variables sup0≤
t
≤
T
−
aT
inf
f∈S
sup0≤
x
≤1|Y
t,T
(x) −f(x)| and inf0≤
t
≤
T−aT
sup0≤
x
≤1|Y
t,T
(x−f(x)| for any given f∈S, where Y
t,T
(x) = (W(t+xa
T
) −W(t)) (2a
T
(log Ta
T
−1 + log log T))−1/2.
We establish a relation between how small the increments are and the functional limit results of Cs?rg{\H o}-Révész increments
for a Wiener process. Similar results for partial sums of i.i.d. random variables are also given.
Received September 10, 1999, Accepted June 1, 2000 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.
Olof J Staffans 《Journal of Mathematical Analysis and Applications》1985,108(1):15-30
Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝0tr(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L∞ space into another weighted L∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations. 相似文献
19.
We consider the asymptotic nonlinear filtering problem dx=f(x)dt + ?1/2 dw,dy=h(x) dt + ? dv and obtain lim?→0 ? log q 2(x,t) = -W(x,t) for unnormalized conditional densities q 2(x,t) using PDE methods. HereW(x,t) is the value function for a deterministic optimal control problem arising in Mortensen's deterministic estimation, and is the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. ijab has also studied this filtering problem, and we extend his large deviation result for certain unnormalized conditional measures. The resulting variational problem corresponds to the above control problem 相似文献
20.
In this note, we prove an ?‐regularity theorem for the Ricci flow. Let (Mn,g(t)) with t ? [?T,0] be a Ricci flow, and let Hx0(y,s) be the conjugate heat kernel centered at some point (x0,0) in the final time slice. By substituting Hx0(?,s) into Perelman's W‐functional, we obtain a monotone quantity Wx0(s) that we refer to as the pointed entropy. This satisfies Wx0(s) ≤ 0, and Wx0(s) = 0 if and only if (Mn,g(t)) is isometric to the trivial flow on Rn. Then our main theorem asserts the following: There exists ? > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mn,g(?T)) such that Wx0(s) ≥ ?? implies |Rm| ≤ r?2 on P? r(x0,0), where r2 ≡ |s| and Pρ(x,t) ≡ Bρ(x,t) × (t?ρ2,t] is our notation for parabolic balls. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of Wx(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mn,g(t),dvolg(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mn,g(t),Hx0(?,t)dvolg(t)). Our log‐Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log‐Sobolev has other consequences as well, including certain average Gaussian upper bounds on the conjugate heat kernel. © 2014 Wiley Periodicals, Inc. 相似文献