Martingale Method Research Articles

Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas

In this paper, we show how the Gordin martingale approximation method fits into the anisotropic Banach space framework. In particular, for the time-one map of a finite horizon planar periodic Lorentz gas, we prove that Hölder observables satisfy statistical limit laws such as the central limit theorem and associated invariance principles. Previously, these properties were known only for a restricted class of observables, excluding for instance velocity.

Pricing double volatility barriers option under stochastic volatility

In this paper, we study a new type of barrier option, which is called ‘double volatility barriers option’. Take the European up-and-out option as an example, the option will become worthless if the volatility of underlying asset becomes too high and reaches or exceeds the barrier. One advantage of this kind of option is to help option holders effectively control volatility risk. To price double volatility barriers option, we use the eigenfunction expansion method to approximate the solution to the partial differential equation that the option price satisfies. We also obtain the estimation formulae for model parameters by the martingale method and the numerical simulations of the option price are provided for the demonstration purpose.

A bivariate Markov modulated intensity model: applications to insurance and credit risk modelling

A class of analytically tractable bivariate Markov modulated point process is presented in this article. The intensities of the bivariate jump process are assumed to be driven by a correlated Markov modulated jump-diffusion processes with dependence among the jumps being modelled using a copula. Following the martingale method, the closed form expressions for the Laplace transforms and moments of the joint process are derived. The proposed model is capable of addressing a variety of problems in the financial world. To exhibit the applicability of the proposed model, the premium of credit default swaps (CDS) with counterparty risk and the probability of surrendering an insurance contract are obtained. The sensitivity of the premium of CDS and surrender probability with respect to various parameters of the model is also demonstrated.

The optimal solution to a principal-agent problem with unknown agent ability

<p style='text-indent:20px;'>We investigate a principal-agent model featured with unknown agent ability. Under the exponential utilities, the necessary and sufficient conditions of the incentive contract are derived by utilizing the martingale and variational methods, and the solutions of the optimal contracts are obtained by using the stochastic maximum principle. The ability uncertainty reduces the principal's ability of incentive provision. It is shown that as time goes by, the information about the ability accumulates, giving the agent less space for belief manipulation, and incentive provision will become easier. Namely, as the contractual time tends to infinity (long-term), the agent ability is revealed completely, the ability uncertainty disappears, and the optimal contracts under known and unknown ability become identical.

First Passage Problems of a Reflected Hyper-Exponential Jump Diffusion Process: A Unified Computational Approach

The extant literature on first passage problems of reflected hyper-exponential jump diffusion processes (RHEPs) lacks efficiently computable formulae for the Laplace transform of the joint distribution of the RHEP and its first passage time, cumulative distribution function of the overshoot, expected cumulative value of the discounted increments of the local time up to the first passage time, expected cumulative discounted value of the RHEP up to the first passage time, and expectation of the first passage time. We combine numerical solutions to ordinary integro-differential equations and martingale methods in a novel manner to derive such expressions. For some of these quantities, our approach can deal with the subtle case in which both the RHEP's overall drift and the discount rate equal zero. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. We illustrate the numerical performance of our methodology through a few examples. We observe that when the RHEP's overall drift and the discount rate are very close to zero rounding errors can make the evaluation of some of our formulae unreliable. In these situations our exact expression for the case of both zero RHEP's overall drift and discount rate can be an effective approximation for the quantities in question that is substantially more efficient than reliably calculating them using their exact expressions and ``Multiprecision Computing. Our research has applications in financial engineering, insurance, economics, and queueing.

Ruin Probabilities for Risk Models with Constant Interest

UDC 519.21 We consider continuous-time risk models with $m$-dependent claim sizes and constant interest rate. Under some special conditions, we obtain the upper bound for the infinite-time ruin probability. Our approach is based on the martingale methods.

