A game or Israeli option is an American style option where both the writer and the holder have th... more A game or Israeli option is an American style option where both the writer and the holder have the right to terminate the contract before the expiration time. shows the fair price for this option can be expressed as the value of a Dynkin game. In general, there are no explicit formulas for fair prices of American and game options and approximations are used for their computations. The paper provides error estimates for binomial approximation of American put options and here we extend the approach of [17] in order to obtain error estimates for binomial approximations of game put options which is more complicated as it requires us to deal with two free boundaries corresponding to the writer and to the holder of the game option.
The paper is primarily concerned with the asymptotic behavior as N → ∞ of averages of nonconventi... more The paper is primarily concerned with the asymptotic behavior as N → ∞ of averages of nonconventional arrays having the form N -1 N n=1 ℓ j=1 T P j (n,N) f j where f j 's are bounded measurable functions, T is an invertible measure preserving transformation and P j 's are polynomials of n and N taking on integer values on integers. It turns out that when T is weakly mixing and P j (n, N ) = p j n + q j N are linear or, more generally, have the form P j (n, N ) = P j (n) + Q j (N ) for some integer valued polynomials P j and Q j then the above averages converge in L 2 but for general polynomials P j the L 2 convergence can be ensured even in the case ℓ = 1 only when T is strongly mixing. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers Λ with positive upper density there exists a subset N Λ of positive integers having uniformly bounded gaps such that for N ∈ N Λ and at least εN, ε > 0 of n's all numbers p j n + q j N, j = 1, ..., ℓ, belong to Λ. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.
Ergodic Theory and Dynamical Systems, Nov 12, 2021
For a ψ-mixing process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple returns {ξ q ... more For a ψ-mixing process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple returns {ξ q i,N (n) ∈ Γ N , i = 1, ..., ℓ} to a set Γ N for n until either a fixed number N or until the moment τ N when another multiple return {ξ q i,N (n) ∈ ∆ N , i = 1, ..., ℓ} takes place for the first time where Γ N ∩ ∆ N = ∅ and q i,N , i = 1, ..., ℓ are certain functions of n taking on nonnegative integer values when n runs from 0 to N . The dependence of q i,N (n)'s on both n and N is the main novelty of the paper. Under some restrictions on the functions q i,N we obtain Poisson distributions limits of N N when counting is until N as N → ∞ and geometric distributions limits when counting is until τ N as N → ∞. We obtain also similar results in the dynamical systems setup considering a ψ-mixing shift T on a sequence space Ω and studying the number of multiple returns {T q i,N (n) ω ∈ A a n , i = 1, ..., ℓ} until the first occurrence of another multiple return {T q i,N (n) ω ∈ A b m , i = 1, ..., ℓ} where A a n , A b m are cylinder sets of length n and m constructed by sequences a, b ∈ Ω, respectively, and chosen so that their probabilities have the same order.
We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomi... more We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black-Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11] and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.
Local limit theorems have their origin in the classical De Moivre-Laplace theorem and they study ... more Local limit theorems have their origin in the classical De Moivre-Laplace theorem and they study the asymptotic behavior as N → ∞ of probabilities having the form P {S N = k} where S N = N n=1 F (ξn) is a sum of an integer valued function F taken on i.i.d. or Markov dependent sequence of random variables {ξ j }. Corresponding results for lattice valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form S N = N n=1 F (ξn, ξ 2n , ..., ξ ℓn ) which continues the recent line of research studying various limit theorems for such expressions.
We provide conditions which yield a strong law of large numbers for expressions of the form 1/N P... more We provide conditions which yield a strong law of large numbers for expressions of the form 1/N P N n=1 F `X(q 1 (n)), • • • , X(q ℓ (n)) ´where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and q i , i > m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q 1 (n), ..., q ℓ (n).
Communications in Mathematical Physics, Apr 1, 1994
The paper deals with large deviation bounds for the proportion of periodic orbits with irregular ... more The paper deals with large deviation bounds for the proportion of periodic orbits with irregular behavior for expansive dynamical systems with specification, in particular, we obtain estimates for large deviations from the equidistribution for closed geodesies on negatively curved manifolds. We derive also large deviation bounds in the averaging principle when the fast motion is the shift along periodic orbits.
