scholarly journals On Markov Modulated Mean-Reverting Price-Difference Models

2008 ◽  
Vol 13 ◽  
pp. 55
Author(s):  
W. P. Malcom ◽  
Lakhdar Aggoun ◽  
Mohamed Al-Lawati
Keyword(s):  
Stochastic Model ◽  
Markov Processes ◽  
Continuous Time ◽  
Price Difference ◽  
Time Dynamics ◽  
Price Process ◽  
Finite State ◽  
The Mean ◽  
New Feature ◽  
Markov Modulated

In this paper we develop a stochastic model incorporating a double-Markov modulated mean-reversion model. Unlike a price process the basis process X can take positive or negative values. This model is based on an explicit discretisation of the corresponding continuous time dynamics. The new feature in our model is that we suppose the mean reverting level in our dynamics as well as the noise coefficient can change according to the states of some finite-state Markov processes which could be the economy and some other unseen random phenomenon.  

scholarly journals Filtering and M-ary Detection of Markov Modulated Mean Reverting Model

2010 ◽  
Vol 15 ◽  
pp. 87
Author(s):  
Lakhdar Aggoun ◽  
Mohamed Al-Lawati ◽  
W.P. Malcolm
Keyword(s):  
Parameter Estimation ◽  
Stochastic Model ◽  
Continuous Time ◽  
Mean Reversion ◽  
Time Dynamics ◽  
Markov Modulated

In an earlier paper we developed a stochastic model incorporating a double-Markov modulated mean-reversion model. The model is based on an explicit discretisation  of the corresponding continuous time dynamics. Here we discuss parameter estimation via the technique of M-ary detection.   

HEDGING OPTIONS IN A DOUBLY MARKOV-MODULATED FINANCIAL MARKET VIA STOCHASTIC FLOWS

2019 ◽  
Vol 22 (08) ◽  
pp. 1950047 ◽  
Author(s):  
TAK KUEN SIU ◽  
ROBERT J. ELLIOTT
Keyword(s):  
Markov Chain ◽  
Financial Market ◽  
Continuous Time ◽  
Regime Switching ◽  
Stochastic Flows ◽  
Finite State ◽  
Markov Modulated ◽  
Appreciation Rate ◽  
The Interest Rate ◽  
Over Time

The hedging of a European-style contingent claim is studied in a continuous-time doubly Markov-modulated financial market, where the interest rate of a bond is modulated by an observable, continuous-time, finite-state, Markov chain and the appreciation rate of a risky share is modulated by a continuous-time, finite-state, hidden Markov chain. The first chain describes the evolution of credit ratings of the bond over time while the second chain models the evolution of the hidden state of an underlying economy over time. Stochastic flows of diffeomorphisms are used to derive some hedge quantities, or Greeks, for the claim. A mixed filter-based and regime-switching Black–Scholes partial differential equation is obtained governing the price of the claim. It will be shown that the delta hedge ratio process obtained from stochastic flows is a risk-minimizing, admissible mean-self-financing portfolio process. Both the first-order and second-order Greeks will be considered.

scholarly journals Sojourn times in finite Markov processes

1989 ◽  
Vol 26 (4) ◽  
pp. 744-756 ◽  
Author(s):  
Gerardo Rubino ◽  
Bruno Sericola
Keyword(s):  
Markov Process ◽  
Asymptotic Behaviour ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Sojourn Time ◽  
Sojourn Times ◽  
Finite State ◽  
Sojourn Time Distributions ◽  
Finite State Space

Sojourn times of Markov processes in subsets of the finite state space are considered. We give a closed form of the distribution of the nth sojourn time in a given subset of states. The asymptotic behaviour of this distribution when time goes to infinity is analyzed, in the discrete time and the continuous-time cases. We consider the usually pseudo-aggregated Markov process canonically constructed from the previous one by collapsing the states of each subset of a given partition. The relation between limits of moments of the sojourn time distributions in the original Markov process and the moments of the corresponding holding times of the pseudo-aggregated one is also studied.

scholarly journals Markov-modulated Ornstein–Uhlenbeck processes

2016 ◽  
Vol 48 (1) ◽  
pp. 235-254 ◽  
Author(s):  
G. Huang ◽  
H. M. Jansen ◽  
M. Mandjes ◽  
P. Spreij ◽  
K. De Turck
Keyword(s):  
Markov Process ◽  
Integration Theory ◽  
Background Process ◽  
Finite State ◽  
The Laplace Transform ◽  
Mean And Variance ◽  
The Mean ◽  
Markov Modulated ◽  
Set Up ◽  
Functional Central Limit

