Parisian ruin over a finite-time horizon

For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) 0}, S, T_u 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval.     

Functional equations and Weierstrass sigma-functions

It is proved that if an entire function f: ? → ? satisfies an equation of the form α ₁(x)β ₁(y) + α ₂(x)β ₂(y) + α ₃(x)β ₃(y), x,y ∈ C, for some α _j , β _j : ? → ? and there exist no \({\widetilde \alpha _j}\) and ?\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az ² + Bz + C) ? σ _Γ(z - z ₁) ? σ _Γ(z - z ₂), where Γ is a lattice in ?; σ _Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z ₁, z ₂ ∈ ?; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).     

Removability of isolated singularities for anisotropic elliptic equations with gradient absorption

We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces L^p(T,X), periodic Besov spaces B_p,q^s(T,X) and periodic Triebel–Lizorkin spaces F_p,q^s(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P₃) in the above three function spaces.     

Output stabilization for a class of uncertain systems

The system

$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$

where v = C*x is an output, u = S*y is a control, A(·) ∈ R ^{n × n} , B(·) ∈ R ^{n × (n–p)}, C ∈ R ^{n × (n–p)}, and D ∈ R ^{n × (n–p)}, is considered. The elements α_ij(·) and β_ij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions

$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$

It is assumed that A(·) ∈ Z ₁ ∪ Z ₃ and A*(·) ∈ Z ₁ ∪ Z ₃, where Z ₁ is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z ₃ differs from Z _t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.

     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

Existence theorems for nonlinear differential equations having trichotomy in banach spaces

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation

$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$

(P)

where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f ^d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ _t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay

$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$

(Q)

has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.

     

The Joint Distribution of Running Maximum of a Slepian Process

Consider the Slepian process S defined by S(t) = B(t +?1) ? B(t),t ∈ [0, 1] with B(t), t ∈ ? a standard Brownian motion. In this contribution we analyze the properties between the maximum \(m_{s}=\max \limits _{0\leq u\leq s}S(u)\) and the maximum \(m_{t}=\max \limits _{0\leq u\leq t}S(u)\) for 0 ≤ s < t ≤?1 fixed. Explicit integral expressions are obtained for the joint distribution function between m _s and m _t and the distribution function of the partial maximum m _s . Further, we apply our results for the determination of the moments of m _s .     

Ergodic theorems with arithmetical weights

We prove that the divisor function d(n) counting the number of divisors of the integer n is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system (X, A, ν, τ) and any f ∈ L ^p (ν), p > 1, the limit

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Sigma _{k = 1}^nd\left( k \right)}}\sum\limits_{k = 1}^n {d\left( k \right)f\left( {{\tau ^k}x} \right)} $$

exists ν-almost everywhere. The proof is based on Bourgain’s method, namely the circle method based on the shift model. Using more elementary ideas we also obtain similar results for other arithmetical functions, like the θ(n) function counting the number of squarefree divisors of n and the generalized Euler totient function J _s (n) = Σ _d|n d ^s μ(n/d), s > 0.

     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

Stabilization of a Class of Uncertain Systems

We consider the problem to synthesize a stabilizing control u synthesis for systems \(\frac{{dx}}{{dt}} = Ax + Bu\) where A ∈ ?^n×n and B ∈ ?^n×m, while the elements α_i,j(·) of the matrix A are uniformly bounded nonanticipatory functionals of arbitrary nature. If the system is continuous, then the elements of the matrix B are continuous and uniformly bounded functionals as well. If the system is pulse-modulated, then the elements of the matrix B are differentiable uniformly bounded functions of time. It is assumed that k isolated uniformly bounded elements \({\alpha _{{i_l},{j_l}}}\left( \cdot \right)\) satisfying the condition \(\mathop {\inf }\limits_{\left( \cdot \right)} \left| {{\alpha _{{i_l},{j_l}}}\left( \cdot \right)} \right|{\alpha _ - } > 0,\quad l \in \overline {1,k}\) are located above the main diagonal of the matrix A(·), where G _k is the set of all isolated elements of the system, J₁ is the set of indices of rows of matrix A(·) containing isolated elements, and J₂ is the set of indices of its rows free of isolated elements. It is assumed that other elements located above the main diagonal are sufficiently small provided that their row indices belong to J₁, i.e., \(\mathop {\sup }\limits_{\left( \cdot \right)} \left| {{\alpha _{i,j}}\left( \cdot \right)} \right| < \delta ,\quad {\alpha _{i,j}} \notin {G_k},\quad i \in {J_1},\quad j > i\). All other elements located above the main diagonal are uniformly bounded. The relation u = S(·)x is satisfied in the continuous case, while the relation u = ξ(t) is satisfied in the pulse-modulated case; here the components of the vector ξ are outputs of synchronous pulse elements. Constructing a special quadratic Lyapunov function, one can determine a matrix S(·) such that the closed system becomes globally exponentially stable in the continuous case. In the pulse-modulated case, input pulses are synthesized such that the system becomes globally asymptotically stable.     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convexweakly compact subset of X. Under some suitable restriction, we prove that every asymptoticallyregular semigroup T = {T(t) : t ∈¸ S} of selfmappings on C satisfying

${\mathop {\lim \inf }\limits_{S \mathrel\backepsilon t \to \infty } }{\left| {{\left\| {T(t)} \right\|}} \right|} < {\text{WCS}}(X)$

has a common fixed point, where WCS(X) is the weakly convergent sequence coefficient of X, and\({\left| {{\left\| {T(t)} \right\|}} \right|}\) is the exact Lipschitz constant of T(t).     