It is known since that the slow motion X ε in the timescaled multidimensional averaging setup con... more It is known since that the slow motion X ε in the timescaled multidimensional averaging setup converges weakly as ε → 0 to a diffusion process provided EB(x, ξ(s)) ≡ 0 where ξ is a sufficiently fast mixing stochastic process. In this paper we show that both X ε and a family of diffusions Ξ ε can be redefined on a common sufficiently rich probability space so that where all Ξ ε , ε > 0 have the same diffusion coefficients but underlying Brownian motions may change with ε. This is the first strong approximation result both in the above setup and at all when the limit is a nontrivial multidimensional diffusion. We obtain also a similar result for the corresponding discrete time averaging setup which was not considered before at all. As an application we consider Dynkin's games with path dependent payoffs involving a diffusion and obtain error estimates for computation of values of such games by means of such discrete time approximations which provides a more effective computational tool than the standard discretization of the diffusion itself.
For a ψ-mixing stationary process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple re... more For a ψ-mixing stationary process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple recurrencies {ξ q i (n) ∈ Γ N , i = 1, ..., ℓ} to a set Γ N for n until the moment τ N (which we call a hazard) when another multiple recurrence {ξ q i (n) ∈ ∆ N , i = 1, ..., ℓ} takes place for the first time where Γ N ∩ ∆ N = ∅ and q i (n) < q i+1 (n), i = 1, ..., ℓ are nonnegative increasing functions taking on integer values on integers. It turns out that if P {ξ 0 ∈ Γ N } and P {ξ 0 ∈ ∆ N } decay in N with the same speed then N N converges weakly to a geometrically distributed random variable. We obtain also a similar result in the dynamical systems setup considering a ψ-mixing shift T on a sequence space Ω and study the number of multiple recurrencies {T q i (n) ω ∈ A b n , i = 1, ..., ℓ} until the first occurence of another multiple recurrence {T q i (n) ω ∈ A a m , i = 1, ..., ℓ} where A a m , A b n are cylinder sets of length m and n constructed by sequences a, b ∈ Ω, respectively, and chosen so that their probabilities have the same order. This work is motivated by a number of papers on asymptotics of numbers of single and multiple returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".
The paper introduces and studies hedging for game (Israeli) style extension of swing options cons... more The paper introduces and studies hedging for game (Israeli) style extension of swing options considered as multiple exercise derivatives. Assuming that the underlying security can be traded without restrictions we derive a formula for valuation of multiple exercise options via classical hedging arguments. Introducing the notion of the shortfall risk for such options we study also partial hedging which leads to minimization of this risk.
Stochastic Processes and their Applications, Aug 1, 2016
We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form ξ N = 1 √ N N... more We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form ξ N = 1 √ N N n=1 (F (X(q 1 (n)), ..., X(q ℓ (n))) -F ) where X(n) is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, F = F d(µ × ... × µ), µ is the distribution of X(0) and q i (n) = in for 1 ≤ i ≤ k while for i > k they are positive functions taking integer values on integers with some growth conditions which are satisfies, for instance, when they are polynomials of increasing degrees. Our setup is similar to where a nonconventional functional central limit theorem was obtained and the present paper provides estimates for the convergence speed. As a part of the study we provide answers for the crucial question on positivity of the limiting variance lim N→∞ Var(ξ N ) which was not studied in . Extensions to the continuous time case will be discussed as well. As in our results are applicable to stationary processes generated by some classes of sufficiently well mixing Markov chains and dynamical systems.
Probability Theory and Related Fields, May 12, 2009
The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337-3... more The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337-349, 1987) and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1/N where T is a weakly mixing measure preserving transformation, f i 's are bounded measurable functions and q i 's are polynomials taking on integer values on the integers. Motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1/ where X i 's are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, F = Fd(μ 0 × μ 1 × • • • × μ ), μ j is the distribution of X j (0), and q i 's are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q i 's are polynomials of growing degrees. When F(x 0 , x 1 , . . . , x ) = x 0 x 1 x 2 • • • x exponentially fast α-mixing already suffices. This result can be applied in the case when X i (n) = T n f i where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as in the case when X i (n) = f i (ξ n ) where ξ n is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.
