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 £ TWx,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
In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T ‐periodic solutions for a class of nonlinear n ‐th order differential equations with delays of the form x(n)(t) + f (x(n‐ 1)(t)) + g (t, x (t ‐ τ (t))) = p (t). 相似文献
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/nu = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
For a pair of points x, y in a compact, Riemannian manifold M let nt(x, y) (resp. st(x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block these geodesic segments). We study relationships
between the growth rates of nt(x, y) and st(x, y) as t → ∞. We obtain lower bounds on st(x, y) in terms of the topological entropy h(M) and the fundamental group π1(M). For instance, we show that if h(M) > 0 then st grows exponentially, with the rate at least h(M)/2. This strengthens earlier results on blocking of geodesics (Burns and Gutkin Discrete Contin Dyn Syst 21:403–413, 2008;
Lafont and Schmidt Geom Topol 11:867–887, 2007), and puts them in a new perspective.
相似文献
The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X1, X2, …, Xn) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ1, Θ2, …, Θn). Let θx* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θx* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θx* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X1, X2, …, Xn is considered for a location vector Θ = (Θ1, Θ2, …, Θn) as x gets large along a path, γ, in Rn. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞. 相似文献
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 inff∈S sup0≤x≤1|Yt,T(x) −f(x)| and inf0≤t≤T−aT sup0≤x≤1|Yt,T(x−f(x)| for any given f∈S, where Yt,T(x) = (W(t+xaT) −W(t)) (2aT(log TaT−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
Let P(n) denote the largest prime factor of an integer n (N(x) = x (2+O(?{log2x/logx} ) )ò2xr(logx/logt) [(logt)/(t2)] d t,N(x) = x \left(2+O\left(\sqrt{\log_{2}\,x/\!\log x}\,\right) \right)\int_2^x\rho(\log x/\!\log t) {\log t\over t^2} {\rm d} t, 相似文献
In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E.
[(x)\dot] = f (x) + eg (x, t) + e2[^(g)] (x, t, e){\dot{x} = f (x) + \varepsilon g (x, t) + \varepsilon^{2}\widehat{g} (x, t, \varepsilon)}
, where
x ? W ì \mathbbRn{x \in \Omega \subset \mathbb{R}^n}
,
g,[^(g)]{g,\widehat{g}}
are T periodic functions of t and there is a0 ∈ Ω such that f ( a0) = 0 and f ′( a0) is a nilpotent matrix. When n = 3 and f (x) = (0, q (x3) , 0) we get results on existence and stability of periodic orbits. We apply these results in a non ideal mechanical system:
the Centrifugal Vibrator. We make a stability analysis of this dynamical system and get a characterization of the Sommerfeld
Effect as a bifurcation of periodic orbits. 相似文献
Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that
?n £ xf(n)=o(x)\sum_{n\le x} f(n)=o(x)
. We propose sufficient conditions on the function f to deduce the equality
?n £ xf(s(n))=o(x)\sum_{n\le x} f(s(n))=o(x)
. Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))α, where α is a positive real number. 相似文献
We study here a new kind of modified Bernstein polynomial operators on L1(0, 1) introduced by J. L. Durrmeyer in [4]. We define for f integrable on [0, 1] the modified Bernstein polynomial Mnf: Mnf(x) = (n + 1) ∑nk = oPnk(x)∝10Pnk(t) f(t) dt. If the derivative drf/dxr with r 0 is continuous on [0, 1], dr/dxrMnf converge uniformly on [0,1] and supxε[0,1] ¦Mnf(x) − f(x)¦ 2ωf(1/trn) if ωf is the modulus of continuity of f. If f is in Sobolev space Wl,p(0, 1) with l 0, p 1, Mnf converge to f in wl,p(0, 1). 相似文献
Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?0 = 1 and c(x) ? c0 = 1 for |x| ? δ, and |γn(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γn(x) = c?2(x) ? 1. The samples are insonified by plane pulses s(x · θ0 – t) where x = |θ0| = 1 and the scattered pulse is shown to have the form |x|?1es(|x| – t, θ, θ0) in the far field, where x = |x| θ. The response es(τ, θ, θ0) is measurable. The goal of the work is to construct the sample parameters γn and γ? from es(τ, θ, θ0) for suitable choiches of s, θ and θ0. In the limiting case of constant density: γ?(x)? 0 it is shown that Where δ represents the Dirac δ and S2 is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x). 相似文献
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H10(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) = ∫s0f(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
In this paper we use a theorem of Crandall and Pazy to provide the product integral representation of the nonlinear evolution operator associated with solutions to the semilinear Volterra equation: x(?)(t) = W(t, τ) ?(0) + ∝τtW(t, s)F(s, xs(?)) ds.Here the kernel W(t, s) is a linear evolution operator on a Banach space X; I is an interval of the form [?r, 0] or (?∞, 0] and F is a nonlinear mapping of R × C(I, X) into X. The abstract theory is applied to examples of partial functional differential equations. 相似文献
We suppose that M is a closed subspace of l∞(J, X), the space of all bounded sequences {x(n)}n?J ? X, where J ? {Z+,Z} and X is a complex Banach space. We define the M-spectrum σM(u) of a sequence u ? l∞(J,X). Certain conditions will be supposed on both M and σM(u) to insure the existence of u ? M. We prove that if u is ergodic, such that σM(u,) is at most countable and, for every λ ? σM(u), the sequence e?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σrk=1ak(u(n + k) ? u(n)) + Σs ? Z?(n ? s)u(s) = h(n), n?J, where ψ?l∞(Z,X) such that ψ|J? M, A is the generator of a C0-semigroup of linear bounded operators {T(t)}t>0 on X, h ? M, ? ? l1(Z) and ak?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M. 相似文献