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Динамическое уравнение для средних - История изменений
2024-03-29T04:55:03Z
История изменений этой страницы в вики
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WikiSysop в 17:45, 15 марта 2010
2010-03-15T17:45:24Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Предыдущая</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Версия 17:45, 15 марта 2010</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l12" >Строка 12:</td>
<td colspan="2" class="diff-lineno">Строка 12:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> x(t+dt) = x + a(x,t)\,dt + b(x,t)\,\varepsilon\, \sqrt{dt}. </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> x(t+dt) = x + a(x,t)\,dt + b(x,t)\,\varepsilon\, \sqrt{dt}. </math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<del class="diffchange diffchange-inline">EQN</del>)'''</div></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<ins class="diffchange diffchange-inline">3.1</ins>)'''</div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Значение процесса <math>\textstyle x=x(t)</math> и гауссова величина <math>\textstyle \varepsilon</math> являются двумя ''независимыми'' случайными величинами. В результате вычисления () возникает новое случайное число <math>\textstyle x(t+dt)</math>. Чтобы найти его среднее значение необходимо проинтегрировать левую часть () с марковской плотностью <math>\textstyle P(x_0,t_0\Rightarrow x+dt, t+dt)</math>. Эквивалентный результат получится при усреднении правой части с <math>\textstyle P(x_0,t_0\Rightarrow x, t)\cdot P(\varepsilon)</math>, где <math>\textstyle P(\varepsilon)</math> &mdash; гауссова плотность вероятности. Так как <math>\textstyle x</math> и <math>\textstyle \varepsilon</math> независимы и <math>\textstyle \left\langle \varepsilon\right\rangle =0</math>, то усреднение последнего слагаемого в () даёт ноль, поэтому:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Значение процесса <math>\textstyle x=x(t)</math> и гауссова величина <math>\textstyle \varepsilon</math> являются двумя ''независимыми'' случайными величинами. В результате вычисления (<ins class="diffchange diffchange-inline">3.1</ins>) возникает новое случайное число <math>\textstyle x(t+dt)</math>. Чтобы найти его среднее значение необходимо проинтегрировать левую часть (<ins class="diffchange diffchange-inline">3.1</ins>) с марковской плотностью <math>\textstyle P(x_0,t_0\Rightarrow x+dt, t+dt)</math>. Эквивалентный результат получится при усреднении правой части с <math>\textstyle P(x_0,t_0\Rightarrow x, t)\cdot P(\varepsilon)</math>, где <math>\textstyle P(\varepsilon)</math> &mdash; гауссова плотность вероятности. Так как <math>\textstyle x</math> и <math>\textstyle \varepsilon</math> независимы и <math>\textstyle \left\langle \varepsilon\right\rangle =0</math>, то усреднение последнего слагаемого в (<ins class="diffchange diffchange-inline">3.1</ins>) даёт ноль, поэтому:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>\left\langle x(t+dt)\right\rangle = \left\langle x(t)\right\rangle + \left\langle a(x(t), t)\right\rangle \,dt.</math></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>\left\langle x(t+dt)\right\rangle = \left\langle x(t)\right\rangle + \left\langle a(x(t), t)\right\rangle \,dt.</math></center></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l23" >Строка 23:</td>
<td colspan="2" class="diff-lineno">Строка 23:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> \frac{d\left\langle x\right\rangle }{dt}= \dot{\left\langle x\right\rangle } = \left\langle a(x, t)\right\rangle . </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> \frac{d\left\langle x\right\rangle }{dt}= \dot{\left\langle x\right\rangle } = \left\langle a(x, t)\right\rangle . </math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<del class="diffchange diffchange-inline">EQN</del>)'''</div></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<ins class="diffchange diffchange-inline">3.2</ins>)'''</div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Если <math>\textstyle a(x,t)=\alpha(t)+\beta(t)\, x</math>, то () имеет ту же форму, что и детерминированное уравнение:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Если <math>\textstyle a(x,t)=\alpha(t)+\beta(t)\, x</math>, то (<ins class="diffchange diffchange-inline">3.2</ins>) имеет ту же форму, что и детерминированное уравнение:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>\dot{\left\langle x\right\rangle } = \alpha(t)+\beta(t)\, \left\langle x\right\rangle .</math></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>\dot{\left\langle x\right\rangle } = \alpha(t)+\beta(t)\, \left\langle x\right\rangle .</math></center></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32" >Строка 32:</td>