Abstract In this paper we consider an Ornstein–Uhlenbeck (OU) process (M(t))t≥0 whose parameters are determined by an external Markov process (X(t))t≥0 on a finite state space {1, . . ., d}; this process is usually referred to as Markov-modulated Ornstein–Uhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t1, . . ., tK. Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t + u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

FUNCTIONS OF MARKOV PROCESSES AND ALGEBRAIC MEASURES

1992 ◽  
Vol 04 (01) ◽  
pp. 39-64 ◽  
Author(s):  
M. FANNES ◽  
B. NACHTERGAELE ◽  
L. SLEGERS
Keyword(s):  
Markov Processes ◽  
Invariant Measures ◽  
Canonical Representation ◽  
Positive Matrix ◽  
Translation Invariant ◽  
State Spaces ◽  
Finite State ◽  
The Mean ◽  
Class Of Functions ◽  
Translation Invariant Measures

We introduce a class of translation-invariant measures on the set {0, …, q−1}ℤ determined by a set of q d-dimensional matrices. They are algebraic in the sense that their densities are obtained by applying a functional to products of the defining matrices. Positivity of probabilities is assured by assuming a positivity structure on the algebra of defining matrices. Restricting attention to the usual positivity notion of positive matrix elements, a detailed analysis leads to a canonical representation theorem that solves the parametrization problem. Furthermore, we show that the class of algebraic measures coincides with the class of functions of Markov processes with finite state spaces. Our main result consists in the detailed study of the asymptotics of the conditional probabilities from which we derive a formula for the mean entropy.

scholarly journals Sojourn times in finite Markov processes

1989 ◽  
Vol 26 (04) ◽  
pp. 744-756 ◽  
Author(s):  
Gerardo Rubino ◽  
Bruno Sericola
Keyword(s):  
Markov Process ◽  
Asymptotic Behaviour ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Sojourn Time ◽  
Sojourn Times ◽  
Finite State ◽  
Sojourn Time Distributions ◽  
Finite State Space

Sojourn times of Markov processes in subsets of the finite state space are considered. We give a closed form of the distribution of the nth sojourn time in a given subset of states. The asymptotic behaviour of this distribution when time goes to infinity is analyzed, in the discrete time and the continuous-time cases. We consider the usually pseudo-aggregated Markov process canonically constructed from the previous one by collapsing the states of each subset of a given partition. The relation between limits of moments of the sojourn time distributions in the original Markov process and the moments of the corresponding holding times of the pseudo-aggregated one is also studied.

COMPARISONS OF DEPENDENCE FOR STATIONARY MARKOV PROCESSES

2000 ◽  
Vol 14 (3) ◽  
pp. 299-315 ◽  
Author(s):  
Taizhong Hu ◽  
Xiaoming Pan
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Waiting Time ◽  
Continuous Time ◽  
Queueing Theory ◽  
Stationary Markov Process ◽  
Stationary Waiting Time ◽  
Markov Modulated ◽  
Stationary Workload ◽  
Monotonicity Results

Results and conditions which quantify the decrease in dependence with lag for a stationary Markov process and enable one to compare the dependence for two stationary Markov processes are obtained. The notions of dependence used in this article are the supermodular ordering and the concordance ordering. Both discrete-time and continuous-time Markov processes are considered. Some applications of the main results are given. In queueing theory, the monotonicity results of the waiting time of the nth customer as well as the stationary waiting time in an MR/GI/1 queue and the stationary workload in a Markov-modulated queue are established, thus strengthening previous results while simplifying their derivation. This article is a continuation of those by Fang et al. [7] and Hu and Joe [10].

A canonical representation for aggregated Markov processes

1998 ◽  
Vol 35 (02) ◽  
pp. 313-324 ◽  
Author(s):  
Bret Larget
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Continuous Time ◽  
Markov Models ◽  
Sufficient Conditions ◽  
Canonical Representation ◽  
Regularity Conditions ◽  
Deterministic Function ◽  
Finite State ◽  
Canonical Representations

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.

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