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<title>Compound Poisson process</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Compound Poisson process</span></span>

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<p>A <b>compound Poisson process</b> is a continuous-time <a href="Stochastic_process" title="Stochastic process">stochastic process</a> with jumps. The jumps arrive randomly according to a <a href="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 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 \lambda >0}">

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

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</math></span><img src="./eea25afc0351140f919cf791c49c1964b8b081de.svg" 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}" loading="lazy"></span> and jump size distribution <i>G</i>, is a process <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 \{\,Y(t):t\geq 0\,\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\,Y(t):t\geq 0\,\}}</annotation>

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</math></span><img src="./b01d016577ddce2aa3cc0499c37e98df71d71269.svg" 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\,\}}" loading="lazy"></span> given by

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<annotation encoding="application/x-tex">{\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}</annotation>

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</math></span><img src="./5942ff0f7afd66a0a9ea1b606110ef037fa80c97.svg" 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}}" loading="lazy"></span></dd></dl>

<p>where, <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 \{\,N(t):t\geq 0\,\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\,N(t):t\geq 0\,\}}</annotation>

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</math></span><img src="./18f326258f6cd11097009b5ecab15fcb78bcb88e.svg" 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\,\}}" loading="lazy"></span> is the counting variable of a <a href="Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a> with rate <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 \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" 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 }" loading="lazy"></span>, and <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 \{\,D_{i}:i\geq 1\,\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\,D_{i}:i\geq 1\,\}}</annotation>

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</math></span><img src="./69d5c68480b38d42d642adf67ee6ea6e51d385d1.svg" 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\,\}}" loading="lazy"></span> are independent and identically distributed random variables, with distribution function <i>G</i>, which are also independent of <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 \{\,N(t):t\geq 0\,\}.\,}">

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<annotation encoding="application/x-tex">{\displaystyle \{\,N(t):t\geq 0\,\}.\,}</annotation>

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</math></span><img src="./68be5a8887db5b783aa56afcc42fb2a9f82b13a2.svg" 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\,\}.\,}" loading="lazy"></span>

</p><p>When <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 D_{i}}">

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

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</math></span><img src="./9f07b53d3212e08ca316a536c8aac0bbefa79ee1.svg" 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}}" loading="lazy"></span> are non-negative integer-valued random variables, then this compound Poisson process is known as a <b>stuttering Poisson process.</b>

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<div class="mw-heading mw-heading2"><h2 id="Properties_of_the_compound_Poisson_process">Properties of the compound Poisson process</h2></div>

<p>The <a href="Expected_value" title="Expected value">expected value</a> of a compound Poisson process can be calculated using a result known as <a href="Wald's_equation" title="Wald's equation">Wald's equation</a> as:

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<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>

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</math></span><img src="./ac136fbc1dcdeac1cb4b6f6c0b4a7bc032e18f10.svg" 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).}" loading="lazy"></span></dd></dl>

<p>Making similar use of the <a href="Law_of_total_variance" title="Law of total variance">law of total variance</a>, the <a href="Variance" title="Variance">variance</a> can be calculated as:

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<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>

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</math></span><img src="./5a818cc242b7003a3d5f043f431fdf57801e9734.svg" 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}}}" loading="lazy"></span></dd></dl>

<p>Lastly, using the <a href="Law_of_total_probability" title="Law of total probability">law of total probability</a>, the <a href="Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> can be given as follows:

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<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>

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</math></span><img src="./cad06d8edca1d55ac45254d3cc4d6b7041a5ffe1.svg" 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)}" loading="lazy"></span></dd></dl>

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<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>

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</math></span><img src="./5b8480ad2cecd8cd45d38ad108824ed88fda17cc.svg" 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}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Exponentiation_of_measures">Exponentiation of measures</h2></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.

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<annotation encoding="application/x-tex">{\displaystyle \mu (A)=\Pr(D\in A).\,}</annotation>

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</math></span><img src="./b50aecdf78cf46b123c3db3d4ee859895944fe64.svg" 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).\,}" loading="lazy"></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="Probability_distribution" title="Probability distribution">probability distribution</a> of <i>Y</i>(<i>t</i>) is the measure

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<annotation encoding="application/x-tex">{\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,}</annotation>

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</math></span><img src="./869ad48fd9a8cd9e7745237d2ae13184da3b2ee1.svg" 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}))\,}" loading="lazy"></span></dd></dl>

<p>where the exponential exp(<i>ν</i>) of a finite measure <i>ν</i> on <a href="Borel_set" title="Borel set">Borel subsets</a> of the <a href="Real_number" title="Real number">real line</a> is defined by

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<annotation encoding="application/x-tex">{\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}}</annotation>

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</math></span><img src="./44821bf9cce7ee17cb85e23a4620b707951c8757.svg" 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!}}" loading="lazy"></span></dd></dl>

<p>and

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<mtext> factors</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}}</annotation>

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</math></span><img src="./5abad2086c46b8fefb4cc961412d3aba7e0d62c4.svg" 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}}}}" loading="lazy"></span></dd></dl>

<p>is a <a href="Convolution" title="Convolution">convolution</a> of measures, and the series converges <a href="Convergence_of_random_variables" title="Convergence of random variables">weakly</a>.

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<ul><li><a href="Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a></li>

<li><a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li>

<li><a href="Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson distribution</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous Poisson process</a></li>

<li><a href="Campbell's_formula" class="mw-redirect" title="Campbell's formula">Campbell's formula</a> for the <a href="Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> of a compound Poisson process</li></ul>

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</style><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="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="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="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="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="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="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="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="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="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="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="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="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="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="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="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="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="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="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="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="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="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">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</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="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="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="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_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="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="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="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="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="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="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="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 href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-12-23" href="https://en.wikipedia.org/wiki/?title=Compound_Poisson_process&oldid=1264709457">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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