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<title>GARCH-Modelle</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">GARCH-Modelle</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>GARCH-Modelle</b> (GARCH, <a href="Akronym" title="Akronym">Akronym</a> für: <i><b>G</b>eneralized <b>A</b>uto<b>R</b>egressive <b>C</b>onditional <b>H</b>eteroscedasticity</i>, <a href="Deutsche_Sprache" title="Deutsche Sprache">deutsch</a> <i>verallgemeinerte autoregressive bedingte <a href="Heteroskedastizit%C3%A4t" class="mw-redirect" title="Heteroskedastizität">Heteroskedastizität</a></i>) bzw. <b>verallgemeinerte autoregressive Modelle mit bedingter Heteroskedastizität</b> oder auch <b>verallgemeinerte autoregressive bedingt heteroskedastische Zeitreihenmodelle</b> sind <a href="Stochastik" title="Stochastik">stochastische</a> Modelle zur <a href="Zeitreihenanalyse" title="Zeitreihenanalyse">Zeitreihenanalyse</a>, die eine Verallgemeinerung der <a href="ARCH-Modell" class="mw-redirect" title="ARCH-Modell">ARCH-Modelle</a> (<i><b>a</b>uto<b>r</b>egressive <b>c</b>onditional <b>h</b>eteroscedasticity</i>) sind. Sie werden beispielsweise in der <a href="%C3%96konometrie" title="Ökonometrie">Ökonometrie</a> bei der Analyse der Renditen von Aktienkursen zur Modellierung des <a href="Volatilit%C3%A4t" title="Volatilität">Volatilitätsclusterings</a> verwendet. GARCH-Modelle wurden 1986 von <a href="Tim_Bollerslev" title="Tim Bollerslev">Tim Bollerslev</a> auf der Grundlage des ARCH-Modells von <a href="Robert_F._Engle" title="Robert F. Engle">Robert F. Engle</a> (1982) entwickelt.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Eine Zeitreihe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{t})_{t\in \mathbb {Z} }}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{t})_{t\in \mathbb {Z} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1866e7bd1ece543a64e54bd7015700a60dfe5f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.983ex; height:2.843ex;" alt="{\displaystyle (x_{t})_{t\in \mathbb {Z} }}" loading="lazy"></span> heißt GARCH(<i>p</i>,<i>q</i>)-Zeitreihe, wenn sie rekursiv definiert ist durch<sup id="cite_ref-kreiss_1-0" class="reference"><a href="#cite_note-kreiss-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{t}&=\sigma _{t}\epsilon _{t}\\\sigma _{t}^{2}&=a_{0}+a_{1}x_{t-1}^{2}+\dotsb +a_{p}x_{t-p}^{2}+b_{1}\sigma _{t-1}^{2}+\dotsb +b_{q}\sigma _{t-q}^{2},\end{aligned}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f028aead5e066dbcf84ca3131dc10e9260635c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.225ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}x_{t}&=\sigma _{t}\epsilon _{t}\\\sigma _{t}^{2}&=a_{0}+a_{1}x_{t-1}^{2}+\dotsb +a_{p}x_{t-p}^{2}+b_{1}\sigma _{t-1}^{2}+\dotsb +b_{q}\sigma _{t-q}^{2},\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\dotsc ,a_{p},b_{1},\dotsc ,b_{q}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},\dotsc ,a_{p},b_{1},\dotsc ,b_{q}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/193f31b3d237fe1b0474565bc37fa21469bdf807.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.001ex; height:2.843ex;" alt="{\displaystyle a_{0},\dotsc ,a_{p},b_{1},\dotsc ,b_{q}}" loading="lazy"></span> reelle, nichtnegative Parameter mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{p}\neq 0}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/869c96f62d353edc9d7d7c9fee3f38c38f577ca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.55ex; height:2.843ex;" alt="{\displaystyle a_{p}\neq 0}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{q}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle (\epsilon _{t})_{t\in \mathbb {Z} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8af147c277259de91f4c37cafa79381cee5e0fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle (\epsilon _{t})_{t\in \mathbb {Z} }}" loading="lazy"></span> aus <a href="Unabh%C3%A4ngig_identisch_verteilte_Zufallsvariablen" class="mw-redirect" title="Unabhängig identisch verteilte Zufallsvariablen">unabhängigen identisch verteilten</a> Zufallsvariablen mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (\epsilon _{t})=0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\epsilon _{t})=0}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (\epsilon _{t})=1}</annotation>

