Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing

We introduce a method for generating (W_x,T^(m,s),m_x,T^(m,s),M_x,T^(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , where W_x,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, m_x,T^(m,s) = inf_{0 ￡ t ￡ T}W_x,t^(m,s)m_{x,T}^{(\mu,\sigma)} = {\rm inf}_{0\leq t \leq T}W_{x,t}^{(\mu,\sigma)} and M_x,T^(m,s) = sup_{0 ￡ t ￡ T} W_x,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 (W_x,T^(m,s),m_x,T^(m,s),M_x,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.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T) and a set A ? XA\in\mathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)} , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d^*(E)>0 and for all {n ? \mathbbZ\colon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}     

Uniform decay estimates for solutions of the Yamabe equation

We study positive solutions u of the Yamabe equation c_m Du-s( x) u+k( x) u^\fracm+2m-2=0{c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}, when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L ^Γ-norm of u, for some G 3 \tfrac2mm-2{\Gamma\geq\tfrac{2m}{m-2}}. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.     

Plane partitions IV: A conjecture of Mills—Robbins—Rumsey

In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ _n (x, m) The special caseZ _n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ _n (2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation ofZ _n (1,m) given previously Many results for the₃ F ₂ hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation ofZ _n (2,m) as a determinant $$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{i = 0}^l {\left( {\begin{array}{*{20}c} {m + j + t} \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {m + i} \\ {m + t} \\ \end{array} } \right)} } \right)_{0 \leqq ij \leqq n - 1} $$ Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ _n (2,m) In the final analysis, one would like a combinatorial explanation ofZ _n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture     

On the product of the distances of a point from the vertices of a polytope

Letx ₁,...,x _m be points in the solid unit sphere ofE _n and letx belong to the convex hull ofx ₁,...,x _m. Then . This implies that all such products are bounded by (2/m) ^m (m −1) ^m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z ₀)| where andp′(z ₀)=0.     

The evaluation of a quartic integral via Schwinger, Schur and Bessel

We provide additional methods for the evaluation of the integral

Some ratio inequalities for iterated stochastic integrals

Let X = (X_t, ?_t) be a continuous local martingale with quadratic variation 〈X〉 and X₀ = 0. Define iterated stochastic integrals I_n(X) = (I_n(t, X), ?_t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I₀(t, X) = 1 and I₁(t, X) = X_t. Let (??^x_t(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = sup_x∈? ??^x_t(X) and X* = sup_t≥0 |X_t|. In this paper, we shall establish various ratio inequalities for I_n(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants c_n,p and C_n,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant C_n,p,γ depending only on n, p and γ, where U_n is one of 〈I_n(X)〉^1/2_∞ and I*_n(X), and V one of the three random variables X*, 〈X〉^1/2_∞ and ??*_∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }

Let \mathbbZ_p^m \mathbb{Z}_{p^m } be the ring of integers modulo p ^m , where p is a prime and m ⩾ 1. The general linear group GL _n ( \mathbbZ_p^m \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A _n := \mathbbZ_p^m \mathbb{Z}_{p^m } [x ₁, …, x _n ]. Denote by A_n^{GL₂ (\mathbbZ_pm )} A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The Structure of Finite c-Local Rings

For a commutative ring R, assume that c is a nonzero element of Z(R) with the property that cZ(R) = {0}. A local ring R is called c-local if Z(R)² = {0, c}, Z(R)³ = {0}, and xZ(R) = {0} implies x ∈ {0, c}. For any finite c-local ring (R, 𝔪), it is proved that the ideal m has a minimal generating set which has a c-partition. The structure and classification up to isomorphism of all finite commutative c-local rings with order greater than 2⁵ are determined.     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

Plane partitions IV: A conjecture of Mills—Robbins—Rumsey

In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ _n (x, m) The special caseZ _n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ _n (2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation ofZ _n (1,m) given previously Many results for the₃ F ₂ hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation ofZ _n (2,m) as a determinant $$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{t = 0}^1 {\left( {\mathop {m + j + t}\limits_t } \right)} \left( {\mathop {m + t}\limits_{m + t} } \right)} \right)_{0 \leqq ij \leqq n - 1} $$ Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ _n (2,m) In the final analysis, one would like a combinatorial explanation ofZ _n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture     

The Gaussian Isoperimetric Inequality and Transportation

Any probability measure on ^d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e ^–(x) dx be a probability measure on ^d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.     

On divisible weighted Dynkin diagrams and reachable elements

Let e be a nilpotent element of a complex simple Lie algebra $ \mathfrak{g} Let e be a nilpotent element of a complex simple Lie algebra \mathfrakg \mathfrak{g} . The weighted Dynkin diagram of e, D(e) \mathcal{D}(e) , is said to be divisible if D(e)

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

Reducibility of polynomials a0(x) + a1 (x)y + a2 (x) y2 modulo p

Let f=a₀(x)+a₁(x)y+a₂(x)y² ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 c_m·H(f)^e_mp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e₁ = 4,e₂ = 6, e₃ = 6 [2/3]{2}\over{3} and e_m = 2 m for m S 4. We also show that the exponents e_m are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.     

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit