Robert Kertz - Academia.edu (original) (raw)

Papers by Robert Kertz

Stochastic Processes and their Applications, Jul 1, 1985

A procedure is given for proving strictness of some sharp, infinite-sequence martingale inequalit... more A procedure is given for proving strictness of some sharp, infinite-sequence martingale inequalities, which arise from sharp, finite-sequence martingale inequalities attained by degenerating extremal distributions. The procedure is applied to obtain strictness of the sharp inequalities of Cox and Kemperman P(lX,I > 1 for some i = 1,2,. .) C (ln 2))' sup E i: X, n ,=" and of Cox (sharp form of Burkholder's inequality) for all nontrivial martingale difference sequences X0, X,,

Annals of Probability, 1991

Advances in Applied Probability, Sep 1, 1988

Journal of Differential Equations, 1979

A, we let Us,?(t) represent a strongly continuous semigroup generated by orA + v$?. We show that ... more A, we let Us,?(t) represent a strongly continuous semigroup generated by orA + v$?. We show that under appropriate simultaneous convergence of LY and 7, Ua,,Jt) converges strongly to a strongly continuous semigroup U(t), having infinitesimal operator characterized through PA(VA)'f where t = min{j > 0, PA(VA)' # 0). We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations.

Mathematics of Operations Research, Aug 1, 1982

In countably additive gambling models with general utility functions, plans for play are construc... more In countably additive gambling models with general utility functions, plans for play are constructed which persist in being conditionally ϵ-optimal along every history and which are conditionally optimal whenever possible. Such plans are formed by piecing together plans which are known to be good for the gambler at single time periods. Verification of the optimality properties of these plans uses transient renewal theoretic arguments.

For k-armed Bernoulli bandits with discounting, sharp comparisons are given between average optim... more For k-armed Bernoulli bandits with discounting, sharp comparisons are given between average optimal rewards for a gambler and for a 'perfectly informed' gambler, over natural collections of prior distributions. Some of these comparisons are proved under general discounting, and others under non-increasing discount sequences. Connections are made between these comparisons and the concept of 'regret' in the minimax approach to bandit processes. Identification of extremal cases in the sharp comparisons is emphasized. A discrete-time k-armed bandit is a sequential decision problem in which a gambler selects one from among k stochastic processes to observe at each of a possibly infinite number of stages. The gambler's objective is to maximize the expected discounted sum of the observations. We consider k Bernoulli processes or arms (labeled 1, - . - , k) with parameters-probabilities of success-01, * * *, Ok. So the gambler's objective is to maximize the expected discounted number of successes, where the discount factor at time m is am-, ?0. The gambler does not know e = (01,- -, Ok) precisely and so regards it as a random variable; its distribution function is G. For the gambler, the sequence of successes and failures associated with arm i is therefore not a sequence of independent Bernoulli trials; rather, the trials are independent conditional on 0i and so are exchangeable. The gambler chooses an arm, say arm i, at stage m and receives cam for a success. The gambler observes whether the result is success or failure and thereby gets some information about Oi, the amount of information depending on G, and possibly also about other O's, again depending on G. The probability of success on arm i is E(Oi I G). The gambler proceeds sequentially, perhaps forsaking potential immediate gain for the possibility of learning something about 0 that will improve chances of later gain.

Lecture notes-monograph series, 2000

The set of probability measures on M with the stochastic order and the set of right-tail integrab... more The set of probability measures on M with the stochastic order and the set of right-tail integrable probability measures on M with the convex order form complete lattices. Connections of these lattice structures to martingale theory and to the Hardy-Littlewood maximal function are exhibited.

