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<name>Wilfrid Kendall</name>
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<subtitle>The purpose of this blog is to support the work of a small research team which will soon start work on an EPSRC-funded project "Probabilistic coupling and nilpotent diffusions".</subtitle>
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<rights>(C) 2019 Wilfrid Kendall</rights>
<updated>2019-08-19T21:54:19Z</updated>
<entry>
<title>Coupling Polynomial Stratonovich Integrals: the two-dimensional Brownian case by Wilfrid KendallWilfrid Kendallhttps://blogs.warwick.ac.uk/nilpotent/entry/coupling_polynomial_stratonovich/2017-05-29T13:14:37Z2017-05-29T11:34:35Z<p class="answer">Writing about web page <a href="https://arxiv.org/abs/1705.01600" title="Related external link: https://arxiv.org/abs/1705.01600">https://arxiv.org/abs/1705.01600</a></p>
<p>Sayan Banerjee and I have recently posted <a href="https://arxiv.org/abs/1705.01600">an article on arXiv</a> entitled "Coupling Polynomial Stratonovich Integrals: the two-dimensional Brownian case", which describes rather general couplings for polynomial stochastic integrals driven by two-dimensional Brownian motions. From the abstract:<br />
"We show how to build an immersion coupling of a two-dimensional Brownian motion <span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-1"><span class="mrow" id="MathJax-Span-2"><span class="mo" id="MathJax-Span-3">(</span><span class="msubsup" id="MathJax-Span-4"><span class="mi" id="MathJax-Span-5">W</span><span class="mn" id="MathJax-Span-6">1</span></span><span class="mo" id="MathJax-Span-7">,</span><span class="msubsup" id="MathJax-Span-8"><span class="mi" id="MathJax-Span-9">W</span><span class="mn" id="MathJax-Span-10">2</span></span><span class="mo" id="MathJax-Span-11">)</span></span></span></nobr></span> along with <span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-12"><span class="mrow" id="MathJax-Span-13"><span class="mrow" id="MathJax-Span-14"><span class="TeXmathchoice" id="MathJax-Span-15"><span class="texatom" id="MathJax-Span-16"><span class="mrow" id="MathJax-Span-17"><span class="mo" id="MathJax-Span-18">(</span></span></span></span><span class="mfrac" id="MathJax-Span-19"><span class="mi" id="MathJax-Span-20">n</span><span class="mn" id="MathJax-Span-21">2</span></span><span class="TeXmathchoice" id="MathJax-Span-22"><span class="texatom" id="MathJax-Span-23"><span class="mrow" id="MathJax-Span-24"><span class="mo" id="MathJax-Span-25">)</span></span></span></span></span><span class="mo" id="MathJax-Span-26">+</span><span class="mi" id="MathJax-Span-27">n</span><span class="mo" id="MathJax-Span-28">=</span><span class="mstyle" id="MathJax-Span-29"><span class="mrow" id="MathJax-Span-30"><span class="mfrac" id="MathJax-Span-31"><span class="mn" id="MathJax-Span-32">1</span><span class="mn" id="MathJax-Span-33">2</span></span></span></span><span class="mi" id="MathJax-Span-34">n</span><span class="mo" id="MathJax-Span-35">(</span><span class="mi" id="MathJax-Span-36">n</span><span class="mo" id="MathJax-Span-37">+</span><span class="mn" id="MathJax-Span-38">1</span><span class="mo" id="MathJax-Span-39">)</span></span></span></nobr></span> integrals of the form <span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-40"><span class="mrow" id="MathJax-Span-41"><span class="mo" id="MathJax-Span-42">∫</span><span class="msubsup" id="MathJax-Span-43"><span class="mi" id="MathJax-Span-44">W</span><span class="mi" id="MathJax-Span-45">i</span><span class="mn" id="MathJax-Span-46">1</span></span><span class="msubsup" id="MathJax-Span-47"><span class="mi" id="MathJax-Span-48">W</span><span class="mi" id="MathJax-Span-49">j</span><span class="mn" id="MathJax-Span-50">2</span></span><span class="mo" id="MathJax-Span-51">∘</span><span class="mi" id="MathJax-Span-52">d</span><span class="msubsup" id="MathJax-Span-53"><span class="mi" id="MathJax-Span-54">W</span><span class="mn" id="MathJax-Span-55">2</span></span></span></span></nobr></span>, where <span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-56"><span class="mrow" id="MathJax-Span-57"><span class="mi" id="MathJax-Span-58">j</span><span class="mo" id="MathJax-Span-59">=</span><span class="mn" id="MathJax-Span-60">1</span><span class="mo" id="MathJax-Span-61">,</span><span class="mo" id="MathJax-Span-62">…</span><span class="mo" id="MathJax-Span-63">,</span><span class="mi" id="MathJax-Span-64">n</span></span></span></nobr></span> and <span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-65"><span class="mrow" id="MathJax-Span-66"><span class="mi" id="MathJax-Span-67">i</span><span class="mo" id="MathJax-Span-68">=</span><span class="mn" id="MathJax-Span-69">0</span><span class="mo" id="MathJax-Span-70">,</span><span class="mo" id="MathJax-Span-71">…</span><span class="mo" id="MathJax-Span-72">,</span><span class="mi" id="MathJax-Span-73">n</span><span class="mo" id="MathJax-Span-74">−</span><span class="mi" id="MathJax-Span-75">j</span></span></span></nobr></span> for some fixed <span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-76"><span class="mrow" id="MathJax-Span-77"><span class="mi" id="MathJax-Span-78">n</span></span></span></nobr></span>. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)), and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also selected integral functionals of the diffusions." </p><p class="answer">Writing about web page <a href="https://arxiv.org/abs/1705.01600" title="Related external link: https://arxiv.org/abs/1705.01600">https://arxiv.org/abs/1705.01600</a></p>