Transactions of the American Mathematical Society, 1996
I start with random base expansions of numbers from the interval [0, 1] and, more generally, vect... more I start with random base expansions of numbers from the interval [0, 1] and, more generally, vectors from [0, 1] d , which leads to random expanding transformations on the d-dimensional torus T d . As in the classical deterministic case of Besicovitch and Eggleston I find the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets "invariant" with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg. In place of the usual entropy which emerges (as explained in Billingsley's book) in the Besicovitch-Eggleston and Furstenberg cases, the relativised entropy of random expanding transformations comes into play in my setup. I also extend to the case of random transformations the Bowen-Ruelle formula for the Hausdorff dimension of repellers.
The paper deals with the fast-slow motions setups in the continuous time where Σ and b are smooth... more The paper deals with the fast-slow motions setups in the continuous time where Σ and b are smooth matrix and vector functions and ξ is a stationary vector stochastic process with weakly dependent terms and such that Eξ(0) = 0. The assumptions imposed on the process ξ allow applications to a wide class of observables g in the dynamical systems setup so that ξ can be taken in the form ξ where F is either a flow or a diffeomorphism with some hyperbolicity and g is a vector function. In this paper we show that both X ε and a family of diffusions Ξ ε can be redefined on a common sufficiently rich probability space so that E sup 0≤t≤T |X ε (t) -Ξ ε (t)| p ≤ Cε δ , p ≥ 1 for some C, δ > 0 and all ε > 0, where all Ξ ε , ε > 0 have the same diffusion coefficients but underlying Brownian motions may change with ε.
First, we obtain decay rates of probabilities of tails of polynomials in several independent rand... more First, we obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails. Then we derive stable limit theorems for sums of the form Nt≥n≥1 F X q 1 (n) , . . . , X q ℓ (n) where F is a polynomial, q i (n) is either n -1 + i or ni and Xn, n ≥ 0 is a sequence of independent identically distributed random variables with heavy tails. Our results can be viewed as an extension to the heavy tails case of the nonconventional functional central limit theorem from .
We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces con... more We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces considering both a conventional and a nonconventional setups. Namely, under certain conditions on events Γ 1 , Γ 2 , ... we show that with probability one where q i (n), i = 1, ..., ℓ are integer valued functions satisfying certain assumptions and I Γ denotes the indicator of Γ. When ℓ = 1 (called the conventional setup) this convergence can be established under φ-mixing conditions while when ℓ > 1 (called a nonconventional setup) the stronger ψ-mixing condition is required. These results are extended to shifts T of sequence spaces where n , i = 1, ..., ℓ, n ≥ 1 is a sequence of cylinder sets. As an application we study the asymptotical behavior of maximums of certain logarithmic distance functions and of ( multiple) hitting times of shrinking cylinders.