Journal of Multivariate Analysis, Jun 1, 1986

for some i.i.d.r.v.'s Xi ,..., X, taking values in [0, l] } is precisely the set {(x, y): x Q y <... more for some i.i.d.r.v.'s Xi ,..., X, taking values in [0, l] } is precisely the set {(x, y): x Q y < r,(x); 0 <x < 1 }, where the upper boundary function r, is given in terms of recursively defined functions. The result yields families of inequalities for the "prophet" problem, relating the mortal's value of a game V(X, ,..., X,) to the prophet's value of the game E(max,,,X,). The proofs utilize conjugate duality theory, probabilistic convexity arguments, and functional equation analysis. Asymptotic analysis of the "prophet" regions and inequalities is also given.

Contemporary mathematics, 1999

ABSTRACT The author determines the maximal extent to which prices for lookback options and Americ... more ABSTRACT The author determines the maximal extent to which prices for lookback options and American call options on the same underlying can differ over arbitrage-free complete financial market models. Apart from some common results of mathematical finance, the main mathematical tools are so-called prophet inequalities which relate E[max 0≤n≤N Z n ] and sup{E[Z τ ]:τ≤N a stopping time} for a given collection of non-negative random variables (Z n , n=0,...,N).

Annals of Probability, Dec 1, 1974

Transactions of the American Mathematical Society, 1974

For e > 0 small, let ifÇi) and S(t) be strongly continuous semigroups of linear contractions on a... more For e > 0 small, let ifÇi) and S(t) be strongly continuous semigroups of linear contractions on a Banach space L with infinitesimal operators A(e) and B respectively, where .4(e) = A^ * + eA^ ' + o(e) as e-» 0. Let {b(u); u > o} be a family of linear operators on L satisfying B(e)=B + eïl^ + e2n(2) + o(e2) as e-» 0. Assume that A(e) + e~lB(e) is the infinitesimal operator of a strongly continuous contraction semigroup re(f) on L and that for each f^L,limx^,0\¡Qe~XtS(t)fdt = Pf exists. We give conditions under which T6(t) converges as e-*• 0 to the semigroup generated by the closure of P(A^ + n^) on R(P) n P(4(1)) n PiXI*1*). If /'(4(1) + n(1))/=0,BA=-(^(1) + n(1V. and we let Vf=P(A^ + n(1))ft, then we show that T£(t/e)f converges as e-» 0 to the strongly continuous contraction semigroup generated by the closure of I" ' + V. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.

Journal of Functional Analysis, Feb 1, 1978

ABSTRACT For parameters η, let {B(η)} denote infinitesimal operators of strongly continuous semig... more ABSTRACT For parameters η, let {B(η)} denote infinitesimal operators of strongly continuous semigroups, with resolvents R(λ; B(η)) satisfying λR(λ; B(η)) = P(η) + λV(η) + o(λ). For parameters α, let {A(α)} denote possibly unbounded, linear operators for which {A(α) + B(η)} are infinitesimal operators of strongly continuous semigroups {Uα·η(t)}. For α, η converging simultaneously, we show strong convergence of the semigroups Uα·η(t) to a strongly continuous semigroup U(t), with limiting infinitesimal operator characterized by limα·η ∑jP(η) A(α) × (V(η) A(α))if. We give applications of the abstract perturbation theorems to limit theorems of random evolutions and associated abstract Cauchy problems, in which multiscaling occurs in the convergence.

Publications of The Research Institute for Mathematical Sciences, 1978

Annals of Probability, Oct 1, 1979

Israel Journal of Mathematics, Feb 1, 1992

Let /~ be any probability measure on R with f ]zld~u(z) < co and let/~* denote the associated Har... more Let /~ be any probability measure on R with f ]zld~u(z) < co and let/~* denote the associated Hardy and Littlewood maximal p.m., the p.m. of the Hardy and Littlewood maximal function obtained from ~. Dubins and Gilat [6] showed that ~* is the least upper bound, in the usual stochastic order, of the collection of p.m.'s v on R for which there is a martingale

Israel Journal of Mathematics, Jun 1, 1990

Let μ be any probability measure onR with λ |x|dμ(x)* denote its associated Hardy and Littlewood ... more Let μ be any probability measure onR with λ |x|dμ(x)* denote its associated Hardy and Littlewood maximal p.m. It is shown that for any p.m.v for which μ* in the usual stochastic order, there is a martingale (X t)0≦t≦1 for which sup0≦t≦1 X t andX 1 have respective p.m. 'sv and μ. The proof uses induction and weak convergence arguments; in special cases, explicit martingale constructions are given. These results provide a converse to results of Dubins and Gilat [6]; applications are made to give sharp martingale and ‘prophet’ inequalities.

