Optimal stopping times for solutions of nonlinear stochastic differential equations and their application to one problem of financial mathematics

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random processX _t ,t ∈ [0, T], describing an investor's assets satisfies a nonlinear stochastic differential equation. We determine this switching time τ∈[0,T] as the optimal stopping time for a certain processY _t generated by the processX _t so that the average investor's assets are maximized at the final time, i.e.,EX _T . Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 804–809, June, 1999.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.     

Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence

We establish an estimate for the rate of convergence of a solution of an ordinary stochastic differential equation of order p ≥ 2 with a small parameter in the coefficient of the leading derivative to a solution of a stochastic equation of order p − 1 in the metric ρ(X, Y) = (sup_0≤t≤T M|X(t) − Y(t)|²)^1/2 __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1587–1601, December, 2006.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

Filtration of components of processes of random evolution

The problem of estimation of a nonobservable component θ_t for a two-dimensional process (θ_t, ξ_t) of random evolution (θ _t,ξ_t);x_t, 0≤t≤T, is investigated on the basis of observations of ξ_s. s≤t, where x _t is a homogeneous Markov process with infinitesimal operator Q. Applications to stochastic models of a (B,S)-market of securities is described under conditions of incomplete market. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1701–1705, December, 1998.     

Asymptotic expansions in limit theorems for stochastic processes. II

. For a certain class of families of stochastic processes η^ε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(η^ε[0,T]) are proved of the form

The optimal stopping problem concerned with ultimate maximum of a Lévy process

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.     

Convergence of distributions of integral functionals of extremal random functions

We study the convergence of distributions of integral functionals of random processes of the formU _n (t)=b _n (Z _n (t)-a _n G(t)),t⃛T, where {X=X(t), t⃛T} is a random process,X _n ,n≥1, are independent copies ofX, andZ _n (t)=max_1≤k≤n X _k (t). Ukrainian State Academy of Light Industry, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1201–1209, September, 1999.     

Nonautonomous Stochastic Search in Global Optimization

We present a general method how to prove convergence of a sequence of random variables generated by a nonautonomous scheme of the form X _t =T _t (X _t−1,Y _t ), where Y _t represents randomness, used as an approximation of the set of solutions of the global optimization problem with a continuous cost function. We show some of its applications.     

Some properties of subexponential distributions

The nonnegative random variableX is said to have a subexponential distribution if we have (1-G(t))/(1-F(t))→2 ast→∞, whereF(t)=P{X≤t} andG(t) is the convolution ofF(t) with itself. Conditions on the distribution of independent nonnegative random variablesX andY such that max(X, Y) and min(X, Y) have a subexponential distribution are given. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 138–144, July, 1997. Translated by N. K. Kulman     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

Isometric Embeddings of Families of Special Lagrangian Submanifolds

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g _t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ _t , then (Y,g _t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T ²-symmetry. Mathematics Subject Classification (2000): 53-XX, 53C38.Communicated by N. Hitchin (Oxford)     

Exact Rates of Convergence of Functional Limit Theorems for Csörgő;-Révész Increments of a Wiener Process

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 sup_{0≤ t ≤ T − aT} inf _f∈S sup_{0≤ x ≤1}|Y _t,T (x) −f(x)| and inf_{0≤ t ≤ T−aT} sup_{0≤ 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     

Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field X _i , i ∈ Z ^d from the density is contaminated by measurement errors. In particular, the observations Y _i , i ∈ Z ^d are such that Y _i = X _i + ε _i , where the errors ε _i are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hypothesis testing for the common mean of two normal distributions in the presence of an indifference zone

Summary LetX ₁,...,X _m andY _t,...,Y be independent, random samples from populations which are N(θ,σ _x ² ) and N(θ,σ _y ² ), respectively, with all parameters unknown. In testingH ₀:θ=0 againstH ₁:θ≠0, thet-test based upon either sample is known to be admissible in the two-sample setting. If, however, one testsH ₀ againstH ₁:|θ|≧ε>0, with ε arbitrary, our main results show: (i) the construction of a test which is better than the particulart-test chosen, (ii) eacht-test is admissible under the invariance principle with respect to the group of scale changes, and (iii) there does not exist a test which simultaneously is better than botht-tests.     

On the transition from a Markov chain to a continuous time process

Starting from a real-valued Markov chain X₀,X₁,…,X_n with stationary transition probabilities, a random element {Y(t);t[0, 1]} of the function space D[0, 1] is constructed by letting Y(k/n)=X_k, k= 0,1,…,n, and assuming Y (t) constant in between. Sample tightness criteria for sequences {Y(t);t[0,1]};_n of such random elements in D[0, 1] are then given in terms of the one-step transition probabilities of the underlying Markov chains. Applications are made to Galton-Watson branching processes.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt ₀εℤ such thatf (X, t ₀) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t ₀) is irreducible for all but finitely manyt ₀εℤ unless the curve given byf (X, t)=0 has infinitely many points (x ₀,t ₀) withx ₀εℚ,t ₀εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.     

A global volume lemma and applications

Letf ^t be aC ² Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB _x (ε,t)=[y: dist (f ^s x,f ^s y)≤ε for all 0≤s≤t], then volB _x (ε,t) has the order exp(∫ ₀ ^t φ (f ^s x)ds) in the continuous time case and exp (Σ _s ^t−1 φ (f ^s x)) in the discrete time case, whereφ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in the unstable direction. An application to the theory of large deviations is given. Partially supported by US-Israel BSF. Partially supported by a Darpa grant.     

Randomly fractionally integrated processes

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d _t ) _t∈ℤ of real numbers; if the parameter sequence is constant d _t ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)^−d . They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε _t and B(d)ε _t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X _t ^A = A(d)ε _t and X _t ^B = B(d)ε _t by assuming that d = (d _t , t ∈ ℤ) is a random iid sequence, independent of the noise (ε _t ). In the case where the mean , we show that large sample properties of X ^A and X ^B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X _t ^A ) of a randomly fractionally integrated process X _t ^A with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d _t , this reduces to the standard Hermite rank used in Dobrushin and Major [2]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007.