scholarly journals Integration by Parts and Martingale Representation for a Markov Chain

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Tak Kuen Siu
Keyword(s):  
Markov Chain ◽  
Financial Market ◽  
Continuous Time ◽  
Contingent Claims ◽  
Jump Processes ◽  
Stochastic Integrals ◽  
Integration By Parts ◽  
Martingale Representation ◽  
Finite State ◽  
Change Of Measures

Integration-by-parts formulas for functions of fundamental jump processes relating to a continuous-time, finite-state Markov chain are derived using Bismut's change of measures approach to Malliavin calculus. New expressions for the integrands in stochastic integrals corresponding to representations of martingales for the fundamental jump processes are derived using the integration-by-parts formulas. These results are then applied to hedge contingent claims in a Markov chain financial market, which provides a practical motivation for the developments of the integration-by-parts formulas and the martingale representations.

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 Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. II. Hitting a high level

2005 ◽  
Vol 37 (4) ◽  
pp. 1035-1055 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
High Level ◽  
The Relationship ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v: E→ℝ\{0}, and let (φt)t≥0 be defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 hits level y before hitting 0 and prove weak convergence of the conditioned process as y→∞. In addition, we show the relationship between the conditioning of the process (φt)t≥0 with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (φt)t≥0 with a negative drift to drift to ∞ and the conditioning of it to hit large levels before hitting 0.

A storage model in which the net growth-rate is a Markov chain

1972 ◽  
Vol 9 (01) ◽  
pp. 129-139 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Rate Of Change ◽  
Continuous Time Markov Chain ◽  
Finite Capacity ◽  
Storage Model ◽  
Finite State ◽  
Net Growth Rate ◽  
The Times ◽  
Finite State Space

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

A storage model in which the net growth-rate is a Markov chain

1972 ◽  
Vol 9 (1) ◽  
pp. 129-139 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Rate Of Change ◽  
Continuous Time Markov Chain ◽  
Finite Capacity ◽  
Storage Model ◽  
Finite State ◽  
Net Growth Rate ◽  
The Times ◽  
Finite State Space

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. II. Hitting a high level

2005 ◽  
Vol 37 (04) ◽  
pp. 1035-1055 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
High Level ◽  
The Relationship ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (X t ) t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v: E→ℝ\{0}, and let (φ t ) t≥0 be defined by φ t =∫0 t v(X s )d s. We consider the case in which the process (φ t ) t≥0 is oscillating and that in which (φ t ) t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φ t ) t≥0 on the event that (φ t ) t≥0 hits level y before hitting 0 and prove weak convergence of the conditioned process as y→∞. In addition, we show the relationship between the conditioning of the process (φ t ) t≥0 with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (φ t ) t≥0 with a negative drift to drift to ∞ and the conditioning of it to hit large levels before hitting 0.

Empirical convergence rates for continuous-time Markov chains

2001 ◽  
Vol 38 (1) ◽  
pp. 262-269 ◽  
Author(s):  
Geoffrey Pritchard ◽  
David J. Scott
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Rate Of Convergence ◽  
Continuous Time ◽  
Transition Matrix ◽  
Convergence Rates ◽  
Parametric Model ◽  
Continuous Time Markov Chains ◽  
Finite State ◽  
Second Largest Eigenvalue

We consider the problem of estimating the rate of convergence to stationarity of a continuous-time, finite-state Markov chain. This is done via an estimator of the second-largest eigenvalue of the transition matrix, which in turn is based on conventional inference in a parametric model. We obtain a limiting distribution for the eigenvalue estimator. As an example we treat an M/M/c/c queue, and show that the method allows us to estimate the time to stationarity τ within a time comparable to τ.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (04) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (X t ) t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φ t ) t≥0 be an additive functional defined by φ t =∫0 t v(X s )d s. We consider the case in which the process (φ t ) t≥0 is oscillating and that in which (φ t ) t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φ t ) t≥0 on the event that (φ t ) t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Portfolio Selection with Jumps under Regime Switching

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Lin Zhao
Keyword(s):  
Markov Chain ◽  
Portfolio Selection ◽  
Continuous Time ◽  
Regime Switching ◽  
Jump Processes ◽  
Bank Account ◽  
The Mean ◽  
Mean Variance Portfolio ◽  
Portfolio Selection Model ◽  
Mean Variance

We investigate a continuous-time version of the mean-variance portfolio selection model with jumps under regime switching. The portfolio selection is proposed and analyzed for a market consisting of one bank account and multiple stocks. The random regime switching is assumed to be independent of the underlying Brownian motion and jump processes. A Markov chain modulated diffusion formulation is employed to model the problem.

scholarly journals Marked Continuous-Time Markov Chain Modelling of Burst Behaviour for Single Ion Channels

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Frank G. Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Ion Channels ◽  
Continuous Time ◽  
Total Charge ◽  
Continuous Time Markov Chain ◽  
Open Time ◽  
Finite State ◽  
Gating Process ◽  
Finite State Space ◽  
Number Of Authors

Patch clamp recordings from ion channels often show bursting behaviour, that is, periods of repetitive activity, which are noticeably separated from each other by periods of inactivity. A number of authors have obtained results for important properties of theoretical and empirical bursts when channel gating is modelled by a continuous-time Markov chain with a finite-state space. We show how the use of marked continuous-time Markov chains can simplify the derivation of (i) the distributions of several burst properties, including the total open time, the total charge transfer, and the number of openings in a burst, and (ii) the form of these distributions when the underlying gating process is time reversible and in equilibrium.

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