On the Existence and Application of Continuous-Time Threshold Autoregressions of Order Two

1997 ◽  
Vol 29 (1) ◽  
pp. 205-227 ◽  
Author(s):  
P. J. Brockwell ◽  
R. J. Williams
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Linear Time ◽  
Autoregressive Process ◽  
Initial State ◽  
Share Prices ◽  
Threshold Autoregressive ◽  
Random Time Change ◽  
Time Threshold

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

On the Existence and Application of Continuous-Time Threshold Autoregressions of Order Two

1997 ◽  
Vol 29 (01) ◽  
pp. 205-227 ◽  
Author(s):  
P. J. Brockwell ◽  
R. J. Williams
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Linear Time ◽  
Autoregressive Process ◽  
Initial State ◽  
Share Prices ◽  
Threshold Autoregressive ◽  
Random Time Change ◽  
Time Threshold

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

Approximations and boundary conditions for continuous-time threshold autoregressive processes

1994 ◽  
Vol 31 (4) ◽  
pp. 1103-1109 ◽  
Author(s):  
Rob J. Hyndman
Keyword(s):  
Time Series ◽  
Boundary Conditions ◽  
Continuous Time ◽  
Linear Time ◽  
Autoregressive Processes ◽  
Threshold Autoregressive ◽  
The Past ◽  
Time Threshold ◽  
Non Linear ◽  
The Way

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

Approximations and boundary conditions for continuous-time threshold autoregressive processes

1994 ◽  
Vol 31 (04) ◽  
pp. 1103-1109
Author(s):  
Rob J. Hyndman
Keyword(s):  
Time Series ◽  
Boundary Conditions ◽  
Continuous Time ◽  
Linear Time ◽  
Autoregressive Processes ◽  
Threshold Autoregressive ◽  
The Past ◽  
Time Threshold ◽  
Non Linear ◽  
The Way

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

scholarly journals THE STABILITY OF QUANTUM MARKOV FILTERS

2009 ◽  
Vol 12 (01) ◽  
pp. 153-172 ◽  
Author(s):  
RAMON VAN HANDEL
Keyword(s):  
Differential Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Initial State ◽  
Rank Condition ◽  
Quantum Stochastic Differential Equation ◽  
Absolutely Continuous ◽  
Finite Dimensional ◽  
Initial States ◽  
The Stability

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (3) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

scholarly journals Pricing Defaultable Securities under Actual Probability Measure

2014 ◽  
Vol 2 (4) ◽  
pp. 313-334
Author(s):  
Jianfen Feng ◽  
Dianfa Chen ◽  
Mei Yu
Keyword(s):  
Differential Equation ◽  
Probability Measure ◽  
Stochastic Differential Equation ◽  
Credit Risk ◽  
Discrete Time ◽  
Continuous Time ◽  
Backward Stochastic Differential Equation ◽  
New Approach ◽  
Defaultable Securities

AbstractIn this paper, a new approach is developed to estimate the value of defaultable securities under the actual probability measure. This model gives the price framework by means of the method of backward stochastic differential equation. Such a method solves some problems in most of existing literatures with respect to pricing the credit risk and relaxes certain market limitations. We provide the price of defaultable securities in discrete time and in continuous time respectively, which is favorable to practice to manage real credit risk for finance institutes.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

2015 ◽  
Vol 22 (04) ◽  
pp. 1550022
Author(s):  
Matteo Gregoratti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
The Other ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Proper Formulation

We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.

scholarly journals What is the time value of a stream of investments?

2005 ◽  
Vol 42 (03) ◽  
pp. 861-866 ◽  
Author(s):  
Ragnar Norberg ◽  
Mogens Steffensen
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Stochastic Integral ◽  
Asset Price ◽  
Expected Value ◽  
Integral Expression ◽  
Time Value ◽  
Price Process

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.

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