<td colspan="2" class="diff-lineno">Строка 32:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Поэтому при любой волатильности <math>\textstyle b(x,t)</math> среднее значение процесса с линейным по <math>\textstyle x</math> сносом совпадает с детерминированным решением. Однако в нелинейном случае это не так!</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Поэтому при любой волатильности <math>\textstyle b(x,t)</math> среднее значение процесса с линейным по <math>\textstyle x</math> сносом совпадает с детерминированным решением. Однако в нелинейном случае это не так!</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Абсолютно аналогично, усредняя произвольную функцию <math>\textstyle F=F(x,t)</math>, изменение которой подчиняется лемме Ито ()<del class="diffchange diffchange-inline">, стр. \pageref{process_ito_lemma}</del>, получаем:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Абсолютно аналогично, усредняя произвольную функцию <math>\textstyle F=F(x,t)</math>, изменение которой подчиняется <ins class="diffchange diffchange-inline">[[Лемма Ито|</ins>лемме Ито (<ins class="diffchange diffchange-inline">2.15</ins>)<ins class="diffchange diffchange-inline">]]</ins>, получаем:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> { \;\frac{d\left\langle F(x,t)\right\rangle }{dt} = \left\langle \frac{\partial F}{\partial t} + a(x,t)\cdot \frac{\partial F}{\partial x} + \frac{b^2(x,t)}{2}\cdot \frac{\partial^2 F }{\partial x^2} \right\rangle \; }. </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> { \;\frac{d\left\langle F(x,t)\right\rangle }{dt} = \left\langle \frac{\partial F}{\partial t} + a(x,t)\cdot \frac{\partial F}{\partial x} + \frac{b^2(x,t)}{2}\cdot \frac{\partial^2 F }{\partial x^2} \right\rangle \; }. </math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<del class="diffchange diffchange-inline">EQN</del>)'''</div></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<ins class="diffchange diffchange-inline">3.3</ins>)'''</div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l79" >Строка 79:</td>
<td colspan="2" class="diff-lineno">Строка 79:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>В качестве упражнения предлагается найти среднее для уравнения: <math>\textstyle dx = (\alpha + \beta x)\, dt + (\sigma +\gamma x)\, \delta W</math> (<math>\textstyle \lessdot</math> H).</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>В качестве упражнения предлагается найти среднее для уравнения: <math>\textstyle dx = (\alpha + \beta x)\, dt + (\sigma +\gamma x)\, \delta W</math> (<math>\textstyle \lessdot</math> H).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>\textstyle \bullet</math> Из соотношения () несложно получить уравнение, которому удовлетворяет плотность вероятности <math>\textstyle P(x)</math> в стационарном режиме. Выберем функцию <math>\textstyle F(x)</math>, не зависящую от времени, и положим производную <math>\textstyle \left\langle F(x)\right\rangle </math> равной нулю. Запишем усреднение в явном виде:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>\textstyle \bullet</math> Из соотношения (<ins class="diffchange diffchange-inline">3.3</ins>) несложно получить уравнение, которому удовлетворяет плотность вероятности <math>\textstyle P(x)</math> в стационарном режиме. Выберем функцию <math>\textstyle F(x)</math>, не зависящую от времени, и положим производную <math>\textstyle \left\langle F(x)\right\rangle </math> равной нулю. Запишем усреднение в явном виде:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>\int\limits^{\infty}_{-\infty}P(x)\, \left[ a(x)\cdot \frac{\partial F}{\partial x} + \frac{b^2(x)}{2}\cdot \frac{\partial^2 F }{\partial x^2} \right]\, dx = 0.</math></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>\int\limits^{\infty}_{-\infty}P(x)\, \left[ a(x)\cdot \frac{\partial F}{\partial x} + \frac{b^2(x)}{2}\cdot \frac{\partial^2 F }{\partial x^2} \right]\, dx = 0.</math></center></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l99" >Строка 99:</td>
<td colspan="2" class="diff-lineno">Строка 99:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> \frac{1}{2}\, \frac{P'(x)}{P(x)} =\frac{a(x)}{b^2(x)} - \frac{b'(x)}{b(x)}, </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> \frac{1}{2}\, \frac{P'(x)}{P(x)} =\frac{a(x)}{b^2(x)} - \frac{b'(x)}{b(x)}, </math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<del class="diffchange diffchange-inline">EQN</del>)'''</div></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<ins class="diffchange diffchange-inline">3.4</ins>)'''</div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l108" >Строка 108:</td>
<td colspan="2" class="diff-lineno">Строка 108:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>P(x) = \frac{C}{b^2(x)}\,  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>P(x) = \frac{C}{b^2(x)}\,  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\exp \left\{2\int\frac{a(x)}{b^2(x)} \, dx \right\}{\bigl|} . </math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\exp \left\{2\int\frac{a(x)}{b^2(x)} \, dx \right\}{\bigl|} . </math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<del class="diffchange diffchange-inline">EQN</del>)'''</div></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(<ins class="diffchange diffchange-inline">3.5</ins>)'''</div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l117" >Строка 117:</td>
