Hidden Geometry of Bidirectional Grid-Constrained Stochastic Processes (original) (raw)
Related papers
Bi-Directional Grid Constrained Stochastic Processes' Link to Multi-Skew Brownian Motion
2021
Bi-Directional Grid Constrained (BGC) stochastic processes (BGCSPs) constrain the random movement toward the origin steadily more and more, the further they deviate from the origin, rather than all at once imposing reflective barriers, as does the well-established theory of Itô diffusions with such reflective barriers. We identify that BGCSPs are a variant rather than a special case of the multi-skew Brownian motion (M-SBM). This is because they have their own complexities, such as the barriers being hidden (not known in advance) and not necessarily constant over time. We provide a M-SBM theoretical framework and also a simulation framework to elaborate deeper properties of BGCSPs. The simulation framework is then applied by generating numerous simulations of the constrained paths and the results are analysed. BGCSPs have applications in finance and indeed many other fields requiring graduated constraining, from both above and below the initial position.
Drawdown and Drawup of Bi-Directional Grid Constrained Stochastic Processes
Journal of Mathematics and Statistics, 2020
The Grid Trading Problem (GTP) of mathematical finance, used in portfolio loss minimization, generalized dynamic hedging and algorithmic trading, is researched by examining the impact of the drawdown and drawup of discrete random walks and of Itô diffusions on the Bi-Directional Grid Constrained (BGC) stochastic process for profit Pt and equity Et over time. A comprehensive Discrete Difference Equation (DDE) and a continuous Stochastic Differential Equation (SDE) are derived and proved for the GTP. This allows fund managers and traders the ability to better stress test the impact of volatility to reduce risk and generate positive returns. These theorems are then simulated to complement the theoretical models with charts. Not only does this research extend a rich mathematical problem that can be further researched in its own right, but it also extends the applications into the above areas of finance.
Risk Governance and Control: Financial Markets and Institutions, 2020
Whilst the gambler’s ruin problem (GRP) is based on martingales and the established probability theory proves that the GRP is a doomed strategy, this research details how the semimartingale framework is required for the grid trading problem (GTP) of financial markets, especially foreign exchange (FX) markets. As banks and financial institutions have the requirement to hedge their FX exposure, the GTP can help provide a framework for greater automation of the hedging process and help forecast which hedge scenarios to avoid. Two theorems are adapted from GRP to GTP and prove that grid trading, whilst still subject to the risk of ruin, has the ability to generate significantly more profitable returns in the short term. This is also supported by extensive simulation and distributional analysis. We introduce two absorption barriers, one at zero balance (ruin) and one at a specified profit target. This extends the traditional GRP and the GTP further by deriving both the probability of rui...
Application of Bi-Directional Grid Constrained Stochastic Processes to Algorithmic Trading
Journal of Mathematics and Statistics, 2021
Bi-directional Grid Constrained (BGC) Stochastic Processes (BGCSP) become more constrained the further they drift away from the origin or time axis are examined here. As they drift further away from the time axis, then the greater the likelihood of stopping, as if by two hidden reflective barriers. The theory of BGCSP is applied to a trading environment in which long and short trading is available. The stochastic differential equation of the Grid Trading Problem (GTP) is proposed, proved and its solution is simulated to derive new findings that can lead to further research in this area and the reduction of risk in portfolio management.
Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
The Annals of Probability, 2005
Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T ) = 0 conditioned to stay above the semicircle cT (t) = √ T 2 − t 2 . In the limit of large T , the fluctuation scale of b(t) − cT (t) is T 1/3 and its time-correlation scale is T 2/3 . We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t = τ T , τ ∈ (−1, 1), is only through the second derivative of cT (t) at t = τ T . We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T γ , γ > 1/2. The fluctuation scale is then T (2−γ)/3 . More general conditioning shapes are briefly discussed.
Slow diffusion for a Brownian motion with random reflecting barriers
Stochastic Processes and their Applications, 1996
Let β be a positive number: we consider a particle performing a one dimensional Brownian motion with drift -β, diffusion coefficient 1, and a reflecting barrier at 0. We prove that the time R, needed by the particle to reach a random level X, has the same distribution tail as Γ(α+1) 1/α e 2βX /2β 2 , provided that one of these tails is regularly varying with negative index -α. As a consequence, we discuss the asymptotic behaviour of a Brownian motion with random reflecting barriers, extending some results given by Solomon when X is exponential and α belongs to [1/2, 1].
One-dimensional diffusion processes and their boundaries
1996
It is recalled how one-dimensional homogeneous diffusion processes can be constructed from the Wiener process via a time change and a space transformation. No Lipschitz requirements of the drift coefficient and of the diffusion coefficient as functions of the space variable are needed for this construction to be valid. The process constructed in this way will be the unique weak solution of the corresponding stochastic differential equation. Furthermore, a complete classification of boundary types and boundary behaviour is a direct result of the construction. The boundary behaviour of one-dimensional diffusion processes is illustrated by examples, in particular this boundary behaviour is discussed for a population model recently proposed by Lungu and 0ksendal.
Characterizations and simulations of a class of stochastic processes to model anomalous diffusion
Journal of Physics A-mathematical and Theoretical, 2008
In this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusion. This class, called generalized grey Brownian motion (ggBm), is made up off self-similar with stationary increments processes (H-sssi) and depends on two real parameters alpha in (0,2) and beta in (0,1]. It includes fractional Brownian motion when alpha in (0,2) and beta=1, and time-fractional diffusion stochastic processes when alpha=beta in (0,1). The latters have marginal probability density function governed by time-fractional diffusion equations of order beta. The ggBm is defined through the explicit construction of the underline probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the M-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that ggBm can be represented in terms of the product of a random variable, which is related to the M-Wright function, and an independent fractional Brownian motion. This representation highlights the HHH-{\bf sssi} nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differenital equation of fractional type.
On the dynamics of semimartingales with two reflecting barriers
Journal of Applied Probability, 2013
We consider a semimartingale X which is reflected at an upper barrier T and a lower barrier S, where S and T are also semimartingales such that T is bounded away from S. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers to X, S, and T. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions on X, S, and T, e.g. if we take X, S, and T to be independent martingales (which satisfy some extra technical conditions).