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In this paper, the insider trading model of Xiao and Zhou (Acta Mathematicae Applicatae, 2021) is further studied, in which market makers receive partial information about a static risky asset and an insider stops trading at a random time. With the help of dynamic programming principle, we obtain a unique linear Bayesian equilibrium consisting of insider's trading intensity and market liquidity parameter, instead of none Bayesian equilibrium as before. It shows that (i) as time goes by, both trading intensity and market depth increase exponentially, while residual information decreases exponentially; (ii) with average trading time increasing, trading intensity decrease, but both residual information and insider's expected profit increase, while market depth is a unimodal function with a unique minimum with respect to average trading time; (iii) the less information observed by market makers, the weaker trading intensity and market depth are, but the more both expect profit and residual information are, which is in accord with our economic intuition.

It has been understood that the "local" existence of the Markowitz optimal portfolio or the solution to the local-risk minimization problem is guaranteed by some specific mathematical structures on the underlying assets' price processes known in the literature as "structure conditions." In this paper, we consider a semimartingale market model and an arbitrary random time. This random time may model the default time of a firm, the death time of an insured, or any occurrence time of an event that might somehow impact the market model. By adding additional uncertainty to the market model via this random time, the structure conditions may fail, and hence the Markowitz optimal portfolio and other quadratic-optimal portfolios might fail to exist. Our aim is to investigate the impact of this random time on the structure conditions from different perspectives. Our analysis allows us to conclude that under some mild assumptions on the market model and the random time, the structure conditions remain valid on the one hand. Furthermore, we provide two examples illustrating the importance of these assumptions. On the other hand, we describe the random time models such that these structure conditions are preserved for any market model. These results are elaborated separately for the two contexts of stopping with the random time and incorporating totally a specific class of random times, respectively. [ABSTRACT FROM AUTHOR]

Abstract We work in the setting of the progressive enlargement G of a reference filtration F through the observation of a random time τ. We study an integral representation property for some classes of G -martingales stopped at τ. In the first part, we focus on the case where F is a Poisson filtration and we establish a predictable representation property with respect to three G -martingales. In the second part, we relax the assumption that F is a Poisson filtration and we assume that τ is an F -pseudo-stopping time. We establish integral representations with respect to some G -martingales built from F -martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two G -martingales. [ABSTRACT FROM AUTHOR]

Given a one-dimensional downwards transient diffusion process $X$ , we consider a random time $\unicode[STIX]{x1D70C}$ , the last exit time when $X$ exits a certain level $\ell$ , and detect the optimal stopping time for it. In particular, for this random time $\unicode[STIX]{x1D70C}$ , we solve the optimisation problem $\inf _{\unicode[STIX]{x1D70F}}\mathbb{E}[\unicode[STIX]{x1D706}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70C})_{+}+(1-\unicode[STIX]{x1D706})(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70F})_{+}]$ over all stopping times $\unicode[STIX]{x1D70F}$. We show that the process should stop optimally when it runs below some fixed level $\unicode[STIX]{x1D705}_{\ell }$ for the first time, where $\unicode[STIX]{x1D705}_{\ell }$ is the unique solution in the interval $(0,\unicode[STIX]{x1D706}\ell)$ of an explicitly defined equation. [ABSTRACT FROM AUTHOR]

In this article, we discuss Lie point symmetry of stochastic differential equations driven by Wiener and Poisson processes. The symmetry is obtained by considering infinitesimals involving not only spatial and temporal variables but also that of vector Wiener process variable W( t). This work leads to the derivation of the random time-change formula of Itô Brownian motion in Lie transformation context. [ABSTRACT FROM AUTHOR]