The work treats dynamical systems given by ordinary differential equations in the form where fast... more The work treats dynamical systems given by ordinary differential equations in the form where fast motions Y ε depend on the slow motion X ε (coupled with it) and they are either given by another differential equation or perturbations of an appropriate parametric family of Markov processes with freezed slow variables. In the first case we assume that the fast motions are hyperbolic for each freezed slow variable and in the second case we deal with Markov processes such as random evolutions which are combinations of diffusions and continuous time Markov chains. First, we study large deviations of the slow motion X ε from its averaged (in fast variables Y ε ) approximation Xε . The upper large deviation bound justifies the averaging approximation on the time scale of order 1/ε, called the averaging principle, in the sense of convergence in measure (in the first case) or in probability (in the second case) but our real goal is to obtain both the upper and the lower large deviations bounds which together with some Markov property type arguments (in the first case) or with the real Markov property (in the second case) enable us to study (adiabatic) behavior of the slow motion on the much longer exponential in 1/ε time scale, in particular, to describe its fluctuations in a vicinity of an attractor of the averaged motion and its rare (adiabatic) transitions between neighborhoods of such attractors. When the fast motion Y ε does not depend on the slow one we arrive at a simpler averaging setup studied in numerous papers but the above fully coupled case, which better describes real phenomena, leads to much more complicated problems. Part 1. Hyperbolic Fast Motions 1.1. Introduction 1.2. Main results 1.3. Dynamics of Φ t ε 1.4. Large deviations: preliminaries 1.5. Large deviations: Proof of Theorem 1.2.3 1.6. Further properties of S-functionals 1.7. "Very long" time behavior: exits from a domain 1.8. Adiabatic transitions between basins of attractors 1.9. Averaging in difference equations 1.10. Extensions: stochastic resonance 1.11. Young measures approach to averaging Part 2. Markov Fast Motions 2.1. Introduction 2.2. Preliminaries and main results 2.3. Large deviations 2.4. Verifying assumptions for random evolutions 2.5. Further properties of S-functionals 2.6. "Very long" time behavior: exits from a domain 2.7. Adiabatic transitions between basins of attractors 2.8. Averaging in difference equations 2.9. Extensions: stochastic resonance 2.10. Young measures approach to averaging Bibliography v
we obtained a nonconventional invariance principle (functional central limit theorem) for suffici... more we obtained a nonconventional invariance principle (functional central limit theorem) for sufficiently fast mixing stochastic processes with discrete and continuous time. In this paper we derive a nonconventional invariance principle for sufficiently well mixing random fields.
We start by briefly surveying research on optimal stopping games since their introduction by E.B.... more We start by briefly surveying research on optimal stopping games since their introduction by E.B. Dynkin more than 40 years ago. Recent renewed interest to Dynkin's games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities for the last decade. Among various results on game options we consider error estimates for their discrete approximations, swing game options, game options in markets with transaction costs and other questions.
The usual random walk on a group (homogeneous both in time and in space) is determined by a proba... more The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.
A game or Israeli option is an American style option where both the writer and the holder have th... more A game or Israeli option is an American style option where both the writer and the holder have the right to terminate the contract before the expiration time. shows the fair price for this option can be expressed as the value of a Dynkin game. In general, there are no explicit formulas for fair prices of American and game options and approximations are used for their computations. The paper provides error estimates for binomial approximation of American put options and here we extend the approach of [17] in order to obtain error estimates for binomial approximations of game put options which is more complicated as it requires us to deal with two free boundaries corresponding to the writer and to the holder of the game option.
The paper is primarily concerned with the asymptotic behavior as N → ∞ of averages of nonconventi... more The paper is primarily concerned with the asymptotic behavior as N → ∞ of averages of nonconventional arrays having the form N -1 N n=1 ℓ j=1 T P j (n,N) f j where f j 's are bounded measurable functions, T is an invertible measure preserving transformation and P j 's are polynomials of n and N taking on integer values on integers. It turns out that when T is weakly mixing and P j (n, N ) = p j n + q j N are linear or, more generally, have the form P j (n, N ) = P j (n) + Q j (N ) for some integer valued polynomials P j and Q j then the above averages converge in L 2 but for general polynomials P j the L 2 convergence can be ensured even in the case ℓ = 1 only when T is strongly mixing. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers Λ with positive upper density there exists a subset N Λ of positive integers having uniformly bounded gaps such that for N ∈ N Λ and at least εN, ε > 0 of n's all numbers p j n + q j N, j = 1, ..., ℓ, belong to Λ. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.