<p>Sayan Banerjee and I have recently posted <a href="https://arxiv.org/abs/1705.01600">an article on arXiv</a> entitled "Coupling Polynomial Stratonovich Integrals: the two-dimensional Brownian case", which describes rather general couplings for polynomial stochastic integrals driven by two-dimensional Brownian motions. From the abstract:<br />
"We show how to build an immersion coupling of a two-dimensional Brownian motion <span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-1"><span class="mrow" id="MathJax-Span-2"><span class="mo" id="MathJax-Span-3">(</span><span class="msubsup" id="MathJax-Span-4"><span class="mi" id="MathJax-Span-5">W</span><span class="mn" id="MathJax-Span-6">1</span></span><span class="mo" id="MathJax-Span-7">,</span><span class="msubsup" id="MathJax-Span-8"><span class="mi" id="MathJax-Span-9">W</span><span class="mn" id="MathJax-Span-10">2</span></span><span class="mo" id="MathJax-Span-11">)</span></span></span></nobr></span> along with <span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-12"><span class="mrow" id="MathJax-Span-13"><span class="mrow" id="MathJax-Span-14"><span class="TeXmathchoice" id="MathJax-Span-15"><span class="texatom" id="MathJax-Span-16"><span class="mrow" id="MathJax-Span-17"><span class="mo" id="MathJax-Span-18">(</span></span></span></span><span class="mfrac" id="MathJax-Span-19"><span class="mi" id="MathJax-Span-20">n</span><span class="mn" id="MathJax-Span-21">2</span></span><span class="TeXmathchoice" id="MathJax-Span-22"><span class="texatom" id="MathJax-Span-23"><span class="mrow" id="MathJax-Span-24"><span class="mo" id="MathJax-Span-25">)</span></span></span></span></span><span class="mo" id="MathJax-Span-26">+</span><span class="mi" id="MathJax-Span-27">n</span><span class="mo" id="MathJax-Span-28">=</span><span class="mstyle" id="MathJax-Span-29"><span class="mrow" id="MathJax-Span-30"><span class="mfrac" id="MathJax-Span-31"><span class="mn" id="MathJax-Span-32">1</span><span class="mn" id="MathJax-Span-33">2</span></span></span></span><span class="mi" id="MathJax-Span-34">n</span><span class="mo" id="MathJax-Span-35">(</span><span class="mi" id="MathJax-Span-36">n</span><span class="mo" id="MathJax-Span-37">+</span><span class="mn" id="MathJax-Span-38">1</span><span class="mo" id="MathJax-Span-39">)</span></span></span></nobr></span> integrals of the form <span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-40"><span class="mrow" id="MathJax-Span-41"><span class="mo" id="MathJax-Span-42">∫</span><span class="msubsup" id="MathJax-Span-43"><span class="mi" id="MathJax-Span-44">W</span><span class="mi" id="MathJax-Span-45">i</span><span class="mn" id="MathJax-Span-46">1</span></span><span class="msubsup" id="MathJax-Span-47"><span class="mi" id="MathJax-Span-48">W</span><span class="mi" id="MathJax-Span-49">j</span><span class="mn" id="MathJax-Span-50">2</span></span><span class="mo" id="MathJax-Span-51">∘</span><span class="mi" id="MathJax-Span-52">d</span><span class="msubsup" id="MathJax-Span-53"><span class="mi" id="MathJax-Span-54">W</span><span class="mn" id="MathJax-Span-55">2</span></span></span></span></nobr></span>, where <span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-56"><span class="mrow" id="MathJax-Span-57"><span class="mi" id="MathJax-Span-58">j</span><span class="mo" id="MathJax-Span-59">=</span><span class="mn" id="MathJax-Span-60">1</span><span class="mo" id="MathJax-Span-61">,</span><span class="mo" id="MathJax-Span-62">…</span><span class="mo" id="MathJax-Span-63">,</span><span class="mi" id="MathJax-Span-64">n</span></span></span></nobr></span> and <span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-65"><span class="mrow" id="MathJax-Span-66"><span class="mi" id="MathJax-Span-67">i</span><span class="mo" id="MathJax-Span-68">=</span><span class="mn" id="MathJax-Span-69">0</span><span class="mo" id="MathJax-Span-70">,</span><span class="mo" id="MathJax-Span-71">…</span><span class="mo" id="MathJax-Span-72">,</span><span class="mi" id="MathJax-Span-73">n</span><span class="mo" id="MathJax-Span-74">−</span><span class="mi" id="MathJax-Span-75">j</span></span></span></nobr></span> for some fixed <span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0"><nobr><span class="math" id="MathJax-Span-76"><span class="mrow" id="MathJax-Span-77"><span class="mi" id="MathJax-Span-78">n</span></span></span></nobr></span>. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)), and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also selected integral functionals of the diffusions." </p>