Narrow Results

We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Çinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation.

We establish a noncommutative Blackwell–Ross inequality for supermartingales under a suitable condition which generalizes Khan’s work to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $\beta$, and then give convergence theorems for martingales of $\beta$-conditional expectations. We give the Birkhoff ergodic theorem for $\beta$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $\beta$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on $M$. Finally, the large deviation property of $\beta$-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

We use Young’s and Hölder inequality combined with classical Gronwall’s inequality to derive present a new version of the stochastic of Gronwall’s inequalities with singular kernels.

We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation coefficient in the propagation direction. We discuss the limits of vanishing inner scale and divergent outer scale of the turbulent medium.

This paper derives an approximation for a generalized Langevin equation driven by a force with random oscillation in time and periodic oscillation in space. By a diffusion approximation and the weak convergence of periodic oscillation function, the solution of the generalized Langevin equation is shown to converge in distribution to the solution of a stochastic partial differential equations (SPDEs) driven by time white noise.

The authors establish a kind of inequalities for nonnegative submartingales which depend on two functions Φ and Ψ, and obtain the equivalent conditions for Φ and Ψ such that this kind of inequalities holds. In the case Φ = Ψ ∈ Δ₂, it is proved that this necessary and sufficient condition is equivalent to q_Φ > 1.

The output signal-to-interference (SIR) of conventional matched filter receiver in random environment is considered. When the number of users and the spreading factors tend to infinity with their ratio fixed, the convergence of SIR is showed to be with probability one under finite fourth moment of the spreading sequences. The asymptotic distribution of the SIR is also obtained.

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon–Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding.

We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond.

The paper also contains related characterizations of reflexivity and the infinite tree property.

The purpose of this chapter is to develop certain relatively recent mathematical discoveries known generally as stochastic calculus, or more specifically as Itô’s Calculus and to also illustrate their application in the pricing of options. The mathematical methods of stochastic calculus are illustrated in alternative derivations of the celebrated Black–Scholes–Merton model. The topic is motivated by a desire to provide an intuitive understanding of certain probabilistic methods that have found significant use in financial economics.

In this chapter, we will show that the asymptotic theory for linear regression models with IID observations carries over to ergodic stationary linear time series regression models with Martingale Difference Sequence (MDS) disturbances. Some basic concepts in time series analysis are introduced, and some tests for serial correlation are described.

Conditional probability distribution models have been widely used in economics and finance. In this chapter, we introduce two closely related popular methods to estimate conditional distribution models—Maximum Likelihood Estimation (MLE) and Quasi-MLE (QMLE). MLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is correctly specified, and QMLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is misspecified. Because the score function is an MDS and the dynamic Information Matrix (IM) equality holds when a conditional distribution model is correctly specified, the asymptotic properties of MLE is analogous to those of the OLS estimator when the regression disturbance is an MDS with conditional homoskedasticity, and we can use the Wald test, LM test and Likelihood Ratio (LR) test for hypothesis testing, where the LR test is analogous to the J · F test statistic. On the other hand, when the conditional distribution model is misspecified, the score function has mean zero, but it may no longer be an MDS and the dynamic IM equality may fail. As a result, the asymptotic properties of QMLE are analogous to those of the OLS estimator when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. Robust Wald tests and LM tests can be constructed for hypothesis testing, but the LR test can no longer be used, for a reason similar to the failure of the F-test statistic when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. We discuss methods to test the MDS property of the score function, and the dynamic IM equality, and correct specification of a conditional distribution model.

This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model.

An analogue of the Fourier transform will be introduced for all square integrable continuous martingale processes whose quadratic variation is deterministic. Using this transform we will formulate and prove a stochastic Heisenberg inequality.

Standard use of Cox's regression model and other relative risk regression models for censored survival data requires collection of covariate information on all individuals under study even when only a small fraction of them die or get diseased. For such situations risk set sampling designs offer useful alternatives. For cohort data, methods based on martingale residuals are useful for assessing the fit of a model. Here we introduce grouped martingale residual processes for sampled risk set data, and show that plots of these processes provide a useful tool for checking model-fit. Further we study the large sample properties of the grouped martingale residual processes, and use these to derive a formal goodness-of-fit test to go along with the plots. The methods are illustrated using data on lung cancer deaths in a cohort of uranium miners.

This paper studies a continuous-time market with multiple stocks whose prices are governed by geometric Brownian motions, and admissible investment portfolios are defined via certain square integrability condition. It is proved that, when the investment opportunity set is deterministic (albeit possibly time varying), such a market being arbitrage free is equivalent to the existence of a square integrable (in time) market price of risk, and as a result equivalent to the existence of an equivalent martingale measure. Counterexamples are given to show that these equivalent results are no longer true in a market with a stochastic investment opportunity set.

In the area of response-adaptive design based on randomized urn models, Professor Bai's research has been focused on providing a mathematically rigorous of generalized Friedman's urn model. In a series of papers, matrix recursions and martingale theory were introduced to study randomized urn models. Based on these techniques, some fundamental questions were answered.

This paper studies a very general urn model stimulated by designs in clinical trials, where the number of balls of different types added to the urn at trial n depends on a random outcome directed by the composition at trials 1, 2, …, n - 1. Patient treatments are allocated according to types of balls. We establish the strong consistency and asymptotic normality for both the urn composition and the patient allocation under general assumptions on random generating matrices which determine how balls are added to the urn. Also we obtain explicit forms of the asymptotic variance-covariance matrices of both the urn composition and the patient allocation. The conditions on the nonhomogeneity of generating matrices are mild and widely satisfied in applications. Several applications are also discussed.

Consider a stochastic securities market model with a finite state space and a finite number of trading dates. We study how arbitrage price theory is modified by a no short-selling constraint. The principle of No Arbitrage is characterized by the existence of an equivalent supermartingale measure. If we measure present value as conditional expectations after an equivalent change of measure, then the fundamental value of a security might fall below its market value, leading to the possibility of a price bubble. We show that the Law of One Price holds for marketed claims if and only if there exists an equivalent martingale measure. The latter condition indicates that price bubbles are fragile. Given that the Law of One Price prevails, then a contingent claim has a unique fundamental value if and only if it is the difference of two marketed claims. The main tool for arbitrage analysis in this essay is finite-dimensional LP duality theory.

We discuss the construction of quantum stochastic integrals over the positive plane. Non-commutative representations for the quasi-free CAR and CCR settings are presented extending results obtained previously for martingales over the two parameter plane.