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Given a Wiener process with unknown and unobservable drift, we try to estimate this drift as effectively but also as quickly as possible, in the presence of a quadratic penalty for the estimation error and of a fixed, positive cost per unit of observation time. In a Bayesian framework, where the unobservable drift is assumed to have a known “prior” distribution, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale. We establish structural properties of the solution for the corresponding problem of optimal stopping. In particular, we show that, regardless of the prior distribution, the continuation region is monotonically shrinking in time. Moreover, we provide conditions on the prior distribution that guarantee a one-sided stopping region. Lastly, some concrete prior distributions are studied to illustrate the theoretical results.

В статье обсуждается вариационная характеризация консервативной диффузии как градиентного потока энтропии и дается ее вероятностная интерпретация с помощью анализа возмущений на основе стохастического исчисления. Р. Джорданом, Д. Киндерлерером и Ф. Отто было показано, что для диффузионных процессов типа Ланжевена-Смолуховского поток Фоккера-Планка вероятностных плотностей максимизирует скорость диссипации относительной энтропии, измеряемой расстоянием, пройденным в окружающем пространстве вероятностных мер с конечными вторыми моментами, в смысле квадратичной метрики Васерштейна. Мы получаем новые, основанные на стохастических процессах, версии этих свойств, справедливые вдоль почти каждой траектории диффузионного движения при обратном течении времени, непосредственно используя методологию теории возмущений. Усредняя наши траекторные результаты относительно меры на пространстве траекторий, мы устанавливаем максимальную скорость диссипации энтропии вдоль потока Фоккера-Планка и точно измеряем отклонение от этого максимума, соответствующее любому заданному возмущению. Как следствие нашего траекторного подхода мы выводим HWI-неравенство, связывающее относительную энтропию (H), расстояние Васерштейна (W) и относительную информацию Фишера (I).

We show that “full-bang” control is optimal in a problem which combines features of (i) sequential least-squares estimation with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded control of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary stopping. We develop also the optimal filtering and stopping rules in this context.

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of stochastic control with discretionary stopping involving Walsh diffusions.

This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale $X$. The criterion is minimization of $\mathbf{L}^2$–distance, or "least-squares". This problem has a straightforward and well-known solution when $X$ is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure $P$. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to $P$, under which $X$ behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it "has no gap" (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also "strong", in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of "variance-optimal" and "mean-variance efficient" portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Föllmer, D. Sondermann and most notably M. Schweizer.

Almost twenty years ago, E.R. Fernholz introduced portfolio generating functions which can be used to construct a variety of portfolios, solely in the terms of the individual companies' market weights. I. Karatzas and J. Ruf recently developed another methodology for the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. In this paper, both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions are substituted by path-dependent functionals, which involve the current market weights, as well as additional bounded-variation functions of past and present market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible, and yields improved conditions for outperforming the market portfolio over suitable time-horizons., 45 pages, 3 figures

The capitalization-weighted total relative variation $\sum_{i=1}^d \int_0^\cdot \mu_i (t) \mathrm{d} \langle \log \mu_i \rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $\mu_i (\cdot)$ is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.

Background/Aims: Dialysis membrane has been implicated in selenium (Se) deficiency in hemodialysis (HD). Intradialytic Se removal into dialysate through different membranes was investigated. Methods: We studied 19 patients on standard HD with low-flux polysulfone membrane (group A), 10 patients on standard HD with ethylene vinyl alcohol membrane (group B), 12 patients on hemodiafiltration (HDF; group C) and 16 healthy subjects (control group D). Se was measured in blood before and after dialysis session and in effluent dialysate every hour during session. Results: In all patients together, pre-dialysis serum Se levels were lower than those in control group, but, in a separate analysis, only in standard HD. In all patient groups, there was a net Se removal into dialysate but it was greater in HDF patients who, however, had similar pre-dialysis serum Se levels to those in healthy controls. Conclusion: An intradialytic Se loss was found with all 3 membrane types, but it is not the principal factor for Se depletion in HD.

We study a stochastic differential equation with rank-based characteristics on the plane. We find its flow solutions and characterize coalescence., Scientific and Technological Research Council of Turkey (TÜBİTAK)

We prove the existence of stationary monetary equilibrium with inflation in a “Bewley” model with constant aggregate real variables but with idiosyncratic shocks to the endowments of a continuum of individual agents, when a central bank stands ready to borrow or lend fiat money at a fixed nominal rate of interest and the agents face borrowing constraints. We also find that, in the presence of real micro uncertainty about individual endowments, the rate of inflation is higher (equivalently, the real rate of interest is lower) than it would be in a “certainty-equivalent economy”; to wit, one in which every agent’s endowment is replaced by its expected value. Thus, underlying microeconomic uncertainty and borrowing constraints are shown to generate additional inflation.

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems. The analysis depends crucially on properties of a skew Brownian motion with two-valued drift of the bang-bang type, which we also study in some detail. These properties allow us to compute the transition probabilities of the original planar diffusion, and to study its behavior under time reversal.