<td colspan="2" class="diff-lineno">Строка 117:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>dx = -\beta\cdot(x-\alpha)\,dt + \sigma\,\delta W.</math></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>dx = -\beta\cdot(x-\alpha)\,dt + \sigma\,\delta W.</math></center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Интегрирование в () приводит к следующей плотности вероятности:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Интегрирование в (<ins class="diffchange diffchange-inline">3.5</ins>) приводит к следующей плотности вероятности:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>P(x) = \frac{1}{\sigma}\,\sqrt{\frac{\beta}{\pi}}\,\exp\left\{-\frac{(x-\alpha)^2}{\sigma^2/\beta}\right\},</math></center></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<center><math>P(x) = \frac{1}{\sigma}\,\sqrt{\frac{\beta}{\pi}}\,\exp\left\{-\frac{(x-\alpha)^2}{\sigma^2/\beta}\right\},</math></center></div></td></tr>
</table>
WikiSysop
http://synset.com/wiki/index.php?title=%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%81%D1%80%D0%B5%D0%B4%D0%BD%D0%B8%D1%85&diff=1287&oldid=prev
WikiSysop в 18:23, 21 февраля 2010
2010-02-21T18:23:14Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Предыдущая</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Версия 18:23, 21 февраля 2010</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l105" >Строка 105:</td>
<td colspan="2" class="diff-lineno">Строка 105:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| width="100%"  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|<math> <del class="diffchange diffchange-inline">{ \;</del>P(x) = \frac{C}{b^2(x)}\, \exp \left\{2\int\frac{a(x)}{b^2(x)} \, dx \right\}{\bigl|<del class="diffchange diffchange-inline">}} </del>}. </math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>  | width="90%" align="center"|</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>P(x) = \frac{C}{b^2(x)}\,  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\exp \left\{2\int\frac{a(x)}{b^2(x)} \, dx \right\}{\bigl|} . </math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(EQN)'''</div></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  | <div width="10%" align="right" style="color:#0000CC">'''(EQN)'''</div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>  |}</div></td></tr>
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WikiSysop
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WikiSysop в 18:20, 21 февраля 2010
2010-02-21T18:20:52Z
<p></p>
<a href="http://synset.com/wiki/index.php?title=%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%81%D1%80%D0%B5%D0%B4%D0%BD%D0%B8%D1%85&diff=1286&oldid=1204">Внесённые изменения</a>
WikiSysop
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WikiSysop: Защищена страница «Динамическое уравнение для средних» ([edit=sysop] (бессрочно) [move=sysop] (бессрочно))
2010-02-20T20:27:38Z
<p>Защищена страница «<a href="/wiki/%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%81%D1%80%D0%B5%D0%B4%D0%BD%D0%B8%D1%85" title="Динамическое уравнение для средних">Динамическое уравнение для средних</a>» ([edit=sysop] (бессрочно) [move=sysop] (бессрочно))</p>
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<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">← Предыдущая</td>
<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">Версия 20:27, 20 февраля 2010</td>
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WikiSysop
http://synset.com/wiki/index.php?title=%D0%94%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%81%D1%80%D0%B5%D0%B4%D0%BD%D0%B8%D1%85&diff=1203&oldid=prev
WikiSysop: Новая страница: «{| width="100%" | width="40%"|Порождающий процесс Винера << ! width="20%"|[[Стохастический мир|Оглавление…»
2010-02-20T20:27:24Z
<p>Новая страница: «{| width="100%" | width="40%"|<a href="/wiki/%D0%9F%D0%BE%D1%80%D0%BE%D0%B6%D0%B4%D0%B0%D1%8E%D1%89%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81_%D0%92%D0%B8%D0%BD%D0%B5%D1%80%D0%B0" title="Порождающий процесс Винера">Порождающий процесс Винера</a> << ! width="20%"|[[Стохастический мир|Оглавление…»</p>
<p><b>Новая страница</b></p><div>{| width="100%" <br />
| width="40%"|[[Порождающий процесс Винера]] << <br />
! width="20%"|[[Стохастический мир|Оглавление]] <br />
| width="40%" align="right"| >> [[Процесс Феллера]]<br />
|}<br />
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<br />
<br />
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{| width="100%" <br />
| width="40%"|[[Порождающий процесс Винера]] << <br />
! width="20%"|[[Стохастический мир|Оглавление]] <br />
| width="40%" align="right"| >> [[Процесс Феллера]]<br />
|}<br />
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[[Стохастический мир]] - простое введение в стохастические дифференциальные уравнения</div>
WikiSysop