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</p><p>Bei einem GARCH-Modell hängt also die <a href="Bedingte_Varianz" title="Bedingte Varianz">bedingte Varianz</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}^{2}=\operatorname {Var} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{t}^{2}=\operatorname {Var} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4759e1afa1378f791923c07bb5414788604f87e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.506ex; height:3.176ex;" alt="{\displaystyle \sigma _{t}^{2}=\operatorname {Var} (x_{t}\mid x_{t-1},x_{t-2},\dotsc )}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f279a30bc8eabc788f3fe81c9cfb674e72e858db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle x_{t}}" loading="lazy"></span> von ihrer eigenen Vergangenheit und der Vergangenheit der Zeitreihe ab.

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<div class="mw-heading mw-heading2"><h2 id="Erweiterungen">Erweiterungen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="T-GARCH">T-GARCH</h3></div>

<p>T-GARCH-Modelle sind keine echten GARCH-Modelle, sondern verallgemeinern diese wie folgt:<br>

Mit einer gegebenen Wahrscheinlichkeit p, z. B. p=0.999, entsprechen sie dem „normalen“ GARCH und mit Wahrscheinlichkeit 1-p einem vorher festgelegten Wert. Mit diesen nicht-linearen Modellen können dann zum Beispiel Börsencrashs oder Ähnliches simuliert werden.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="COGARCH">COGARCH</h3></div>

<p><a href="Claudia_Kl%C3%BCppelberg" title="Claudia Klüppelberg">Claudia Klüppelberg</a>, Alexander Lindner und Ross Maller stellten 2004 eine zeitstetige Erweiterung des zeitdiskreten GARCH(1,1)-Prozesses vor. Man beginnt dafür mit den GARCH(1,1)-Gleichungen

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{t}^{2}=a_{0}+a_{1}x_{t-1}^{2}+b_{1}\sigma _{t-1}^{2}=a_{0}+a_{1}\sigma _{t-1}^{2}\epsilon _{t-1}^{2}+b_{1}\sigma _{t-1}^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69e6220f22edea8450a13f92ce15132f14167906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:54.071ex; height:3.343ex;" alt="{\displaystyle \sigma _{t}^{2}=a_{0}+a_{1}x_{t-1}^{2}+b_{1}\sigma _{t-1}^{2}=a_{0}+a_{1}\sigma _{t-1}^{2}\epsilon _{t-1}^{2}+b_{1}\sigma _{t-1}^{2}}" loading="lazy"></span></dd></dl>

<p>und ersetzt die unabhängig identisch verteilten Zufallsvariablen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{t}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} [L,L]_{t}^{\mathrm {d} }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb12354fd1b0514367dc15b7eb19a204c2d9aa5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.932ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} [L,L]_{t}^{\mathrm {d} }}" loading="lazy"></span>, wobei

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<annotation encoding="application/x-tex">{\displaystyle [L,L]_{t}^{\mathrm {d} }=\sum _{s\in [0,t]}(\Delta L_{t})^{2},\quad t\geq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53bcba7cf35ba0974981dc6c589461b14df51594.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:30.058ex; height:6.009ex;" alt="{\displaystyle [L,L]_{t}^{\mathrm {d} }=\sum _{s\in [0,t]}(\Delta L_{t})^{2},\quad t\geq 0}" loading="lazy"></span></dd></dl>

<p>der rein unstetige Teil des <a href="Quadratische_Variation" class="mw-redirect" title="Quadratische Variation">quadratischen Variationsprozesses</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> ist. Man erhält also das System