Background. Active vitamin D is an effective treatment for secondary hyperparathyroidism (SHPT) in chronic kidney disease (CKD) patients often complicated by hypercalcaemia and hyperphosphataemia. Treatment with paricalcitol, a selective vitamin D receptor activator, has shown benefits by adequately reducing parathyroid hormone (PTH) levels with minimal changes in serum calcium (Ca) and phosphorus (P). The purpose of this study is to present data on the use of oral paricalcitol in real-life clinical practice in patients with CKD stage 3–4 and SHPT. Methods. We studied 43 patients, M/F: 25/18, median age: 74 years (47–87), CKD stage 3/4: 16/ 27, with SHPT, who were prescribed oral paricalcitol at recommended doses for 6 months. Monthly measurements of serum intact PTH (iPTH), Ca, P, alkaline phosphatase (ALP), haemoglobin, albumin (ALB), lipid profile, proteinuria and 24-h urine creatinine clearance were performed 3 months before and 6 months after treatment initiation. Results. Paricalcitol induced a significant, early and sustained, through the end of follow-up period, decrease in iPTH and ALP levels and an increase in serum ALB. No significant increase in Ca and P levels as well as in Ca × P product was observed during the study period. No significant changes were found in protein excretion, kidney function and the other measured parameters between baseline and last evaluation. Paricalcitol final median dose was 5 μg/week ranging between 3 and 7 μg/week. Conclusions. In the context of real-life clinical practice, oral paricalcitol for 6 months is an effective, well-tolerated treatment of SHPT in CKD stage 3–4 with minimal effects on calcium and phosphorus metabolism.

For given nonnegative constants g, h, ρ, σ with ρ 2 + σ 2 = 1 and g + h > 0, we construct a diffusion process (X 1(·), X 2(·)) with values in the plane and infinitesimal generator $${\begin{array}{ll}\fancyscript{L}=\mathbf{1}_{\{ x_1 > x_2\}}\left(\frac{\rho^2}2{\frac{\partial^2}{\partial x{_1^2}}} +\frac{\sigma^2}{2}{\frac{\partial^2}{\partial x{_2^2}}}-h\frac{\partial}{\partial x_1} +g \frac{\partial}{\partial{x_2}}\right)\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathbf{1}_{\{ x_1\le x_2\}}\left(\frac{\sigma^2}{2}{\frac{\partial^2}{\partial x{_1^2}}} +\frac{\rho^2}{2} {\frac{\partial^2}{\partial x{_2^2}}}+g\frac{\partial}{\partial x_1} - h \frac{\partial}{\partial{x_2}}\right),\,\,\,\,\,\,\,\,\,\,\,\, (0.1)\end{array}}$$ and discuss its realization in terms of appropriate systems of stochastic differential equations. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation $$\begin{array}{ll}{\rm d}Z(t) = f \big(Z (t)\big){\rm d}M( t) + {\rm d}N(t), \quad Z(0) = \xi\end{array}$$ driven by suitable continuous, orthogonal semimartingales M(·) and N(·) and with f(·) of bounded variation, which we study here in detail; and those of a one-dimensional diffusion Y(·) with bang-bang drift $${dY(t) = -\lambda {\rm sign} \big( Y (t) \big) {\rm d}t + {\rm d}W (t), Y(0)=y}$$ driven by a standard Brownian motion W(·). We also show that the planar diffusion (X 1(·), X 2(·)) can be represented in terms of this process Y(·), its local time L Y (·) at the origin, and an independent standard Brownian motion Q(·), in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.

The monetary and fiscal control of a simple economy without outside randomness is studied here from the microeconomic basis of a strategic market game. The government’s bureaucracy is treated as a public good that provides services at a cost. A conventional public good is also considered.

Wiener chaos expansions have been used by Lototsky and Rozovskii for constructing solutions to forward stochastic evolution equations. In this work we propose a similar methodology for studying linear backward stochastic evolution equations (BSEEs) and introduce generalized solutions of BSEEs in weighted Wiener chaos spaces.

It holds in great generality that a plan is optimal for a dynamic programming problem, if and only if it is “thrifty” and “equalizing.” An alternative characterization of an optimal plan, that applies in many economic models, is that the plan must satisfy an appropriate Euler equation and a transversality condition. Here we explore the connections between these two characterizations.

Background: Optimal dialysate calcium concentration (DCa) has not been determined under less hypercalcemic vitamin D analogues and Ca-free phosphate (P) binders. Methods: Twelve hyperparathyroidic hemodialysis patients under sevelamer hydrochloride were treated with paricalcitol in three periods (A, B and C) under DCa of 3.5, 3 and 2.5 mEq/l, respectively, in a 3-way open-label randomized crossover study. Results: Serum parathyroid hormone decreased in all periods. Under DCa = 2.5 mEq/l, there was a need for longer treatment duration, higher paricalcitol doses and increased sevelamer doses for higher serum P levels. No differences in serum Ca were observed. Ca × P values followed P changes. Episodes of Ca × P >55 mg2/dl2 were more frequent in the C period. Conclusions: Under paricalcitol and sevelamer, serum parathyroid hormone decreased without Ca increase under any DCa. DCa of 2.5 mEq/l resulted in higher paricalcitol doses, increased serum P levels and more frequent high Ca × P episodes and may not be optimal with the new medications.

