COGARCH Volatility Dynamics and Its Stochastic Differential Form

# COGARCH (continuous GARCH) model definition


# COGARCH (continuous GARCH) model definition












In 2004 in their paper Kluppelberg, Lindner and Maller proposed a continuous vesrion of GARCH(1,1) model (COGARCH). Below is the derivation of model formulae as presented in the original paper:

- Suppose standard GARCH(1,1) model: $$ Y_n=\epsilon_n\sigma_n, \\ \sigma^2_n=\beta+\lambda Y^2_{n-1}+\delta\sigma^2_{n-1}, \\ n\in \mathbb{N} $$

- Take $\delta\ge0$ and $\lambda\ge0$. From previous we obtain: $$ \sigma^2_n=\beta+\lambda Y^2_{n-1}+\delta\sigma^2_{n-1}=\beta+(\delta+\lambda\epsilon^2_{n-1})\sigma^2_{n-1} $$ where $\epsilon_{n-1}$ and $\sigma^2_{n-1}$ are independent.



- The summation in previous can be written as: $$ \beta\sum^{n-1}_{i=0}\prod^{n-1}_{j=i+1}(\delta +\lambda\epsilon^2_j) = \beta\int_0^n\exp\Biggl(\sum^{n-1}_{j=\lfloor s \rfloor+1}\ln(\delta +\lambda\epsilon^2_j)\Biggr)ds $$

- Now switching to continuous time. In order to preserve single source of randomness as in discrete GARCH(1,1) model authors suggest to replace the noise variable $\epsilon_j$ by increments of Levy process $\Delta L=L_t-L_{t-}, \ t\ge0$, where $L$ is (càdlàg) Levy process.

- Now keeping $0<\delta<0$ and $\lambda\ge0$ and with (4) in mind define an auxiliary (càdlàg) process $(X_t)_{t\ge0}$ by: $$ X_t=-t\ln\delta-\sum_{0<s\le t}(\ln(1+(\lambda/\delta)\Delta L_s^2)), \ t\ge0 $$

- This $(X_t)_{t\ge0}$ process will help us to rewrite volatility process from (3) in more compact way as left-continuous process: $$ \sigma^2_t=\Biggl(\beta\int_0^te^{X_s}ds+\sigma_0^2\Biggr)e^{-X_{t-}}, \ t\ge0 $$ $(X_t)_{t\ge0}$ process here is a spectrally negative Levy process of bounded variation with drift $\gamma_{X,0}=-\ln\delta$, gaussian component $\tau^2_X=0$ and Levy measure $\Pi_X((-\infty,-x])=\Pi_L(\{y\in\mathbb{R}:|y|\ge\sqrt{(e^{x}-1)\delta/\lambda}\})$.

- Finally define the intergrated continuous time GARCH (COGARCH) process $(G_t)_{t\ge0}$ as the càdlàg process satisfying: $$ dG_t=\sigma_tdL_t, \ t\ge0, \ G_0=0 $$

Last three equations in above derivation define the COGARCH model (as presented in the paper of Kluppelberg, Lindner and Maller). However, if we take a look at the corresponding Wikipedia page, we will see a slightly different definition of COGARCH model using stochastic differential equations: $$ dG_t=\sigma_{t-}dL_t \\ d\sigma^2_t=(\beta-\lambda\sigma^2_t)dt+\delta\sigma^2_{t-}d[L,L]^d_t $$ where: $$ [L,L]^d_t=\sum_{s\in[0,t]}(\Delta L_t^2) $$ is the purely discontinuous part of the quadratic variation process of $L$.

I have no problem understanding formulas from original paper of Kluppelberg, Lindner and Maller, however I fail to understand how volatility process $\sigma^2_t$ in wikipedia version is derived. I believe I'm missing some basic detail here but I still can't figure it out. Any clarification will be appreciated.

UPDATE: In the original paper I've found the proposition (with proof) that volatility process from (7) above satisfies the stochastic differential equation: $$ d\sigma^2_{t+}=\beta dt+\sigma^2_te^{X_{t-}}d(e^{-X_t}) $$

and we have: $$ d\sigma^2_{t+}=\beta t+\ln\delta\int_0^t\sigma^2_sds+(\lambda/\delta)\sum_{0<s<t}\sigma^2_s(\Delta _sL)^2+\sigma^2_0 $$

This seems similar to wikipedia version of volatility process but still not the same.