Ergodic Theory and Dynamical Systems, Nov 12, 2021
For a ψ-mixing process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple returns {ξ q ... more For a ψ-mixing process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple returns {ξ q i,N (n) ∈ Γ N , i = 1, ..., ℓ} to a set Γ N for n until either a fixed number N or until the moment τ N when another multiple return {ξ q i,N (n) ∈ ∆ N , i = 1, ..., ℓ} takes place for the first time where Γ N ∩ ∆ N = ∅ and q i,N , i = 1, ..., ℓ are certain functions of n taking on nonnegative integer values when n runs from 0 to N . The dependence of q i,N (n)'s on both n and N is the main novelty of the paper. Under some restrictions on the functions q i,N we obtain Poisson distributions limits of N N when counting is until N as N → ∞ and geometric distributions limits when counting is until τ N as N → ∞. We obtain also similar results in the dynamical systems setup considering a ψ-mixing shift T on a sequence space Ω and studying the number of multiple returns {T q i,N (n) ω ∈ A a n , i = 1, ..., ℓ} until the first occurrence of another multiple return {T q i,N (n) ω ∈ A b m , i = 1, ..., ℓ} where A a n , A b m are cylinder sets of length n and m constructed by sequences a, b ∈ Ω, respectively, and chosen so that their probabilities have the same order.
We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomi... more We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black-Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11] and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.
Local limit theorems have their origin in the classical De Moivre-Laplace theorem and they study ... more Local limit theorems have their origin in the classical De Moivre-Laplace theorem and they study the asymptotic behavior as N → ∞ of probabilities having the form P {S N = k} where S N = N n=1 F (ξn) is a sum of an integer valued function F taken on i.i.d. or Markov dependent sequence of random variables {ξ j }. Corresponding results for lattice valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form S N = N n=1 F (ξn, ξ 2n , ..., ξ ℓn ) which continues the recent line of research studying various limit theorems for such expressions.
We provide conditions which yield a strong law of large numbers for expressions of the form 1/N P... more We provide conditions which yield a strong law of large numbers for expressions of the form 1/N P N n=1 F `X(q 1 (n)), • • • , X(q ℓ (n)) ´where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and q i , i > m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q 1 (n), ..., q ℓ (n).
Communications in Mathematical Physics, Apr 1, 1994
The paper deals with large deviation bounds for the proportion of periodic orbits with irregular ... more The paper deals with large deviation bounds for the proportion of periodic orbits with irregular behavior for expansive dynamical systems with specification, in particular, we obtain estimates for large deviations from the equidistribution for closed geodesies on negatively curved manifolds. We derive also large deviation bounds in the averaging principle when the fast motion is the shift along periodic orbits.
It is known since that the slow motion X ε in the timescaled multidimensional averaging setup con... more It is known since that the slow motion X ε in the timescaled multidimensional averaging setup converges weakly as ε → 0 to a diffusion process provided EB(x, ξ(s)) ≡ 0 where ξ is a sufficiently fast mixing stochastic process. In this paper we show that both X ε and a family of diffusions Ξ ε can be redefined on a common sufficiently rich probability space so that where all Ξ ε , ε > 0 have the same diffusion coefficients but underlying Brownian motions may change with ε. This is the first strong approximation result both in the above setup and at all when the limit is a nontrivial multidimensional diffusion. We obtain also a similar result for the corresponding discrete time averaging setup which was not considered before at all. As an application we consider Dynkin's games with path dependent payoffs involving a diffusion and obtain error estimates for computation of values of such games by means of such discrete time approximations which provides a more effective computational tool than the standard discretization of the diffusion itself.
For a ψ-mixing stationary process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple re... more For a ψ-mixing stationary process ξ 0 , ξ 1 , ξ 2 , ... we consider the number N N of multiple recurrencies {ξ q i (n) ∈ Γ N , i = 1, ..., ℓ} to a set Γ N for n until the moment τ N (which we call a hazard) when another multiple recurrence {ξ q i (n) ∈ ∆ N , i = 1, ..., ℓ} takes place for the first time where Γ N ∩ ∆ N = ∅ and q i (n) < q i+1 (n), i = 1, ..., ℓ are nonnegative increasing functions taking on integer values on integers. It turns out that if P {ξ 0 ∈ Γ N } and P {ξ 0 ∈ ∆ N } decay in N with the same speed then N N converges weakly to a geometrically distributed random variable. We obtain also a similar result in the dynamical systems setup considering a ψ-mixing shift T on a sequence space Ω and study the number of multiple recurrencies {T q i (n) ω ∈ A b n , i = 1, ..., ℓ} until the first occurence of another multiple recurrence {T q i (n) ω ∈ A a m , i = 1, ..., ℓ} where A a m , A b n are cylinder sets of length m and n constructed by sequences a, b ∈ Ω, respectively, and chosen so that their probabilities have the same order. This work is motivated by a number of papers on asymptotics of numbers of single and multiple returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".