This paper studies the habit-forming preference problem of maximizing total expected utility from consumption net of the standard of living, a weighted average of past consumption. We describe the effective state space of the corresponding optimal wealth and standard of living processes, identify the associated value function as a generalized utility function, and exploit the interplay between dynamic programming and Feynman-Kac results via the theory of random fields and stochastic partial differential equations (SPDEs). The resulting value random field of the optimization problem satisfies a nonlinear, backward SPDE of parabolic type, widely referred to as the stochastic Hamilton-Jacobi-Bellman equation. The dual value random field is characterized further in terms of a backward parabolic SPDE which is linear. Progressively measurable versions of stochastic feedback formulae for the optimal portfolio and consumption choices are obtained as well.

We argue that real uncertainty itself causes long-run nominal inflation. Consider an infinite horizon cash-in-advance economy with a representative agent and real uncertainty, modeled by independent, identically distributed endowments. Suppose the central bank fixes the nominal rate of interest. We show that the equilibrium long-run rate of inflation is strictly higher, on almost every path of endowment realizations, than it would be if the endowments were constant. Indeed, we present an explicit formula for the long-run rate of inflation, based on the famous Fisher equation. The Fisher equation says the short-run rate of inflation should equal the nominal rate of interest less the real rate of interest. The long-run Fisher equation for our stochastic economy is similar, but with the rate of inflation replaced by the harmonic mean of the growth rate of money.

Stationary equilibria are constructed for a series of nonstochastic production economies in which the decisions of producers, wage earners, shareholders, and savers modulate, via a “production function”, the endowment variables in an additive manner. The efficiency of each model is compared to that of a single agent who produces for personal consumption.

We develop a martingale approach for continuous-time stochastic control with discretionary stopping. The relevant Dynamic Programming Equation and Maximum Principle are presented. Necessary and sufficient conditions are provided for the optimality of a control strategy; these are analogues of the "equalization" and "thriftiness" conditions introduced by Dubins and Savage (1976) in a related, discrete-time context. The existence of a thrifty control strategy is established.

Over the long term, the returns on smaller stocks are likely to be higher than the returns on larger stocks. This phenomenon has been called size effect, and a number of explanations have been proposed to account for it. Here we show that the difference in return between the larger and the smaller stocks can be accounted for by a liquidity premium for the smaller stocks, and we estimate the value of this premium using structural parameters for the capital distribution of the U.S. stock market during the 1990s

A change in the arrival rate of a Poisson process sometimes necessitates immediate action. If the change time is unobservable, then the design of online change detection procedures becomes important and is known as the Poisson disorder problem. Formulated and partially solved by Davis [Banach Center Publ., 1 (1976) 65–72], the standard Poisson problem addresses the tradeoff between false alarms and detection delay costs in the most useful way for applications. In this paper we solve the standard problem completely and describe efficient numerical methods to calculate the policy parameters.

The game approach to the theory of optimal stopping assumes two players, the “controller” and the “stopper”. The reward of the game is a nonnegative process Y with RCLL paths on a time-horizon [0,T...

We provide simple, easy-to-test criteria for the existence of relative arbitrage in equity markets. These criteria postulate essentially that the excess growth rate of the market portfolio, a positive quantity that can be estimated or even computed from a given market structure, be ‘‘sufficiently large’’. We show that conditions which satisfy these criteria are manifestly present in the U.S. equity market. We then construct examples of abstract markets in which the criteria hold. These abstract markets allow us to isolate conditions similar to those prevalent in actual markets, and to construct explicit portfolios under these conditions. We study in some detail a specific example of an abstract market which is volatility-stabilized, in that the return from the market portfolio has constant drift and variance rates while the smallest stocks are assigned the largest volatilities. A rather interesting probabilistic structure emerges, in which time changes and the asymptotic theory for planar Brownian motion play crucial roles. The largest stock and the overall market grow at the same, constant rate, though individual stocks fluctuate widely.

We formulate a stochastic control problem with a general information structure, and show that an optimal law exists and is characterized as the unique solution of a recursive stochastic equation. For a special information structure of the “signal-plus-noise” type and with quadratic cost-functions, this recursive equation is solved for the value function of the control problem. This value function is then shown to satisfy the Mortensen equation of Dynamic Programming in function-space.

A new characterization of excessive functions for arbitrary one-dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function” and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth-fit in the context of optimal stopping. The results are illustrated in detail on a number of non-trivial, concrete optimal stopping problems, both old and new.

A process X is observed continuously in time; it behaves like Brownian motion with drift, which changes from zero to a known constant ϑ>0 at some time τ that is not directly observable. It is important to detect this change when it happens, and we attempt to do so by selecting a stopping rule T* that minimizes the “expected miss” E|T−τ| over all stopping rules T. Assuming that τ has an exponential distribution with known parameter λ>0 and is independent of the driving Brownian motion, we show that the optimal rule T* is to declare that the change has occurred, at the first time t for which