arXiv subjects

Robert J. Elliott

Publications and source records attributed to Robert J. Elliott.

At least 19 recordsLinked to original sources

arXiv2025

Stochastic control for Backward Stochastic Differential Equations with semi-Markov chain noises

In this paper, we extend the results of Elliott and Yang \cite{elliott3} and discuss the control of a stochastic process for which the driving noise is provided by a martingale associated with a semi-Markov Chain. An existence and a comparison theorem are obtained. In our discrete time setting, adjoint processes are provided by backward stochastic difference equations. Technical results from partial differential equation theory to establish a verification theorem are not required.

math.PR↗

arXiv2021

Regime Switching Entropic Risk Measures on Crude Oil Pricing

This paper introduces a new type of risk measures, namely regime switching entropic risk measures, and study their applicability through simulations. The state of the economy is incorporated into the entropic risk formulation by using a Markov chain. Closed formulae of the risk measure are obtained for futures on crude oil derivatives. The applicability of these new types of risk measures is based on the study of the risk aversion parameter and the convenience yield. The numerical results show a term structure and a mean-reverting behavior of the convenience yield.

q-fin.RM↗

arXiv2021

A stochastic control approach to bid-ask price modelling

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 on a Markov chain. A Girsanov theorem for Markov chains is implemented for the change of coefficients, including the diffusion coefficient which cannot be changed by the usual Girsanov theorem for Brownian motion. The price of a European type asset is then determined using an Esscher transform and a system of partial differential equations. A dynamic programming principle and a maximum/minimum principle associated with the stochastic control problem are then derived to model bid and ask prices. These prices are not quotes of traders or market makers but represent estimates in our model on which reasonable quantities could be traded.

q-fin.MF↗

arXiv2021

Backward stochastic differential equations with regime-switching and sublinear expectations

This paper introduces a backward stochastic differential equation driven by both Brownian motion and a Markov chain (BSDEBM). Regime-switching is also incorporated through its driver. The existence and uniqueness of the solution of the BSDEBM are proved. A comparison theorem is also derived. Filtration consistent sublinear expectations are defined and characterized as solutions to the BSDEBM. The bid and ask prices are then represented using sublinear expectations.

math.PR↗

arXiv2019

Semimartingale dynamics and estimation for a semi-Markov chain

We consider a finite state discrete time process X. Without loss of generality the finite state space can be identified with the set of unit vectors {e1, e2, . . . , eN} with ei = (0, . . . , 0, 1, 0, . . . , 0)0 2 RN. For a Markov chain the times the process stays in any state are geometrically distributed. This condition is relaxed for a semi-Markov chain. We first derive the semimartingale dynamics for a semi-Markov chain. We then consider the situation where the chain is observed in noise. We suggest how to estimate the occupation times in the states and derive filters and smoothers for quantities associated with the chain.

math.PR↗

arXiv2017

A level-1 Limit Order book with time dependent arrival rates

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 conditional diffusion limit of the price process is the so-called Brownian meander.

q-fin.TR↗

arXiv2015

Reflected Backward Stochastic Differential Equations for a Finite State Markov Chain Model and Applications to American Options

In this paper, we introduce a new kind of reflected backward stochastic differential equations (RBSDEs) driven by a martingale, in a Markov chain model, but not driven by Brownian motion, and give existence and uniqueness results for the new equations. Then we discuss American options in a finite state Markov chain model, in the presence of a stochastic discount function (SDF) and using the theory of the new RBSDEs. We show that there exists a constrained super-hedging strategy for an American option, which is unique in our framework as the solution to an RBSDE.

math.PR↗

arXiv2015

On Anticipated backward stochastic differential equations with Markov chain noise

In 2013, Lu and Ren \cite {luren} considered anticipated backward stochastic differential equations driven by finite state, continuous time Markov chain noise and established the existence and uniqueness of the solutions of these equations and a scalar comparison theorem. In this paper, we provide an estimate for their solutions and study the duality between these equations and stochastic differential delayed equations with Markov chain noise. Finally we derive another comparison theorem for these solutions depending only on the two drivers.

math.PR↗

arXiv2015

Some Properties of Reflected Backward Stochastic Differential Equations for a Finite State Markov Chain Model

In this paper, we provide an estimate for the solutions of reflected backward stochastic differential equations (RBSDEs) driven by a Markov chain, derive a continuous dependence property for their solutions with respect to the parameters of the equations, and show similar properties for solutions of backward stochastic differential equations (BSDEs). We finally establish a comparison result for the solutions of RBSDEs driven by a Markov chain.