The paper introduces and studies hedging for game (Israeli) style extension of swing options cons... more The paper introduces and studies hedging for game (Israeli) style extension of swing options considered as multiple exercise derivatives. Assuming that the underlying security can be traded without restrictions we derive a formula for valuation of multiple exercise options via classical hedging arguments. Introducing the notion of the shortfall risk for such options we study also partial hedging which leads to minimization of this risk.
Stochastic Processes and their Applications, Aug 1, 2016
We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form ξ N = 1 √ N N... more We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form ξ N = 1 √ N N n=1 (F (X(q 1 (n)), ..., X(q ℓ (n))) -F ) where X(n) is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, F = F d(µ × ... × µ), µ is the distribution of X(0) and q i (n) = in for 1 ≤ i ≤ k while for i > k they are positive functions taking integer values on integers with some growth conditions which are satisfies, for instance, when they are polynomials of increasing degrees. Our setup is similar to where a nonconventional functional central limit theorem was obtained and the present paper provides estimates for the convergence speed. As a part of the study we provide answers for the crucial question on positivity of the limiting variance lim N→∞ Var(ξ N ) which was not studied in . Extensions to the continuous time case will be discussed as well. As in our results are applicable to stationary processes generated by some classes of sufficiently well mixing Markov chains and dynamical systems.
Probability Theory and Related Fields, May 12, 2009
The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337-3... more The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337-349, 1987) and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1/N where T is a weakly mixing measure preserving transformation, f i 's are bounded measurable functions and q i 's are polynomials taking on integer values on the integers. Motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1/ where X i 's are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, F = Fd(μ 0 × μ 1 × • • • × μ ), μ j is the distribution of X j (0), and q i 's are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q i 's are polynomials of growing degrees. When F(x 0 , x 1 , . . . , x ) = x 0 x 1 x 2 • • • x exponentially fast α-mixing already suffices. This result can be applied in the case when X i (n) = T n f i where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as in the case when X i (n) = f i (ξ n ) where ξ n is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.
Transactions of the American Mathematical Society, 1996
I start with random base expansions of numbers from the interval [0, 1] and, more generally, vect... more I start with random base expansions of numbers from the interval [0, 1] and, more generally, vectors from [0, 1] d , which leads to random expanding transformations on the d-dimensional torus T d . As in the classical deterministic case of Besicovitch and Eggleston I find the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets "invariant" with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg. In place of the usual entropy which emerges (as explained in Billingsley's book) in the Besicovitch-Eggleston and Furstenberg cases, the relativised entropy of random expanding transformations comes into play in my setup. I also extend to the case of random transformations the Bowen-Ruelle formula for the Hausdorff dimension of repellers.
The paper deals with the fast-slow motions setups in the continuous time where Σ and b are smooth... more The paper deals with the fast-slow motions setups in the continuous time where Σ and b are smooth matrix and vector functions and ξ is a stationary vector stochastic process with weakly dependent terms and such that Eξ(0) = 0. The assumptions imposed on the process ξ allow applications to a wide class of observables g in the dynamical systems setup so that ξ can be taken in the form ξ where F is either a flow or a diffeomorphism with some hyperbolicity and g is a vector function. In this paper we show that both X ε and a family of diffusions Ξ ε can be redefined on a common sufficiently rich probability space so that E sup 0≤t≤T |X ε (t) -Ξ ε (t)| p ≤ Cε δ , p ≥ 1 for some C, δ > 0 and all ε > 0, where all Ξ ε , ε > 0 have the same diffusion coefficients but underlying Brownian motions may change with ε.