An elementary derivation of moments of Hawkes processes

AbstractHawkes processes have been widely used in many areas, but their probability properties can be quite difficult. In this paper an elementary approach is presented to obtain moments of Hawkes processes and/or the intensity of a number of marked Hawkes processes, in which the detailed outline is given step by step; it works not only for all Markovian Hawkes processes but also for some non-Markovian Hawkes processes. The approach is simpler and more convenient than usual methods such as the Dynkin formula and martingale methods. The method is applied to one-dimensional Hawkes processes and other related processes such as Cox processes, dynamic contagion processes, inhomogeneous Poisson processes, and non-Markovian cases. Several results are obtained which may be useful in studying Hawkes processes and other counting processes. Our proposed method is an extension of the Dynkin formula, which is simple and easy to use.

Optimal stopping for the exponential of a Brownian bridge

AbstractWe study the problem of stopping a Brownian bridgeXin order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

Optimal retirement and portfolio selection with consumption ratcheting

The purpose of this paper is to study the optimal retirement and consumption/investment decisions of an infinitely lived agent who does not tolerate any decline in his/her consumption throughout his/her lifetime. The agent receives labor income but suffers disutility from working until retirement. The agent’s optimization problem combines features of both singular control and optimal stopping. We use the martingale method and study the dual problem, which can be decoupled into a singular control problem and an optimal stopping problem. We provide a closed-form solution of the optimal strategies for the von Neumann–Morgenstern utility function. We show that the coefficient of relative risk aversion implied by the optimal portfolio (i.e., the implied coefficient of relative risk aversion, ICRRA) is a constant value smaller than 1. Moreover, we show that the ICRRA is independent of the agent’s felicity utility function and depends only on the subjective discount rate and market parameters.

Sample Paths Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian Motion

We analyze the effect of additive fractional noise with Hurst parameter H > {1}/{2} on fast-slow systems. Our strategy is based on sample paths estimates, similar to the approach by Berglund and Gentz in the Brownian motion case. Yet, the setting of fractional Brownian motion does not allow us to use the martingale methods from fast-slow systems with Brownian motion. We thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold. In this setting, we provide a neighborhood, tailored to the fast-slow structure of the system, that contains the process with high probability. We prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood. We also illustrate our results in an example arising in climate modeling, where time-correlated noise processes have become of greater relevance recently.

Implicit Incentives for Fund Managers with Partial Information

We study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. The estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. We solve the problem via the martingale method and, with a concavification procedure, we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy.

Exponential bounds of ruin probabilities for non-homogeneous risk models

Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.

A phase transition for preferential attachment models with additive fitness

Preferential attachment models form a popular class of growing networks, where incoming vertices are preferably connected to vertices with high degree. We consider a variant of this process, where vertices are equipped with a random initial fitness representing initial inhomogeneities among vertices and the fitness influences the attractiveness of a vertex in an additive way. We consider a heavy-tailed fitness distribution and show that the model exhibits a phase transition depending on the tail exponent of the fitness distribution. In the weak disorder regime, one of the old vertices has maximal degree irrespective of fitness, while for strong disorder the vertex with maximal degree has to satisfy the right balance between fitness and age. Our methods use martingale methods to show concentration of degree evolutions as well as extreme value theory to control the fitness landscape.

Mathematical Modeling of Main Classes of Stochastic Productive Systems

Introduction. The article deals with mathematical models of two main classes of processes in stochastic productive systems. For a multistage system, conditions of belonging to a “just-in-time” class or a class with infinite support of the time distribution function for productive operations are determined. Materials and Methods. Descriptions and investigations of models are carried out by trajectory (martingale) methods. For “just-in-time” systems and multistage stochastic productive systems, terms and methods of random walks in a random environment and birth and death processes are used. The results are formulated as descriptions of intensity characteristics of equalizers of point counting processes. Results. Two theorems are given and proved; they justify the proposed classification of the mathematical models of productive systems. The criteria of the belonging of the stochastic productive system to the class “just-in-time” are given. A theorem on the incompatibility of groups of “just-in-time” systems and systems infinite support of the time distribution for operations is proved. Discussion and Conclusion. The results show the feasibility of analyzing stochastic productive systems by martingale methods. The descriptions of terms of intensities of the equalizers time of productive processes admit generalization.

