Jump to navigation Jump to search

General

Display information for equation id:math.225934.4 on revision:225934

* Page found: Geometric Brownian motion (eq math.225934.4)

(force rerendering)

Cannot find the equation data in the database. Fetching from revision text.

Occurrences on the following pages:

Hash: 72947b669cbb5a77430e2b1d7e1dd8a7

TeX (original user input):

 S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right).

TeX (checked):

S_{t}=S_{0}\exp \left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_{t}\right).

LaTeXML (experimental; uses MathML) rendering

MathML (7.139 KB / 1.187 KB) :

S t = S 0 exp ( ( μ - σ 2 2 ) t + σ W t ) . subscript 𝑆 𝑡 subscript 𝑆 0 𝜇 superscript 𝜎 2 2 𝑡 𝜎 subscript 𝑊 𝑡 {\displaystyle S_{t}=S_{0}\exp\left(\left(\mu-{\frac{\sigma^{2}}{2}}\right)t+% \sigma W_{t}\right).}
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle S_{t}=S_{0}\exp\left(\left(\mu-{\frac{\sigma^{2}}{2}}\right)t+%&#10;\sigma W_{t}\right).}" display="inline">
  <semantics id="p1.1.m1.1a">
    <mrow id="p1.1.m1.1.20" xref="p1.1.m1.1.20.2.cmml">
      <mrow id="p1.1.m1.1.20.2" xref="p1.1.m1.1.20.2.cmml">
        <msub id="p1.1.m1.1.20.2.1" xref="p1.1.m1.1.20.2.1.cmml">
          <mi id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">S</mi>
          <mi id="p1.1.m1.1.2.1" xref="p1.1.m1.1.2.1.cmml">t</mi>
        </msub>
        <mo id="p1.1.m1.1.3" xref="p1.1.m1.1.3.cmml">=</mo>
        <mrow id="p1.1.m1.1.20.2.2" xref="p1.1.m1.1.20.2.2.cmml">
          <msub id="p1.1.m1.1.20.2.2.2" xref="p1.1.m1.1.20.2.2.2.cmml">
            <mi id="p1.1.m1.1.4" xref="p1.1.m1.1.4.cmml">S</mi>
            <mn id="p1.1.m1.1.5.1" xref="p1.1.m1.1.5.1.cmml">0</mn>
          </msub>
          <mo id="p1.1.m1.1.20.2.2.1" xref="p1.1.m1.1.20.2.2.1.cmml"></mo>
          <mrow id="p1.1.m1.1.20.2.2.3.2" xref="p1.1.m1.1.20.2.2.3.1.cmml">
            <mi id="p1.1.m1.1.6" xref="p1.1.m1.1.6.cmml">exp</mi>
            <mo id="p1.1.m1.1.20.2.2.3.2a" xref="p1.1.m1.1.20.2.2.3.1.cmml"></mo>
            <mrow id="p1.1.m1.1.20.2.2.3.2.2" xref="p1.1.m1.1.20.2.2.3.1.cmml">
              <mo id="p1.1.m1.1.7" xref="p1.1.m1.1.20.2.2.3.1.cmml">(</mo>
              <mrow id="p1.1.m1.1.20.2.2.3.2.1" xref="p1.1.m1.1.20.2.2.3.2.1.cmml">
                <mrow id="p1.1.m1.1.20.2.2.3.2.1.1" xref="p1.1.m1.1.20.2.2.3.2.1.1.cmml">
                  <mrow id="p1.1.m1.1.20.2.2.3.2.1.1.2" xref="p1.1.m1.1.20.2.2.3.2.1.1.2.2.cmml">
                    <mo id="p1.1.m1.1.8" xref="p1.1.m1.1.20.2.2.3.2.1.1.2.2.cmml">(</mo>
                    <mrow id="p1.1.m1.1.20.2.2.3.2.1.1.2.2" xref="p1.1.m1.1.20.2.2.3.2.1.1.2.2.cmml">
                      <mi id="p1.1.m1.1.9" xref="p1.1.m1.1.9.cmml">μ</mi>
                      <mo id="p1.1.m1.1.10" xref="p1.1.m1.1.10.cmml">-</mo>
                      <mstyle displaystyle="true" id="p1.1.m1.1.11" xref="p1.1.m1.1.11.cmml">
                        <mfrac id="p1.1.m1.1.11a" xref="p1.1.m1.1.11.cmml">
                          <msup id="p1.1.m1.1.11.2" xref="p1.1.m1.1.11.2.cmml">
                            <mi id="p1.1.m1.1.11.2.1" xref="p1.1.m1.1.11.2.1.cmml">σ</mi>
                            <mn id="p1.1.m1.1.11.2.2.1" xref="p1.1.m1.1.11.2.2.1.cmml">2</mn>
                          </msup>
                          <mn id="p1.1.m1.1.11.3" xref="p1.1.m1.1.11.3.cmml">2</mn>