Advances in Applied Probability, Jun 1, 1980

Journal of Applied Probability, Dec 1, 1997

It is demonstrated that for each n 2 there exists a universal constant, c n , such that for any s... more It is demonstrated that for each n 2 there exists a universal constant, c n , such that for any sequence of independent random variables fX r ; r 1g with nite variances, E max 1 i n X i ] ? sup T EX T c n p n ? 1 max 1 i n p Var (X i), where the supremum is over all stopping times T, 1 T n. Furthermore, c n 1=2 and lim inf n!1 c n 0:439485 : : :.

Advances in Applied Probability, Mar 1, 1999

Let X 1 ; X 2 ; be a sequence of i.i.d. random variables with distribution function F (x), and le... more Let X 1 ; X 2 ; be a sequence of i.i.d. random variables with distribution function F (x), and let X 1;n ; : : : ; X n;n be the sequence of order statistics of X 1 ; : : : ; X n. For a sequence (c n) n 1 of positive constants the smallest t o-line counting random variable is de ned by N e (c n) := maxfj n : X 1;n + + X j;n c n g. In this paper, an asymptotic joint distributional comparison is given between the o-line count N e (c n) and on-line counts N n for`good' sequential (on-line) policies satisfying the sum constraint P j 1 X j I(j n) c n. Speci cally, for such policies , under appropriate conditions on the distribution function F (x) and the constants (c n) n 1 , we nd sequences of positive constants (B n)

Transactions of the American Mathematical Society, 1983

If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and i... more If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V(X0,X¡,...) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v): x = V(X0, Xx,.. .,Xn) and y = £(maxyS"X¡) for some X0,..., Xn] is precisely the set C"= {(x,y):x<y<x(\ + n(\-*'/")); 0 « x « l}; and the set of ordered pairs {(x, y): x = V(X0, X,,...) and y = £(sup" X") for some X0, X,,...} is precisely the set X C= UQ. »=i As a special case, if A"0, X,,... is a martingale with EX0 = x, then £(max7tí" X) =c x + nx(\-x'/n) and £(sup" Xlt) « x-x\n x, and both inequalities are sharp. 1. Introduction. The subject of this paper is comparisons between the expected supremum of a uniformly bounded process and the optimal expected return (using stop rules) of the process. Let X0, Xx,... be random variables (on some common probability space (ß, &, P)) taking values in [0,1] and let V(X0, Xx,...) denote the value (supremum, over stop rules t, of EX,) of the process XQ, Xx,... (for a formal definition, see §2). The first main result of this paper (Theorem 3.2) gives a complete description of the possible values of the ordered pairs (V(X0,...,Xn), £(maxys;"Ay)) for all processes uniformly bounded in [0,1], and the second main result (Theorem 4.2) gives the corresponding result for infinite sequences. Comparisons of the value V(X0, Xx,...) and £(sup" Xn) have been called "prophet" problems because of the natural identification of E(supn Xn) with the optimal expected return of a prophet or player endowed with complete foresight. Such comparisons for sequences of independent random variables have been given by Krengel and Sucheston [12,13], Garling, and Dvoretzky (both in [13]), and Hill and Kertz [8-10]. Extending these results, Hill [7] has shown that (1) the set of ordered pairs ((x, y): x = V(X0, Xx,...) and y-£(sup" Xn) for some sequence of independent random variables X0, Xx,... taking values in [0,1]} is precisely the set ((x, y): x <y =£ 2x-x2;0 =£ x < 1}.