math.PR↗

arXiv2014

Backward Stochastic Differential Equations with Continuous Coefficients in a Markov Chain Model and with Applications to European Options

In this paper we discuss backward stochastic differential equations with Markov chain noise, having continuous drivers. We obtain the existence of a solution which is possibly not unique. Moreover, we show there is a minimal solution for this kind of equation and derive the corresponding comparison result. This is applied to pricing of European options in a market with Markov chain noise.

math.PR↗

arXiv2014

Comparison and converse comparison theorems for backward stochastic differential equations with Markov chain noise

Comparison and converse comparison theorems are important parts of the research on backward stochastic differential equations. In this paper, we obtain comparison results for one dimensional backward stochastic differential equations with Markov chain noise, extending and generalizing previous work under natural and simplified hypotheses, and establish a converse comparison theorem for the same type of equation after giving the definition and properties of a type of nonlinear expectation: $f$-expectation.

math.PR↗

arXiv2013

Quadratic hedging schemes for non-Gaussian GARCH models

We propose different schemes for option hedging when asset returns are modeled using a general class of GARCH models. More specifically, we implement local risk minimization and a minimum variance hedge approximation based on an extended Girsanov principle that generalizes Duan's (1995) delta hedge. Since the minimal martingale measure fails to produce a probability measure in this setting, we construct local risk minimization hedging strategies with respect to a pricing kernel. These approaches are investigated in the context of non-Gaussian driven models. Furthermore, we analyze these methods for non-Gaussian GARCH diffusion limit processes and link them to the corresponding discrete time counterparts. A detailed numerical analysis based on S&P 500 European Call options is provided to assess the empirical performance of the proposed schemes. We also test the sensitivity of the hedging strategies with respect to the risk neutral measure used by recomputing some of our results with an exponential affine pricing kernel.

q-fin.PR↗

arXiv2013

Filters and smoothers for self-exciting Markov modulated counting processes

We consider a self-exciting counting process, the parameters of which depend on a hidden finite-state Markov chain. We derive the optimal filter and smoother for the hidden chain based on observation of the jump process. This filter is in closed form and is finite dimensional. We demonstrate the performance of this filter both with simulated data, and by analysing the `flash crash' of 6th May 2010 in this framework.

q-fin.CP↗

arXiv2013

Computational Dynamic Market Risk Measures in Discrete Time Setting

Different approaches to defining dynamic market risk measures are available in the literature. Most are focused or derived from probability theory, economic behavior or dynamic programming. Here, we propose an approach to define and implement dynamic market risk measures based on recursion and state economy representation. The proposed approach is to be implementable and to inherit properties from static market risk measures.

q-fin.RM↗

arXiv2012

Existence, uniqueness and comparisons for BSDEs in general spaces

We present a theory of backward stochastic differential equations in continuous time with an arbitrary filtered probability space. No assumptions are made regarding the left continuity of the filtration, of the predictable quadratic variations of martingales or of the measure integrating the driver. We present conditions for existence and uniqueness of square-integrable solutions, using Lipschitz continuity of the driver. These conditions unite the requirements for existence in continuous and discrete time and allow discrete processes to be embedded with continuous ones. We also present conditions for a comparison theorem and hence construct time consistent nonlinear expectations in these general spaces.

math.PR↗

arXiv2010

Time consistency and moving horizons for risk measures

We consider portfolio selection when decisions based on a dynamic risk measure are affected by the use of a moving horizon, and the possible inconsistencies that this creates. By giving a formal treatment of time consistency which is independent of Bellman's equations, we show that there is a new sense in which these decisions can be seen as consistent.

q-fin.RM↗

arXiv2010

Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions

Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expectations in the spirit of [Dynamically consistent nonlinear evaluations and expectations (2005) Shandong Univ.]. We prove basic properties of these expectations and show their applications to dynamic risk measures on such spaces. In particular, we prove comparison theorems for scalar and vector valued solutions to BSDEs, and discuss arbitrage and risk measures in the scalar case.

q-fin.CP↗

arXiv2010

A general theory of Finite State Backward Stochastic Difference Equations

By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions in their own right, not as approximations to the continuous case. We establish the existence and uniqueness of solutions under weaker assumptions than are needed in the continuous time setting, and also establish a comparison theorem for these solutions. The conditions of this theorem are shown to approximate those required in the continuous time setting. We also explore the relationship between the driver $F$ and the set of solutions; in particular, we determine under what conditions the driver is uniquely determined by the solution. Applications to the theory of nonlinear expectations are explored, including a representation result.

math.PR↗