Despite de absence of the usage of a time scale in several geomorphology works, it represents a pivotal variable on the understanding of morphodynamic processes. With the addition of new methodologies to landform analysis by means of geochronological, paleoclimatological and neotectonic empirically constructed studies, the concept of time acquired a more complex perspective, aimed at the elucidation of geomorphological problems at distinct spatial scales. Regarding the understanding of formative events cycles, the emergence of recurring periodic cycles must be addressed: tectonic cycles, Mylankovitich orbital cycles, Dansgaard_Oeschger cycles, ENSO cycles, annual season variability. It is important to consider that the method inherent error must be taken into account, for instance, a 10% error in a time scale of 103 years is equivalent to 102 years of the safety interval, thus generating substantial hindrances to the comparison of geochronological data obtained within a 102 years resolution. One of the main implications regarding the way time is addressed in geomorphology is reflected on the equilibrium conceptions. In the case of semi-arid environments the very notion of equilibrium transcends the temporal observation scales of geomorphology. [ABSTRACT FROM AUTHOR]

We consider a fractional counting process with jumps of amplitude 1,2,..., k, with k ∈ N, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability. [ABSTRACT FROM AUTHOR]

Abstract: We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time . Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions. [Copyright &y& Elsevier]

Kac's conjecture relating the solution of wave and telegraph equations in higher dimensions through a Poisson-process-driven random time is established through the concepts of stochastic calculus. New expression is derived for the probability density function of the random time. We demonstrate how the relationship between the solution of a lossy wave- and that of a lossless wave equation can be exploited to derive some statistical identities. Relevance of the results presented to the study of pulse propagation in a dispersive medium characterized by a Lorentz or Drude model is discussed and new evolution equations for 2-D Maxwell's equations are presented for the Drude medium. It is shown that the computational time required for updating the electric field using the stochastic technique is expected to go up as O(\sqrtt). [ABSTRACT FROM PUBLISHER]

Abstract: The present research is motivated by the recent results of Jeanblanc and Song (2011) . Our aim is to demonstrate, with the help of multiplicative systems introduced in Meyer (1979) , that for any given positive -submartingale such that , there exists a random time on some extension of the filtered probability space such that the Azéma submartingale associated with coincides with . Pertinent properties of this construction are studied and it is subsequently extended to the case of several correlated random times with the predetermined univariate conditional distributions. [Copyright &y& Elsevier]

Abstract: Some systems work for a job with random working times. It would be useful for such systems that they are checked at the completion of working times to detect their failures, which is called a random inspection policy. Two inspection policies, where a system is checked at periodic or successive times and also at every completion of working times, are considered. The total expected costs until the detection of failure are obtained, and when the random working time is exponential, optimal inspection policies which minimize them are derived analytically. Furthermore, the inspection policy where the system is checked only at every completion of N th working time is proposed. Finally, as one of modified random inspection models, the backup model where the system goes back to the latest checking time when it has failed is taken up and analyzed, by using the inspection policy. [Copyright &y& Elsevier]

Variable reinforcement schedules are used to arrange the availability of reinforcement following varying response ratios or intervals of time. Random reinforcement schedules are subtypes of variable reinforcement schedules that can be used to arrange the availability of reinforcement at a constant probability across number of responses or time. Generating schedule values for variable and random reinforcement schedules can be difficult. The present article describes the steps necessary to write macros in Microsoft Excel that will generate variable-ratio, variable-interval, variable-time, random-ratio, random-interval, and random-time reinforcement schedule values. [ABSTRACT FROM AUTHOR]

An arbitrage-free CEV economy driven by Brownian motion in independent, continuous random time is introduced. European options are priced by the no-arbitrage principle as conditional averages of their classical CEV values over the CEV-modified random time to maturity. A novel representation of the classical CEV price is used to investigate the asymptotics of the average implied volatility. It is shown that the average implied volatility of the at-the-money call option is lower and of deep out-of-the-money call options, under appropriate sufficient conditions, greater than the implied CEV volatilities. Unlike in the classical CEV model, the shape of the out-of-the-money tail can be both downward and upward sloping depending on the tails of random time. The model is implemented in limit lognormal time. Its multiscaling law is shown to imply a term structure of implied volatility that is qualitatively more sensitive to changes in the time to maturity than is the classical CEV model. [ABSTRACT FROM AUTHOR]

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility. [ABSTRACT FROM AUTHOR]