Compound Poisson process

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data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Compound_Poisson_process" title="Special:EditPage/Compound Poisson process">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Compound+Poisson+process%22">"Compound Poisson process"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Compound+Poisson+process%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Compound+Poisson+process%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Compound+Poisson+process%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Compound+Poisson+process%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Compound+Poisson+process%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">September 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <b>compound Poisson process</b> is a continuous-time <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> with jumps. The jumps arrive randomly according to a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}"></span> and jump size distribution <i>G</i>, is a process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\,Y(t):t\geq 0\,\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\,Y(t):t\geq 0\,\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01d016577ddce2aa3cc0499c37e98df71d71269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.559ex; height:2.843ex;" alt="{\displaystyle \{\,Y(t):t\geq 0\,\}}"></span> given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </munderover> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5942ff0f7afd66a0a9ea1b606110ef037fa80c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.987ex; height:7.676ex;" alt="{\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}"></span></dd></dl> <p>where, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\,N(t):t\geq 0\,\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\,N(t):t\geq 0\,\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f326258f6cd11097009b5ecab15fcb78bcb88e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.849ex; height:2.843ex;" alt="{\displaystyle \{\,N(t):t\geq 0\,\}}"></span> is the counting variable of a <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> with rate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\,D_{i}:i\geq 1\,\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>≥</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\,D_{i}:i\geq 1\,\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d5c68480b38d42d642adf67ee6ea6e51d385d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.824ex; height:2.843ex;" alt="{\displaystyle \{\,D_{i}:i\geq 1\,\}}"></span> are independent and identically distributed random variables, with distribution function <i>G</i>, which are also independent of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\,N(t):t\geq 0\,\}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mspace width="thinmathspace" /> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\,N(t):t\geq 0\,\}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68be5a8887db5b783aa56afcc42fb2a9f82b13a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.883ex; height:2.843ex;" alt="{\displaystyle \{\,N(t):t\geq 0\,\}.\,}"></span> </p><p>When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span> are non-negative integer-valued random variables, then this compound Poisson process is known as a <b>stuttering Poisson process.</b> </p> <div class="mw-heading mw-heading2"><h2 id="Properties_of_the_compound_Poisson_process">Properties of the compound Poisson process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compound_Poisson_process&action=edit&section=1" title="Edit section: Properties of the compound Poisson process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Expected_value" title="Expected value">expected value</a> of a compound Poisson process can be calculated using a result known as <a href="/wiki/Wald%27s_equation" title="Wald's equation">Wald's equation</a> as: </p> <dl><dd><span class="mwe-math-element"><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} (Y(t))=\operatorname {E} (D_{1}+\cdots +D_{N(t)})=\operatorname {E} (N(t))\operatorname {E} (D_{1})=\operatorname {E} (N(t))\operatorname {E} (D)=\lambda t\operatorname {E} (D).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>t</mi> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (Y(t))=\operatorname {E} (D_{1}+\cdots +D_{N(t)})=\operatorname {E} (N(t))\operatorname {E} (D_{1})=\operatorname {E} (N(t))\operatorname {E} (D)=\lambda t\operatorname {E} (D).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac136fbc1dcdeac1cb4b6f6c0b4a7bc032e18f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:77.687ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} (Y(t))=\operatorname {E} (D_{1}+\cdots +D_{N(t)})=\operatorname {E} (N(t))\operatorname {E} (D_{1})=\operatorname {E} (N(t))\operatorname {E} (D)=\lambda t\operatorname {E} (D).}"></span></dd></dl> <p>Making similar use of the <a href="/wiki/Law_of_total_variance" title="Law of total variance">law of total variance</a>, the <a href="/wiki/Variance" title="Variance">variance</a> can be calculated as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {var} (Y(t))&=\operatorname {E} (\operatorname {var} (Y(t)\mid N(t)))+\operatorname {var} (\operatorname {E} (Y(t)\mid N(t)))\\[5pt]&=\operatorname {E} (N(t)\operatorname {var} (D))+\operatorname {var} (N(t)\operatorname {E} (D))\\[5pt]&=\operatorname {var} (D)\operatorname {E} (N(t))+\operatorname {E} (D)^{2}\operatorname {var} (N(t))\\[5pt]&=\operatorname {var} (D)\lambda t+\operatorname {E} (D)^{2}\lambda t\\[5pt]&=\lambda t(\operatorname {var} (D)+\operatorname {E} (D)^{2})\\[5pt]&=\lambda t\operatorname {E} (D^{2}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mi>λ</mi> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>λ</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>λ</mi> <mi>t</mi> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {var} (Y(t))&=\operatorname {E} (\operatorname {var} (Y(t)\mid N(t)))+\operatorname {var} (\operatorname {E} (Y(t)\mid N(t)))\\[5pt]&=\operatorname {E} (N(t)\operatorname {var} (D))+\operatorname {var} (N(t)\operatorname {E} (D))\\[5pt]&=\operatorname {var} (D)\operatorname {E} (N(t))+\operatorname {E} (D)^{2}\operatorname {var} (N(t))\\[5pt]&=\operatorname {var} (D)\lambda t+\operatorname {E} (D)^{2}\lambda t\\[5pt]&=\lambda t(\operatorname {var} (D)+\operatorname {E} (D)^{2})\\[5pt]&=\lambda t\operatorname {E} (D^{2}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a818cc242b7003a3d5f043f431fdf57801e9734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.78ex; margin-bottom: -0.224ex; width:55.373ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {var} (Y(t))&=\operatorname {E} (\operatorname {var} (Y(t)\mid N(t)))+\operatorname {var} (\operatorname {E} (Y(t)\mid N(t)))\\[5pt]&=\operatorname {E} (N(t)\operatorname {var} (D))+\operatorname {var} (N(t)\operatorname {E} (D))\\[5pt]&=\operatorname {var} (D)\operatorname {E} (N(t))+\operatorname {E} (D)^{2}\operatorname {var} (N(t))\\[5pt]&=\operatorname {var} (D)\lambda t+\operatorname {E} (D)^{2}\lambda t\\[5pt]&=\lambda t(\operatorname {var} (D)+\operatorname {E} (D)^{2})\\[5pt]&=\lambda t\operatorname {E} (D^{2}).