scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

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.

A Brownian motion with two reflecting barriers and Markov-modulated speed

2004 ◽  
Vol 41 (04) ◽  
pp. 1237-1242 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Linear Algebra ◽  
Stationary Distribution ◽  
The State ◽  
Long Run ◽  
Markov Modulated ◽  
Two Barriers ◽  
Reflecting Barriers

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.

scholarly journals DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

2019 ◽  
Vol 34 (2) ◽  
pp. 235-257
Author(s):  
Peter Spreij ◽  
Jaap Storm
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Regime Switching ◽  
Limit Theorems ◽  
Counting Process ◽  
Limit Behavior ◽  
Diffusion Approximations ◽  
Default Rate ◽  
Markov Modulated ◽  
Rapid Switching

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

A Brownian motion with two reflecting barriers and Markov-modulated speed

2004 ◽  
Vol 41 (4) ◽  
pp. 1237-1242 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Linear Algebra ◽  
Stationary Distribution ◽  
The State ◽  
Long Run ◽  
Markov Modulated ◽  
Two Barriers ◽  
Reflecting Barriers

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.

scholarly journals Markov Chains Conditioned Never to Wait Too Long at the Origin

2009 ◽  
Vol 46 (03) ◽  
pp. 812-826
Author(s):  
Saul Jacka
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
The State ◽  
Coin Tossing ◽  
First Time ◽  
Vague Convergence ◽  
Irreducible Markov Chain

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain,X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit asT→∞ in the cases where either the state space is finite orXis transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.

scholarly journals Pricing Options with Credit Risk in Markovian Regime-Switching Markets

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jinzhi Li ◽  
Shixia Ma
Keyword(s):  
Markov Chain ◽  
Credit Risk ◽  
Regime Switching ◽  
Stock Price ◽  
Arrival Rate ◽  
Reduced Form ◽  
European Option ◽  
Pricing Options ◽  
Markov Modulated ◽  
The Interest Rate

This paper investigates the valuation of European option with credit risk in a reduced form model when the stock price is driven by the so-called Markov-modulated jump-diffusion process, in which the arrival rate of rare events and the volatility rate of stock are controlled by a continuous-time Markov chain. We also assume that the interest rate and the default intensity follow the Vasicek models whose parameters are governed by the same Markov chain. We study the pricing of European option and present numerical illustrations.

PRICING CREDIT DERIVATIVES IN A MARKOV-MODULATED REDUCED-FORM MODEL

2013 ◽  
Vol 16 (04) ◽  
pp. 1350018 ◽  
Author(s):  
TAMAL BANERJEE ◽  
MRINAL K. GHOSH ◽  
SRIKANTH K. IYER
Keyword(s):  
Markov Chain ◽  
Regime Switching ◽  
Credit Rating ◽  
Credit Derivatives ◽  
Reduced Form ◽  
New Family ◽  
Default Intensity ◽  
High Volatility ◽  
Markov Modulated ◽  
The Impact

Numerous incidents in the financial world have exposed the need for the design and analysis of models for correlated default timings. Some models have been studied in this regard which can capture the feedback in case of a major credit event. We extend the research in the same direction by proposing a new family of models having the feedback phenomena and capturing the effects of regime switching economy on the market. The regime switching economy is modeled by a continuous time Markov chain. The Markov chain may also be interpreted to represent the credit rating of the firm whose bond we seek to price. We model the default intensity in a pool of firms using the Markov chain and a risk factor process. We price some single-name and multi-name credit derivatives in terms of certain transforms of the default and loss processes. These transforms can be calculated explicitly in case the default intensity is modeled as a linear function of a conditionally affine jump diffusion process. In such a case, under suitable technical conditions, the price of credit derivatives are obtained as solutions to a system of ODEs with weak coupling, subject to appropriate terminal conditions. Solving the system of ODEs numerically, we analyze the credit derivative spreads and compare their behavior with the nonswitching counterparts. We show that our model can easily incorporate the effects of business cycle. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low floating interest rate, high default intensity rate, and high volatility. We also model the effects of firm restructuring on the credit spread, in case of a default.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (01) ◽  
pp. 241-259
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (1) ◽  
pp. 241-259 ◽  
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

scholarly journals Stochastic Lotka-Volterra systems under regime switching with jumps

Filomat ◽  
2014 ◽  
Vol 28 (9) ◽  
pp. 1907-1928 ◽  
Author(s):  
Ruihua Wu ◽  
Xiaoling Zou ◽  
Ke Wang ◽  
Meng Liu
Keyword(s):  
Markov Chain ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
Volterra Model ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Stochastic Permanence ◽  
Color Noise ◽  
Volterra Systems ◽  
The Moment

A stochastic Lotka-Volterra model with Markovian switching driven by jumps is proposed and investigated. In the model, the white noise, color noise and jumping noise are taken into account at the same time. This model is more feasible and applicable. Firstly, sufficient conditions for stochastic permanence and extinction are presented. Then the moment average in time and the asymptotic pathwise properties are estimated. Our results show that these properties have close relations with the jumps and the stationary probability distribution of the Markov chain. Finally, several numerical simulations are provided to illustrate the effectiveness of the results.

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