First, we obtain decay rates of probabilities of tails of polynomials in several independent rand... more First, we obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails. Then we derive stable limit theorems for sums of the form Nt≥n≥1 F X q 1 (n) , . . . , X q ℓ (n) where F is a polynomial, q i (n) is either n -1 + i or ni and Xn, n ≥ 0 is a sequence of independent identically distributed random variables with heavy tails. Our results can be viewed as an extension to the heavy tails case of the nonconventional functional central limit theorem from .
We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces con... more We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces considering both a conventional and a nonconventional setups. Namely, under certain conditions on events Γ 1 , Γ 2 , ... we show that with probability one where q i (n), i = 1, ..., ℓ are integer valued functions satisfying certain assumptions and I Γ denotes the indicator of Γ. When ℓ = 1 (called the conventional setup) this convergence can be established under φ-mixing conditions while when ℓ > 1 (called a nonconventional setup) the stronger ψ-mixing condition is required. These results are extended to shifts T of sequence spaces where n , i = 1, ..., ℓ, n ≥ 1 is a sequence of cylinder sets. As an application we study the asymptotical behavior of maximums of certain logarithmic distance functions and of ( multiple) hitting times of shrinking cylinders.
The work treats dynamical systems given by ordinary differential equations in the form where fast... more The work treats dynamical systems given by ordinary differential equations in the form where fast motions Y ε depend on the slow motion X ε (coupled with it) and they are either given by another differential equation or perturbations of an appropriate parametric family of Markov processes with freezed slow variables. In the first case we assume that the fast motions are hyperbolic for each freezed slow variable and in the second case we deal with Markov processes such as random evolutions which are combinations of diffusions and continuous time Markov chains. First, we study large deviations of the slow motion X ε from its averaged (in fast variables Y ε ) approximation Xε . The upper large deviation bound justifies the averaging approximation on the time scale of order 1/ε, called the averaging principle, in the sense of convergence in measure (in the first case) or in probability (in the second case) but our real goal is to obtain both the upper and the lower large deviations bounds which together with some Markov property type arguments (in the first case) or with the real Markov property (in the second case) enable us to study (adiabatic) behavior of the slow motion on the much longer exponential in 1/ε time scale, in particular, to describe its fluctuations in a vicinity of an attractor of the averaged motion and its rare (adiabatic) transitions between neighborhoods of such attractors. When the fast motion Y ε does not depend on the slow one we arrive at a simpler averaging setup studied in numerous papers but the above fully coupled case, which better describes real phenomena, leads to much more complicated problems. Part 1. Hyperbolic Fast Motions 1.1. Introduction 1.2. Main results 1.3. Dynamics of Φ t ε 1.4. Large deviations: preliminaries 1.5. Large deviations: Proof of Theorem 1.2.3 1.6. Further properties of S-functionals 1.7. "Very long" time behavior: exits from a domain 1.8. Adiabatic transitions between basins of attractors 1.9. Averaging in difference equations 1.10. Extensions: stochastic resonance 1.11. Young measures approach to averaging Part 2. Markov Fast Motions 2.1. Introduction 2.2. Preliminaries and main results 2.3. Large deviations 2.4. Verifying assumptions for random evolutions 2.5. Further properties of S-functionals 2.6. "Very long" time behavior: exits from a domain 2.7. Adiabatic transitions between basins of attractors 2.8. Averaging in difference equations 2.9. Extensions: stochastic resonance 2.10. Young measures approach to averaging Bibliography v
we obtained a nonconventional invariance principle (functional central limit theorem) for suffici... more we obtained a nonconventional invariance principle (functional central limit theorem) for sufficiently fast mixing stochastic processes with discrete and continuous time. In this paper we derive a nonconventional invariance principle for sufficiently well mixing random fields.
We start by briefly surveying research on optimal stopping games since their introduction by E.B.... more We start by briefly surveying research on optimal stopping games since their introduction by E.B. Dynkin more than 40 years ago. Recent renewed interest to Dynkin's games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities for the last decade. Among various results on game options we consider error estimates for their discrete approximations, swing game options, game options in markets with transaction costs and other questions.
The usual random walk on a group (homogeneous both in time and in space) is determined by a proba... more The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.
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Papers by Yuri Kifer