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Lines changed: 86 additions & 39 deletions
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"metadata": {
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"colab": {
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"provenance": [],
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"authorship_tag": "ABX9TyPpNfNl0H6Q30YMECZxlTxw",
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"authorship_tag": "ABX9TyN0GHycEaD9ydo/oniivwC7",
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"include_colab_link": true
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},
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"kernelspec": {
@@ -1540,103 +1540,150 @@
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{
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"cell_type": "markdown",
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"source": [
1543-
"For the first term, we repeat the same argument as in the previous case.\n",
1544-
"\n",
1545-
"Looking at the second term, since we exclude $i=j$, if the first summation contains $\\Delta N$ terms, the second one sums over $\\Delta N-1$ terms. Therefore, the second summation contains $\\Delta N(\\Delta N-1)$ terms.\n",
1546-
"\n",
1547-
"We assume that increments associated with distinct transactions are statistically independent, that is, the outcome (gain or loss) of one exchange does not affect the probabilities of gain or loss of any other exchange. Therefore, the expectation value of the product can be decomposed into the product of the expectation values:\n",
1543+
"For the first term, we repeat the reasoning from the previous case. Since we avoid the case where $i=j$, if the first summation involves $\\Delta N$ terms, the second sums over $\\Delta N$ values ​​for $i$ and $\\Delta N-1$ values ​​for $j$ (for example), thereby excluding instances where $i=j$. Thus, the second summation consists of $\\Delta N(\\Delta N-1)$ terms. Denoting the ensemble average of the terms $\\langle \\Delta(w)_j \\Delta(w)_i \\rangle$ as $\\langle \\Delta\\Delta(w)_E \\rangle$, we write:\n",
15481544
"\n",
15491545
"\\begin{equation}\n",
1550-
"\\left\\langle\\Delta(w)_{i}\\Delta(w)_{j}\\right\\rangle\n",
1546+
"\\left\\langle\n",
1547+
"\\left(\\Delta w\\right)^{2}\n",
1548+
"\\right\\rangle\n",
15511549
"=\n",
1552-
"\\left\\langle\\Delta(w)_{i}\\right\\rangle\n",
1553-
"\\left\\langle\\Delta(w)_{j}\\right\\rangle .\n",
1550+
"\\left\\langle \\Delta N\\right\\rangle\n",
1551+
"\\left\\langle\n",
1552+
"\\Delta(w)^{2}_{E}\n",
1553+
"\\right\\rangle\n",
1554+
"+\n",
1555+
"\\left\\langle\n",
1556+
"\\Delta N(\\Delta N-1)\n",
1557+
"\\right\\rangle\n",
1558+
"\\left\\langle\n",
1559+
"\\Delta\\Delta(w)_{E}\n",
1560+
"\\right\\rangle .\n",
15541561
"\\end{equation}\n",
15551562
"\n",
1556-
"Furthermore, since all transactions follow the same probability distribution, the indices $i$ and $j$ are merely labels (dummy indices); therefore, each individual expectation value corresponds to the same average value per event, denoted by $\\left\\langle\\Delta(w)_{E}\\right\\rangle$. Hence, the cross term ($i\\neq j$) simplifies to\n",
1563+
"However, now\n",
15571564
"\n",
15581565
"\\begin{equation}\n",
1559-
"\\left\\langle\\Delta(w)_{E}\\right\\rangle^{2}.\n",
1566+
"\\left\\langle\n",
1567+
"\\Delta N(\\Delta N-1)\n",
1568+
"\\right\\rangle\n",
15601569
"\\end{equation}\n",
15611570
"\n",
1562-
"Note that the same reasoning cannot be applied to the first term, since there we have the same variable with $i=j$, and therefore the variables are obviously not independent. Thus,\n",
1571+
"is slightly more complicated than\n",
15631572
"\n",
15641573
"\\begin{equation}\n",
1565-
"\\left\\langle \\left(\\Delta w\\right)^{2}\\right\\rangle\n",
1566-
"=\n",
1567-
"\\left\\langle \\Delta N\\right\\rangle\n",
1568-
"\\left\\langle \\Delta(w)^{2}_{E}\\right\\rangle\n",
1569-
"+\n",
1570-
"\\left\\langle \\Delta N\\left(\\Delta N-1\\right)\\right\\rangle\n",
1571-
"\\left\\langle \\Delta(w)_{E}\\right\\rangle^{2}.\n",
1574+
"\\left\\langle\\Delta N\\right\\rangle=\\lambda\\Delta t.\n",
15721575
"\\end{equation}\n",
15731576
"\n",
1574-
"However, now the term $\\left\\langle \\Delta N\\left(\\Delta N-1\\right)\\right\\rangle$ is slightly more complicated than $\\left\\langle \\Delta N\\right\\rangle=\\lambda\\Delta t$. For the moment, we will use the factorial moments of the Poisson distribution ([Wikipedia](https://en.wikipedia.org/wiki/Factorial_moment_generating_function)). Using the notation\n",
