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We propose a useful approach for investigating the statistical properties of foreign currency exchange rates. Our approach is based on queueing theory, particularly, the so-called renewal-reward theorem. For the first passage processes of…

We introduce a new stochastic duration model for transaction times in asset markets. We argue that widely accepted rules for aggregating seemingly related trades mislead inference pertaining to durations between unrelated trades: while any…

Market participants regularly send bid and ask quotes to exchange-operated limit order books. This creates an optimization challenge where their potential profit is determined by their quoted price and how often their orders are…

Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian…

An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient.…

In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization…

For the pedestrian observer, financial markets look completely random with erratic and uncontrollable behavior. To a large extend, this is correct. At first approximation the difference between real price changes and the random walk model…

In the first part of this paper (Sections 1-4), we study a standard exchange economy model with Cobb-Douglas type consumers and give a necessary and sufficient condition for the existence of an odd period cycle in the Walras-Samuelson…

Reinforcement learning can interact with the environment and is suitable for applications in decision control systems. Therefore, we used the reinforcement learning method to establish a foreign exchange transaction, avoiding the…

This paper studies an interacting particle system of interest in econophysics inspired from a model introduced in the physics literature. The original model consists of the customers of a single bank characterized by their capital, and the…

We demonstrate effectiveness of the first-order algorithm from [Milstein, Tretyakov. Theory Prob. Appl. 47 (2002), 53-68] in application to barrier option pricing. The algorithm uses the weak Euler approximation far from barriers and a…

We have studied numerically the statistical mechanics of the dynamic phenomena, including money circulation and economic mobility, in some transfer models. The models on which our investigations were performed are the basic model proposed…

This paper develops a model for the bid and ask prices of a European type asset by formulating a stochastic control problem. The state process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend…

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first…

The problem of escape of a Brownian particle in a cusp-shaped metastable potential is of special importance in nonadiabatic and weakly-adiabatic rate theory for electron transfer (ET) reactions. Especially, for the weakly-adiabatic…

We study a classical Bayesian statistics problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the $0$-$1$ loss function and a constant cost of observation per unit of time for general prior…

We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some…

We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the…

We study boundary inference at $H=3/4$ for mixed fractional Brownian motion and mixed fractional Ornstein--Uhlenbeck models under high-frequency observation. This boundary is economically important because it separates the critical and…

We begin by exploring the intuition of Brownian motion by explaining its birth through the observations of Robert Brown and later through Bachelier's work on its applications to the financial market and finally its rigorous and concretized…