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} G_{t}=\sigma _{t-}\,\mathrm {d} L_{t}}</annotation>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \sigma _{t}^{2}=(\beta -\eta \sigma _{t}^{2})\,\mathrm {d} t+\varphi \sigma _{t-}^{2}\,\mathrm {d} [L,L]_{t}^{\mathrm {d} }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ab1cc1447e352d622fef39f78df4774ff5063cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.942ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} \sigma _{t}^{2}=(\beta -\eta \sigma _{t}^{2})\,\mathrm {d} t+\varphi \sigma _{t-}^{2}\,\mathrm {d} [L,L]_{t}^{\mathrm {d} }}" loading="lazy"></span></dd></dl>

<p>von <a href="Stochastische_Differentialgleichung" title="Stochastische Differentialgleichung">stochastischen Differentialgleichungen</a>, wobei sich die positiven Parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/693ad9f934775838bd72406b41ada4a59785d7ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bbf42ecda092975c9c69dae84e16182ba5fe2e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9af2720c91be489f57ecde4bb651b95e113d0144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}" loading="lazy"></span> bestimmen lassen. Hat man nun eine Anfangsbedingung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{0},\sigma _{0}^{2})}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G_{0},\sigma _{0}^{2})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f0ca86314bb42671bc7c5de1f2bcc6fd3c7a42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.109ex; height:3.176ex;" alt="{\displaystyle (G_{0},\sigma _{0}^{2})}" loading="lazy"></span> gegeben, so hat das obige System eine pfadweise eindeutige Lösung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G_{t},\sigma _{t}^{2})_{t\geq 0}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (G_{t},\sigma _{t}^{2})_{t\geq 0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c721c48db8e5875b4315af654d0dbd6b17f08cf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.807ex; height:3.176ex;" alt="{\displaystyle (G_{t},\sigma _{t}^{2})_{t\geq 0}}" loading="lazy"></span>, die dann als COGARCH-Modell (<i><b>co</b>ntinuous-time <b>GARCH</b></i>) bezeichnet wird.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="NARCH-Modell" title="NARCH-Modell">NARCH-Modell</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>T. Bollerslev: <i>Generalized Autoregressive Conditional Heteroskedasticity.</i> In: <i>Journal of Econometrics.</i> Vol. 31, No. 3, 1986, S. 307–327, <a href="https://doi.org/10.1016/0304-4076(86)90063-1" class="extiw external" title="doi:10.1016/0304-4076(86)90063-1">doi:10.1016/0304-4076(86)90063-1</a>.</li>

<li>J. Franke, W. Härdle, C. M. Hafner: <i>Statistics of Financial Markets: An Introduction.</i> 2. Auflage. Springer, Berlin / Heidelberg / New York 2008, ISBN 978-3-540-76269-0.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-kreiss-1"><span class="mw-cite-backlink"><a href="#cite_ref-kreiss_1-0">↑</a></span> <span class="reference-text">Jens-Peter Kreiß, Georg Neuhaus: <i>Einführung in die Zeitreihenanalyse.</i> Springer-Verlag, Berlin / Heidelberg 2006, ISBN 3-540-25628-8, S. 298f.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1671">Dissertation zu T-GARCH</a></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">C. Klüppelberg, A. Lindner, R. Maller: <cite style="font-style:italic">A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour</cite>. In: <cite style="font-style:italic">Journal of Applied Probability</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>41</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>3</span>, 2004, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>601–622</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1239/jap%2F1091543413">10.1239/jap/1091543413</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:GARCH-Modelle&rft.atitle=A+continuous-time+GARCH+process+driven+by+a+L%C3%A9vy+process%3A+stationarity+and+second-order+behaviour&rft.au=C.+Kl%C3%BCppelberg%2C+A.+Lindner%2C+R.+Maller&rft.date=2004&rft.doi=10.1239%2Fjap%2F1091543413&rft.genre=journal&rft.issue=3&rft.jtitle=Journal+of+Applied+Probability&rft.pages=601-622&rft.volume=41" style="display:none"> </span></span>

</li>

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