1577+
"For now, we will use the factorial moment of the Poisson distribution ([Wikipedia]). Using the notation\n",
15751578
"\n",
15761579
"\\begin{equation}\n",
1577-
"\\left(x\\right)_{n}=x(x-1)(x-2)\\dots(x-n+1),\n",
1580+
"(x)_{n}\n",
1581+
"=\n",
1582+
"x(x-1)(x-2)\\dots(x-n+1),\n",
15781583
"\\end{equation}\n",
15791584
"\n",
15801585
"we have:\n",
15811586
"\n",
15821587
"\\begin{equation}\n",
1583-
"\\left(N\\right)_{1}=\\left(N-1+1\\right)=N\n",
1588+
"(N)_{1}\n",
1589+
"=\n",
1590+
"(N-1+1)\n",
1591+
"=\n",
1592+
"N\n",
15841593
"\\end{equation}\n",
15851594
"\n",
1595+
"and\n",
1596+
"\n",
15861597
"\\begin{equation}\n",
1587-
"\\left(N\\right)_{2}=N\\left(N-2+1\\right)=N(N-1).\n",
1598+
"(N)_{2}\n",
1599+
"=\n",
1600+
"N(N-2+1)\n",
1601+
"=\n",
1602+
"N(N-1).\n",
15881603
"\\end{equation}\n",
15891604
"\n",
1590-
"For a Poisson random variable with mean $\\mu$, the factorial moments satisfy\n",
1605+
"For a Poisson distribution, we have\n",
15911606
"\n",
15921607
"\\begin{equation}\n",
1593-
"\\left\\langle \\left(X\\right)_{n}\\right\\rangle=\\mu^{n}.\n",
1608+
"\\left\\langle\n",
1609+
"(X)_{n}\n",
1610+
"\\right\\rangle\n",
1611+
"=\n",
1612+
"\\mu^{n},\n",
15941613
"\\end{equation}\n",
15951614
"\n",
1596-
"Therefore,\n",
1615+
"for a Poisson random variable with mean $\\mu$. Therefore:\n",
15971616
"\n",
15981617
"\\begin{equation}\n",
1599-
"\\left\\langle N\\right\\rangle=\\lambda\\Delta t\n",
1618+
"\\left\\langle N\\right\\rangle\n",
1619+
"=\n",
1620+
"\\lambda\\Delta t\n",
16001621
"\\end{equation}\n",
16011622
"\n",
16021623
"and\n",
16031624
"\n",
16041625
"\\begin{equation}\n",
1605-
"\\left\\langle N(N-1)\\right\\rangle=(\\lambda\\Delta t)^{2}.\n",
1626+
"\\left\\langle\n",
1627+
"N(N-1)\n",
1628+
"\\right\\rangle\n",
1629+
"=\n",
1630+
"(\\lambda\\Delta t)^{2}.\n",
16061631
"\\end{equation}\n",
16071632
"\n",
1608-
"Hence,\n",
1633+
"Thus:\n",
16091634
"\n",
16101635
"\\begin{equation}\n",
1611-
"\\left\\langle \\left(\\Delta w\\right)^{2}\\right\\rangle\n",
1636+
"\\left\\langle\n",
1637+
"\\left(\\Delta w\\right)^{2}\n",
1638+
"\\right\\rangle\n",
16121639
"=\n",
16131640
"\\lambda\\Delta t\n",
1614-
"\\left\\langle \\Delta(w)^{2}_{E}\\right\\rangle\n",
1641+
"\\left\\langle\n",
1642+
"\\Delta(w)^{2}_{E}\n",
1643+
"\\right\\rangle\n",
16151644
"+\n",
16161645
"(\\lambda\\Delta t)^{2}\n",
1617-
"\\left\\langle \\Delta(w)_{E}\\right\\rangle^{2}.\n",
1646+
"\\left\\langle\n",
1647+
"\\Delta\\Delta(w)_{E}\n",
1648+
"\\right\\rangle .\n",
16181649
"\\end{equation}\n",
16191650
"\n",
1620-
"Dividing by $\\Delta t$ and taking the limit,\n",
1651+
"Therefore,\n",
16211652
"\n",
16221653
"\\begin{equation}\n",
16231654
"\\lim_{\\Delta t\\rightarrow0}\n",
1624-
"\\frac{\\left\\langle \\left(\\Delta w\\right)^{2}\\right\\rangle}{\\Delta t}\n",
1655+
"\\frac{\n",
1656+
"\\left\\langle\n",
1657+
"\\left(\\Delta w\\right)^{2}\n",
1658+
"\\right\\rangle\n",
1659+
"}{\\Delta t}\n",
16251660
"=\n",
1626-
"\\lambda\\left\\langle \\Delta(w)^{2}_{E}\\right\\rangle\n",
1661+
"\\lambda\n",
1662+
"\\left\\langle\n",
1663+
"\\Delta(w)^{2}_{E}\n",
1664+
"\\right\\rangle\n",
16271665
"+\n",
16281666
"\\lim_{\\Delta t\\rightarrow0}\n",
16291667
"\\lambda\\Delta t\n",
1630-
"\\left\\langle \\Delta(w)_{E}\\right\\rangle^{2}.\n",
1668+
"\\left\\langle\n",
1669+
"\\Delta\\Delta(w)_{E}\n",
1670+
"\\right\\rangle .\n",
16311671
"\\end{equation}\n",
16321672
"\n",
1633-
"Since the second term vanishes in the limit $\\Delta t\\rightarrow0$, we finally obtain\n",
1673+
"Finally,\n",
16341674
"\n",
16351675
"\\begin{equation}\n",
16361676
"\\lim_{\\Delta t\\rightarrow0}\n",
1637-
"\\frac{\\left\\langle \\left(\\Delta w\\right)^{2}\\right\\rangle}{\\Delta t}\n",
1677+
"\\frac{\n",
1678+
"\\left\\langle\n",
1679+
"\\left(\\Delta w\\right)^{2}\n",
1680+
"\\right\\rangle\n",
1681+
"}{\\Delta t}\n",
16381682
"=\n",
1639-
"\\lambda\\left\\langle \\Delta(w)^{2}_{E}\\right\\rangle .\n",
1683+
"\\lambda\n",
1684+
"\\left\\langle\n",
1685+
"\\Delta(w)^{2}_{E}\n",
1686+
"\\right\\rangle .\n",
16401687
"\\end{equation}"
16411688
],
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"metadata": {

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