A Generalization of Model for Portfolio Selection with Drawdown Constraint on Consumption

In this study, we generalize the results of Arun (2013) on the optimal consumption and investment problem of an infinitely lived agent who does not accept her consumption falling below a fixed proportion of her historically highest level, the so-called drawdown constraint on consumption. We extend the results to a general class of utility functions. We use the martingale method to study the dual problem, which involves the choice of a maximum consumption process. The dual problem can be formulated as a two-dimensional singular control problem, with the free boundary depending on a state variable of the maximum process. We establish the duality theorem and provide semi-closed form solutions regarding the optimal strategies. To highlight our methodology, we present some special cases of utility functions that do not allow for the dimension reduction considered in Arun (2013).

Rough stochastic elasticity of variance and option pricing

This study is concerned with the elasticity of variance for risky assets. We show that the elasticity of variance for S&P500 exhibits short-range correlations. By using asymptotic and martingale methods, we obtain a semi-analytical expression for the option price in the two-scale regime where the constant elasticity of variance is perturbed by a smooth and bounded function of a rapid fractional Ornstein–Uhlenbeck process with Hurst exponent within (0,12). The associated implied volatility is presented and discussed. As a result, the scope of Markov stochastic elasticity of variance model is extended to a non-Markov case.

Optimal Retirement Planning Under Partial Information

The present paper analyses an optimal consumption and investment problem of a retiree with a constant relative risk aversion (CRRA) who faces parameter uncertainty about the financial market. We solve the optimization problem under partial information by making the market observationally complete and consequently applying the martingale method to obtain closed-form solutions to the optimal consumption and investment strategies. Further, we provide some comparative statics and numerical analyses to deeply understand the consumption and investment behavior under partial information. Bearing partial information has little impact on the optimal consumption level, but it makes retirees with a RRA smaller than one invest more riskily, while it makes retirees with a RRA larger than one invest more conservatively.

Optimal Retirement and Portfolio Selection with Consumption Ratcheting

The purpose of this paper is to study the optimal retirement and consumption/investment decisions of an infinitely lived agent who does not tolerate any decline in his/her consumption throughout his/her lifetime. The agent receives labor income but suffers disutility from working until retirement. The agent’s optimization problem combines features of both singular control and optimal stopping. We use the martingale method and study the dual problem, which can be decoupled into a singular control problem and an optimal stopping problem. We provide a closed-form solution of the optimal strategies for the von-Neumann-Morgenstern utility function. We show that the coefficient of relative risk aversion implied by the optimal portfolio (i.e., the implied coefficient of relative risk aversion, ICRRA) is a constant value smaller than 1. Moreover, we show that the ICRRA is independent of the agent's felicity utility function and depends only on the subjective discount rate and market parameters.

Gapped PVBS models for all species numbers and dimensions

Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature [Formula: see text] particle species on [Formula: see text]. The PVBS models were originally introduced as toy models for the classification of ground state phases. A crucial ingredient for this classification is the existence of a spectral gap above the ground state sector. In this work, we derive a spectral gap for PVBS models at arbitrary species number [Formula: see text] and in arbitrary dimension [Formula: see text] in the perturbative regime of small anisotropy parameters. Instead of using the more common martingale method, the proof verifies a finite-size criterion in the spirit of Knabe.

A goodness-of-fit test for copulas based on martingale transformation

This paper proposes an asymptotically distribution-free test for copulas with dynamic marginal distributions, such as GARCH and ARMA processes. The test is based on the empirical copula process with parametrically estimated marginal distributions. By applying the Khmaladze (1982, 1988, 1993) martingale transformation method, the transformed empirical process converges to a standard Gaussian process, so the resulting test statistics are asymptotically distribution-free. Monte Carlo simulations show that the test performs well in finite samples. An empirical application to test copulas between EUR/USD and GBP/USD exchange rates is provided.