Narrow Results

This paper conducts a numerical analysis of the behavior of the average value of the Maximum Local Time, $L_n^{\ast }$, in the Simple Random Walk on the square lattice. It has been established in the literature that the sequence ${\zeta }_n:=\frac{{(logn)}^2}{L_n^{\ast }}$ converges to $\pi$. The author found numerical evidence that the average value of ${\zeta }_n$ ($\overline{{\zeta }_n}$) increases until a certain value of $n$, denoted as $n_c$, after which it decreases and approaches $\pi$. Furthermore, estimates are presented for $n_c$ and $\overline{{\zeta }_{n_c}}$.

We show that any locally finite automata network with global synchronous updates can be emulated by another one , whose structure derives from that of by a simple construction, but whose updates are made asynchronously at its various component automata (e.g. possibly randomly or sequentially, with or without possible simultaneous updates at different nodes). By "emulation", we refer to the existence of a spatial-temporal covering 'local time', allowing one to project the behavior of continuously onto that of . We also show the existence of a spatial-temporal section of the asynchronous automata network's behavior which completely determines the synchronous global state of at every time step.

We give the construction of the asynchronous automata network, establish its freedom from deadlocks, and construct local time functions and spatial-temporal sections relating any posssible behavior of to the single corresponding behavior of on a given input sequence starting from a given initial global state.

This establishes that the behavior of any locally finite synchronous automata network actually can be emulated without the restriction of synchronous update, freeing us from the need of a global clock signal. Local information is sufficient to guarantee that the synchronous behavior of is completely determined by any asynchronous behavior of starting from a corresponding global state and given the same input sequence as . Moreover, the relative passage of corresponding local time at any two nodes in is bounded in a simple way by approximately one-third of the distance between them.

As corollaries, any synchronous generalized cellular automaton or synchronous cellular automaton can be emulated by an asynchronous one of the same type.

Implementation aspects of these asynchronous automata are also discussed, and open problems and research directions are indicated.

We study the optimal timing strategies for trading a mean-reverting price process with a finite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models, including the Ornstein–Uhlenbeck (OU) model, Cox–Ingersoll–Ross (CIR) model, Jacobi model, and inhomogeneous geometric Brownian motion (IGBM) model. We analyze three types of trading strategies: (i) the long–short (long to open, short to close) strategy; (ii) the short–long (short to open, long to close) strategy, and (iii) the chooser strategy whereby the trader has the added flexibility to enter the market by taking either a long or short position, and subsequently close the position. For each strategy, we solve an optimal double stopping problem with sequential deadlines, and determine the optimal timing of trades. Our solution methodology utilizes the local time-space calculus of [Peskir (2005) A change-of-variable formula with local time on curves, Journal of Theoretical Probability18, 499–535] to derive nonlinear integral equations of Volterra-type that uniquely characterize the trading boundaries. The numerical implementation of the integral equations provides examples of the optimal trading boundaries.

In this paper, we study the generalized quadratic covariation of f(B^H) and B^H defined by

We present the expansion of the multifractional Brownian motion (mBm) local time in higher dimensions, in terms of Wick powers of white noises (or multiple Wiener integrals). If a suitable number of kernels is subtracted, they exist in the sense of generalized white noise functionals. Moreover, we show the convergence of the regularized truncated local times for mBm in the sense of Hida distributions.

We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation

We consider an infinite system of non-overlapping globules undergoing Brownian motions in ℝ³. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional stochastic differential equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.

Let denote the local time of Brownian motion. Our main result is to show that for each fixed t

Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a$ and $b$ satisfying $a>-1,$$-1<b<0,$$\left|b\right|<1+a$. In this paper, motivated by the asymptotic property

The functional Itô formula, firstly introduced by Bruno Dupire for continuous semimartingales, might be extended in two directions: different dynamics for the underlying process and/or weaker assumptions on the regularity of the functional. In this paper, we pursue the former type by proving the functional version of the Meyer–Tanaka formula. Following the idea of the proof of the classical time-dependent Meyer–Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we study the running maximum and the max-martingales of Yor and Obłój.

We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter $\frac{3}{4}<H<1$. Using results from infinite ergodic theory, it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.

In this paper we study the grey Brownian motion, namely its representation and local time. First it is shown that grey Brownian motion may be represented in terms of a standard Brownian motion and then using a criterium of S. Berman, Trans. Amer. Math. Soc., 137, 277–299 (1969), we show that grey Brownian motion admits a λ-square integrable local time almost surely (λ denotes the Lebesgue measure). As a consequence we obtain the occupation formula and state possible generalizations of these results.

We study the one-sided limit order book corresponding to limit sell orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described limit order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the unscaled price process and the properly scaled measure-valued limit order book process in the high frequency regime. In particular, we characterize the limiting measure-valued limit order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting limit order book process.

Let B^(µ) denote a Brownian motion with drift µ. In this paper we study two perpetual integral functionals of B^(µ). The first one, introduced and investigated by Dufresne in [5], is

We study three stochastic differential games. In each game, two players control a process X = {X_t, 0 ≤ t < ∞} which takes values in the interval I = (0,1), is absorbed at the endpoints of I, and satisfies a stochastic differential equation

In the first of our games, which is zero-sum, player 𝔄 has a continuous reward function u : [0,1] → ℝ. In addition to α(·), player 𝔄 chooses a stopping rule τ and seeks to maximize the expectation of u(X_τ); whereas player 𝔅 chooses β(·) and aims to minimize this expectation.

In the second game, players 𝔄 and 𝔅 each have continuous reward functions u(·) and v(·), choose stopping rules τ and ρ, and seek to maximize the expectations of u(X_τ) and v(X_ρ), respectively.

In the third game the two players again have continuous reward functions u(·) and v(·), now assumed to be unimodal, and choose stopping rules τ and ρ. This game terminates at the minimum τ∧ρ the stopping rules τ and ρ, and players 𝔄, 𝔅 want to maximize the expectations of u(X_τ∧ρ) and ν(X_τ∧ρ), respectively.

Under mild technical assumptions we show that the first game has a value, and find a saddle point of optimal strategies for the players. The other two games are not zero-sum, in general, and for them we construct Nash equilibria.

We derive alternative representations of the McKean equation for the value of the American put option. Our main result decomposes the value of an American put option into the corresponding European put price and the early exercise premium. We then represent the European put price in a new manner. This representation allows us to alternatively decompose the price of an American put option into its intrinsic value and time value, and to demonstrate the equivalence of our results to the McKean equation.