Closed-form variance swap prices under general affine GARCH models and their continuous-time limits

We consider a general homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space, there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or greater than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.

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Closed-form solution for a class of continuous-time algebraic Riccati equations

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference ◽

10.1109/cdc.2009.5399715 ◽

2009 ◽

Cited By ~ 1

Author(s):

Alejandro J. Rojas

Keyword(s):

Closed Form ◽

Continuous Time ◽

Closed Form Solution ◽

Riccati Equations ◽

Form Solution ◽

Algebraic Riccati Equations

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Linear estimation of self-similar processes via Lamperti's transformation

Journal of Applied Probability ◽

10.1239/jap/1014842548 ◽

2000 ◽

Vol 37 (2) ◽

pp. 429-452 ◽

Cited By ~ 39

Author(s):

Carl J. Nuzman ◽

H. Vincent Poor

Keyword(s):

Brownian Motion ◽

Closed Form ◽

Real Line ◽

Continuous Time ◽

Stationary Processes ◽

Linear Estimation ◽

Positive Real ◽

Self Similar ◽

Infinite Intervals ◽

Positive Real Line

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

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Closed-form solution of the continuous-time Lyapunov matrix equation

IEE Proceedings - Control Theory and Applications ◽

10.1049/ip-cta:19941385 ◽

1994 ◽

Vol 141 (5) ◽

pp. 350-356 ◽

Cited By ~ 4

Author(s):

H.-C. Kim ◽

C.-H. Choi

Keyword(s):

Closed Form ◽

Matrix Equation ◽

Continuous Time ◽

Closed Form Solution ◽

Form Solution ◽

Lyapunov Matrix Equation ◽

Lyapunov Matrix

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THE HEATH–JARROW–MORTON DURATION AND CONVEXITY: A GENERALIZED APPROACH

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024902001687 ◽

2002 ◽

Vol 05 (07) ◽

pp. 695-700 ◽

Cited By ~ 7

Author(s):

MANFRED FRÜHWIRTH

Keyword(s):

Closed Form ◽

Continuous Time ◽

Closed Form Solutions ◽

State Variable ◽

Duration Measure ◽

Convexity Measure

This paper extends the traditional duration measure for continuous-time Heath–Jarrow–Morton models. The result is a general Heath–Jarrow–Morton duration measure based on a zero-coupon yield for an arbitrary maturity as state variable. A convexity measure compatible to this generalized duration is derived. In addition, closed-form solutions are presented for two popular example models.

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