Michael J. Klass

A Diffusion Model for Optimal Portfolio Selection in the Presence of Brokerage Fees

We consider a financial market model with two assets. One has deterministic rate of growth, while the rate of growth of the second asset is governed by a Brownian motion with drift. We can shift money from one asset to another; however, there are losses of money brokerage fees involved in shifting money from the risky to the nonrisky asset. We want to maximize the expected rate of growth of funds. It is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort.

Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables Bt>0 and At, let Yt(λ)=exp{λAt−λ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt≥0At/{Bt(log logBt)1/2} and variants thereof, when EYt(λ)≤1 or when Yt(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with At=Mt and

Affine normability of partial sums of I.I.D. random vectors: A characterization

B_{t}=\sqrt {\langle M\rangle _{t}}

Which I.I.D. Sums are recurrently dominated by their maximal terms

, and sums of conditionally symmetric variables di with At=∑i=1tdi and

A generalization of Burnside's combinatorial lemma

B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}

House-Hunting Without Second Moments

. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in Rm, m≥1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving ∑i=1tdi and ∑i=1tdi2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.

Moment Bounds for Self-Normalized Martingales

SummaryLet X, X1,X2,... be i.i.d. d-dimensional random vectors with partial sums Sn. We identify the collection of random vectors X for which there exist non-singular linear operators Tn and vectors υn∈ℝ d such that {ℒ(Tn(Sn−υn)),n>=1} is tight and has only full weak subsequential limits. The proof is constructive, providing a specific sequence {Tn}. The random vector X is said to be in the generalized domain of attraction (GDOA) of a necessarily operator-stable law γ if there exist {Tn} and {υn} such that ℒ(Tn(Sn−υn))→γ. We characterize the GDOA of every operator-stable law, thereby extending previous results of Hahn and Klass; Hudson, Mason, and Veeh; and Jurek. The characterization assumes a particularly nice form in the case of a stable limit. When γ is symmetric stable, all marginals of X must be in the domain of attraction of a stable law. However, if γ is a nonsymmetric stable law then X may be in the GDOA of γ even if no marginal is in the domain of attraction of any law.

On The Consumption/Distribution Theorem Under The Long-Run Growth Criterion Subject To A Drawdown Constraint

For i.i.d. sequences (Xn) we characterize lim supn→∞Xn/sup1≤i<n|Si|=∞ in terms of the distribution function. Previous results of Kesten(3) and Wittmann(6) are immediate consequences. We also determine the “typical” magnitude of sup1≤i≤n|Si|.

A Series Criterion for the Almost-Sure Growth Rate of the Generalized Diameter of an Increasing Sequence of Random Points

Abstract Let G be a finite group which acts on a set S. We present a method of computing the entire distribution of G-orbits of S (the number of k-element G-orbits of S for all k) in terms of the number of s ϵ S fixed by every σ ϵ H for subgroups H of G, and the Mobius function μ(·, ·) defined on the subgroup lattice of G. We deduce Burnsides lemma as a consequence of our result.

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

Abstract In the house-hunting problem, i.i.d. random variables, X 1, X 2,… are observed sequentially at a cost of c > 0 per observation. The problem is to choose a stopping rule, N, to maximize E(X N − Nc). If the Xs have a finite second moment, the optimal stopping rule is N* = min {n ≥ 1: X n > V*}, where V* satisfies E(X − V*)+ = c. The statement of the problem and its solution requires only the first moment of the X n to be finite. Is a finite second moment really needed? In 1970, Herbert Robbins showed, assuming only a finite first moment, that the rule N* is optimal within the class of stopping rules, N, such that E(X N − Nc)− > −∞, but it is not clear that this restriction of the class of stopping rules is really required. In this article it is shown that this restriction is needed, but that if the expectation is replaced by a generalized expectation, N* is optimal out of all stopping rules assuming only first moments.