Continuous-Time Limit of a Shift Map Yielding an Exactly Solvable Chaotic Differential Equation

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.

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COGARCH Models

Emerging Applications of Differential Equations and Game Theory - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-7998-0134-4.ch005 ◽

2020 ◽

pp. 79-97

Author(s):

Yakup Arı

Keyword(s):

Differential Equation ◽

Time Series ◽

Continuous Time ◽

Likelihood Method ◽

Estimation Methods ◽

Unequally Spaced ◽

Parameter Estimation Methods ◽

Pseudo Maximum Likelihood ◽

Gamma Processes ◽

Methods Of Moments

In this chapter, the features of a continuous time GARCH (COGARCH) process is discussed since the process can be applied as an explicit solution for the stochastic differential equation which is defined for the volatility of unequally spaced time series. COGARCH process driven by a Lévy process is an analogue of discrete time GARCH process and is further generalized to solutions of Lévy driven stochastic differential equations. The Compound Poisson and Variance Gamma processes are defined and used to derive the increments for the COGARCH process. Although there are various parameter estimation methods introduced for COGARCH, this study is focused on two methods which are Pseudo Maximum Likelihood Method and General Methods of Moments. Furthermore, an example is given to illustrate the findings.

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A carrier-borne epidemic with multiple stages of infection

Journal of Applied Probability ◽

10.1017/s0021900200039371 ◽

1991 ◽

Vol 28 (01) ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

J. Gani ◽

Gy. Michaletzky

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Markov Chain ◽

Generating Function ◽

Continuous Time ◽

Probability Generating Function ◽

Death Process ◽

Partial Differential ◽

Multiple Stages

This paper considers a carrier-borne epidemic in continuous time with m + 1 > 2 stages of infection. The carriers U(t) follow a pure death process, mixing homogeneously with susceptibles X 0(t), and infectives Xi (t) in stages 1≦i≦m of infection. The infectives progress through consecutive stages of infection after each contact with the carriers. It is shown that under certain conditions {X 0(t), X 1(t), · ··, Xm (t) U(t); t≧0} is an (m + 2)-variate Markov chain, and the partial differential equation for its probability generating function derived. This can be solved after a transfomation of variables, and the probability of survivors at the end of the epidemic found.

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Dynkin game under g-expectation in continuous time

Arabian Journal of Mathematics ◽

10.1007/s40065-020-00281-2 ◽

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.

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