Hyperbolas over two-dimensional Fibonacci quasilattices

For the number n _s (α, β; X) of points (x ₁ , x ₂) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x ₁ ² ? ??αx ₂ ² ?=?β and such that 0?≤?x ₁? ≤?X, x ₂? ≥?0, the asymptotic formula

$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $

is established, and the coefficient c _s (α, β) is calculated exactly. Using this, we obtain the following result. Let F _m be the Fibonacci numbers, A _i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A _i in the Fibonacci numeral system. Then the number n _s (X) of all solutions (A ₁ , A ₂) of the Diophantine system

$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $

0?≤?A ₁? ≤?X, A ₂? ≥?0, satisfies the asymptotic formula

$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $

Here τ?=?(?1?+?√5)/2 is the golden ratio, and c _s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.

     

On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? ^s , we construct Kronecker-like and van der Corput-like ergodic transformations T _1,Γ and T _2,Γ of [0, 1) ^s . We prove that for admissible lattices Γ, (T _ν,Γ ⁿ (x))_n≥0 is a low discrepancy sequence for all x ∈ [0, 1) ^s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) ^s , for almost all lattices Γ ∈ L _s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L _s ), the following asymptotic formula

$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$

holds with arbitrary small ? > 0, for all x ∈ [0, 1) ^s , and ν ∈ {1, 2}.

     

Representations of the general linear group over symmetry classes of polynomials

Let V be the complex vector space of homogeneous linear polynomials in the variables x₁,..., x _m . Suppose G is a subgroup of S _m , and χ is an irreducible character of G. Let H _d (G, χ) be the symmetry class of polynomials of degree d with respect to G and χ.

For any linear operator T acting on V, there is a (unique) induced operator K _χ (T) ∈ End(H _d (G, χ)) acting on symmetrized decomposable polynomials by

$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$

In this paper, we show that the representation T ? K _χ (T) of the general linear group GL(V) is equivalent to the direct sum of χ(1) copies of a representation (not necessarily irreducible) T ? B _χ ^G (T).

     

An extended class of stabilizable uncertain systems

The system of equations \(\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u\), where A(·) ∈ ?^{n × n}, B(·) ∈ ?^{n × m}, S(·) ∈ R^{n × m}, is considered. The elements of the matrices A(·), B(·), S(·) are uniformly bounded and are functionals of an arbitrary nature. It is assumed that there exist k elements \({\alpha _{{i_i}{j_l}}}\left( \cdot \right)\left( {l \in \overline {1,k} } \right)\) of fixed sign above the main diagonal of the matrix A(·), and each of them is the only significant element in its row and column. The other elements above the main diagonal are sufficiently small. It is assumed that m = n ?k, and the elements β_ij(·) of the matrix B(·) possess the property \(\left| {{\beta _{{i_s}s}}\left( \cdot \right)} \right| = {\beta _0} > 0\;at\;{i_s}\; \in \;\overline {1,n} \backslash \left\{ {{i_1}, \ldots ,{i_k}} \right\}\). The other elements of the matrix B(·) are zero. The positive definite matrix H = {h_ij} of the following form is constructed. The main diagonal is occupied by the positive numbers h_ii = h_i, \({h_{{i_l}}}_{{j_l}}\, = \,{h_{{j_l}{i_l}}}\, = \, - 0.5\sqrt {{h_{{i_l}}}_{{j_l}}} \,\operatorname{sgn} \,{\alpha _{{i_l}}}_{{j_l}}\left( \cdot \right)\). The other elements of the matrix H are zero. The analysis of the derivative of the Lyapunov function V(x) = x*H^–1x yields h_i\(\left( {i \in \overline {1,n} } \right)\) and λ_i ≤ 0 \(\left( {i \in \overline {1,n} } \right)\) such that for S(·) = H^?1ΛB(·), Λ = diag(λ₁, ..., λ_n), the system of the considered equations becomes globally exponentially stable. The control is robust with respect to the elements of the matrix A(·).     

*-regularity of certain generalized group algebras

Let B be a *-semisimple Banach algebra with a bounded approximate identity and \({\alpha: G \longrightarrow {\rm Aut}_{*}(B)}\) (isometric *-automorphisms group of B) an action of a locally group G on B. Let (D, G, γ) be the associated dynamical system, where D = C ₀(G, B) is the Banach *-algebra of all continuous B-valued functions on G vanishing at infinity and the action γ : G → Aut D is given by γ _s (y)(t) = α _s (y(s ^?1 t)) for \({y \in D}\) and \({s, t \in G}\) . Recall that B is said to be *-regular if the natural mapping \({I\in {\rm Prim} \, C^{*}(B) \mapsto I\cap B\in {\rm Prim}_{*}(B)}\) is a homeomorphism under the hull-kernel topology. When G is amenable, we show that if B is *-regular, then the generalized group algebra L ¹(G, D; γ) is *-regular. The converse is also true if we further assume that G is countable discrete. Finally the case of compact groups is studied.     

A Supplement to Hölder’s Inequality. The Resonance Case. I

Suppose that m ≥ 2, numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ₁ ∈ \({L^{{p_1}}}\)(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L ^p (?¹), p ∈ (1, +∞], were introduced by the author in his previous papers.