\end{aligned}}}"></span></dd></dl> <p>Lastly, using the <a href="/wiki/Law_of_total_probability" title="Law of total probability">law of total probability</a>, the <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> can be given as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mo>∣</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad06d8edca1d55ac45254d3cc4d6b7041a5ffe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.134ex; height:5.509ex;" alt="{\displaystyle \Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} (e^{sY})&=\sum _{i}e^{si}\Pr(Y(t)=i)\\[5pt]&=\sum _{i}e^{si}\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)\\[5pt]&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(Y(t)=i\mid N(t)=n)\\[5pt]&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(D_{1}+D_{2}+\cdots +D_{n}=i)\\[5pt]&=\sum _{n}\Pr(N(t)=n)M_{D}(s)^{n}\\[5pt]&=\sum _{n}\Pr(N(t)=n)e^{n\ln(M_{D}(s))}\\[5pt]&=M_{N(t)}(\ln(M_{D}(s)))\\[5pt]&=e^{\lambda t\left(M_{D}(s)-1\right)}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.8em 0.8em 0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>Y</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> </mrow> </msup> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> </mrow> </msup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mo>∣</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> </mrow> </msup> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mo>∣</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> </mrow> </msup> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} (e^{sY})&=\sum _{i}e^{si}\Pr(Y(t)=i)\\[5pt]&=\sum _{i}e^{si}\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)\\[5pt]&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(Y(t)=i\mid N(t)=n)\\[5pt]&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(D_{1}+D_{2}+\cdots +D_{n}=i)\\[5pt]&=\sum _{n}\Pr(N(t)=n)M_{D}(s)^{n}\\[5pt]&=\sum _{n}\Pr(N(t)=n)e^{n\ln(M_{D}(s))}\\[5pt]&=M_{N(t)}(\ln(M_{D}(s)))\\[5pt]&=e^{\lambda t\left(M_{D}(s)-1\right)}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8480ad2cecd8cd45d38ad108824ed88fda17cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -23.671ex; width:63.554ex; height:48.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} (e^{sY})&=\sum _{i}e^{si}\Pr(Y(t)=i)\\[5pt]&=\sum _{i}e^{si}\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)\\[5pt]&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(Y(t)=i\mid N(t)=n)\\[5pt]&=\sum _{n}\Pr(N(t)=n)\sum _{i}e^{si}\Pr(D_{1}+D_{2}+\cdots +D_{n}=i)\\[5pt]&=\sum _{n}\Pr(N(t)=n)M_{D}(s)^{n}\\[5pt]&=\sum _{n}\Pr(N(t)=n)e^{n\ln(M_{D}(s))}\\[5pt]&=M_{N(t)}(\ln(M_{D}(s)))\\[5pt]&=e^{\lambda t\left(M_{D}(s)-1\right)}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Exponentiation_of_measures">Exponentiation of measures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compound_Poisson_process&action=edit&section=2" title="Edit section: Exponentiation of measures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>N</i>, <i>Y</i>, and <i>D</i> be as above. Let <i>μ</i> be the probability measure according to which <i>D</i> is distributed, i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (A)=\Pr(D\in A).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (A)=\Pr(D\in A).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50aecdf78cf46b123c3db3d4ee859895944fe64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.898ex; height:2.843ex;" alt="{\displaystyle \mu (A)=\Pr(D\in A).\,}"></span></dd></dl> <p>Let <i>δ</i><sub>0</sub> be the trivial probability distribution putting all of the mass at zero. Then the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of <i>Y</i>(<i>t</i>) is the measure </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869ad48fd9a8cd9e7745237d2ae13184da3b2ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.082ex; height:2.843ex;" alt="{\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,}"></span></dd></dl> <p>where the exponential exp(<i>ν</i>) of a finite measure <i>ν</i> on <a href="/wiki/Borel_set" title="Borel set">Borel subsets</a> of the <a href="/wiki/Real_number" title="Real number">real line</a> is defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44821bf9cce7ee17cb85e23a4620b707951c8757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.568ex; height:6.843ex;" alt="{\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>ν</mi> <mo>∗</mo> <mo>⋯</mo> <mo>∗</mo> <mi>ν</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> factors</mtext> </mrow> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5abad2086c46b8fefb4cc961412d3aba7e0d62c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:15.973ex; height:5.676ex;" alt="{\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}}"></span></dd></dl> <p>is a <a href="/wiki/Convolution" title="Convolution">convolution</a> of measures, and the series converges <a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">weakly</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compound_Poisson_process&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson distribution</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous Poisson process</a></li> <li><a href="/wiki/Campbell%27s_formula" class="mw-redirect" title="Campbell's formula">Campbell's formula</a> for the <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> of a compound Poisson process</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes" style="font-size:114%;margin:0 4em"><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a class="mw-selflink selflink">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a 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Compound Poisson process - Wikipedia