Stability analysis of Riccati differential equations related to affine diffusion processes

Kim, Kyoung-Kuk

doi:10.1016/j.jmaa.2009.11.020

Cited by 3 publications

(4 citation statements)

References 32 publications

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“…Genesio, Tartaglia, and Vicino (1985) survey methods using a Lyapunov approach; Chiang, Hirsch, and Wu (1988) characterize ∂S in terms of stable submanifolds of unstable equilibria. Kim (2008) establishes a similar result for the quadratic system (2.7).…”

Section: Convergence To Stationaritysupporting

confidence: 66%

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Moment Explosions and Stationary Distributions in Affine Diffusion Models

Glasserman

Kim

2010

Mathematical Finance

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Many of the most widely used models in finance fall within the affine family of diffusion processes. The affine family combines modeling flexibility with substantial tractability, particularly through transform analysis; these models are used both for econometric modeling and for pricing and hedging of derivative securities. We analyze the tail behavior, the range of finite exponential moments, and the convergence to stationarity in affine models, focusing on the class of canonical models defined by Dai and Singleton (2000). We show that these models have limiting stationary distributions and characterize these limits. We show that the tails of both the transient and stationary distributions of these models are necessarily exponential or Gaussian; in the non-Gaussian case, we characterize the tail decay rate for any linear combination of factors. We also give necessary and sufficient conditions for a linear combination of factors to be Gaussian. Our results follow from an investigation into the stability properties of the systems of ordinary differential equations associated with affine diffusions.

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Section: Convergence To Stationaritysupporting

confidence: 66%

“…The origin is, in fact, the unique stable equilibrium (see Kim 2008). We denote its stable manifold by S and call this the stability region of the dynamical system.…”

Section: Define ω = {(T U) ∈ R × W : T ∈ I(u)};mentioning

confidence: 99%

Moment Explosions and Stationary Distributions in Affine Diffusion Models

Glasserman

Kim

2010

Mathematical Finance

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“…Proof. Part of the proof employs arguments in the proof of Theorem 4.1 of Kim [2010], but we include it here for completeness. Instead of (Ric-V), it is more convenient to work with a slightly modified system on r(t) := y(t) − η(w):…”

Section: Stable Regionsmentioning

confidence: 99%

“…Still, there is one case where some of the results in this paper can be obtained to some extent, and this is when A D is invertible. We refer the reader to Kim [2010] for details.…”

Section: Affine Diffusions On Canonical State Spacementioning

confidence: 99%

Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions

Jena

Kim²,

Xing

2012

Stochastic Processes and their Applications

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This paper considers multi-dimensional affine processes with continuous sample paths.By analyzing the Riccati system, which is associated with affine processes via the transform formula, we fully characterize the regions of exponents in which exponential moments of a given process do not explode at any time or explode at a given time. In these two cases, we also compute the longterm growth rate and the explosion rate for exponential moments. These results provide a handle to study implied volatility asymptotics in models where log-returns of stock prices are described by affine processes whose exponential moments do not have an explicit formula.Date: May 15, 2012.

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