                        </mfrac>
                      </mstyle>
                    </mrow>
                    <mo id="p1.1.m1.1.12" xref="p1.1.m1.1.20.2.2.3.2.1.1.2.2.cmml">)</mo>
                  </mrow>
                  <mo id="p1.1.m1.1.20.2.2.3.2.1.1.1" xref="p1.1.m1.1.20.2.2.3.2.1.1.1.cmml"></mo>
                  <mi id="p1.1.m1.1.13" xref="p1.1.m1.1.13.cmml">t</mi>
                </mrow>
                <mo id="p1.1.m1.1.14" xref="p1.1.m1.1.14.cmml">+</mo>
                <mrow id="p1.1.m1.1.20.2.2.3.2.1.2" xref="p1.1.m1.1.20.2.2.3.2.1.2.cmml">
                  <mi id="p1.1.m1.1.15" xref="p1.1.m1.1.15.cmml">σ</mi>
                  <mo id="p1.1.m1.1.20.2.2.3.2.1.2.1" xref="p1.1.m1.1.20.2.2.3.2.1.2.1.cmml"></mo>
                  <msub id="p1.1.m1.1.20.2.2.3.2.1.2.2" xref="p1.1.m1.1.20.2.2.3.2.1.2.2.cmml">
                    <mi id="p1.1.m1.1.16" xref="p1.1.m1.1.16.cmml">W</mi>
                    <mi id="p1.1.m1.1.17.1" xref="p1.1.m1.1.17.1.cmml">t</mi>
                  </msub>
                </mrow>
              </mrow>
              <mo id="p1.1.m1.1.18" xref="p1.1.m1.1.20.2.2.3.1.cmml">)</mo>
            </mrow>
          </mrow>
        </mrow>
      </mrow>
      <mo id="p1.1.m1.1.19" xref="p1.1.m1.1.20.2.cmml">.</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b">
      <apply id="p1.1.m1.1.20.2.cmml" xref="p1.1.m1.1.20">
        <eq id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3"/>
        <apply id="p1.1.m1.1.20.2.1.cmml" xref="p1.1.m1.1.20.2.1">
          <csymbol cd="ambiguous" id="p1.1.m1.1.20.2.1.1.cmml" xref="p1.1.m1.1.20.2.1">subscript</csymbol>
          <ci id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">𝑆</ci>
          <ci id="p1.1.m1.1.2.1.cmml" xref="p1.1.m1.1.2.1">𝑡</ci>
        </apply>
        <apply id="p1.1.m1.1.20.2.2.cmml" xref="p1.1.m1.1.20.2.2">
          <times id="p1.1.m1.1.20.2.2.1.cmml" xref="p1.1.m1.1.20.2.2.1"/>
          <apply id="p1.1.m1.1.20.2.2.2.cmml" xref="p1.1.m1.1.20.2.2.2">
            <csymbol cd="ambiguous" id="p1.1.m1.1.20.2.2.2.1.cmml" xref="p1.1.m1.1.20.2.2.2">subscript</csymbol>
            <ci id="p1.1.m1.1.4.cmml" xref="p1.1.m1.1.4">𝑆</ci>
            <cn type="integer" id="p1.1.m1.1.5.1.cmml" xref="p1.1.m1.1.5.1">0</cn>
          </apply>
          <apply id="p1.1.m1.1.20.2.2.3.1.cmml" xref="p1.1.m1.1.20.2.2.3.2">
            <exp id="p1.1.m1.1.6.cmml" xref="p1.1.m1.1.6"/>
            <apply id="p1.1.m1.1.20.2.2.3.2.1.cmml" xref="p1.1.m1.1.20.2.2.3.2.1">
              <plus id="p1.1.m1.1.14.cmml" xref="p1.1.m1.1.14"/>
              <apply id="p1.1.m1.1.20.2.2.3.2.1.1.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.1">
                <times id="p1.1.m1.1.20.2.2.3.2.1.1.1.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.1.1"/>
                <apply id="p1.1.m1.1.20.2.2.3.2.1.1.2.2.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.1.2">
                  <minus id="p1.1.m1.1.10.cmml" xref="p1.1.m1.1.10"/>
                  <ci id="p1.1.m1.1.9.cmml" xref="p1.1.m1.1.9">𝜇</ci>
                  <apply id="p1.1.m1.1.11.cmml" xref="p1.1.m1.1.11">
                    <divide id="p1.1.m1.1.11.1.cmml" xref="p1.1.m1.1.11"/>
                    <apply id="p1.1.m1.1.11.2.cmml" xref="p1.1.m1.1.11.2">
                      <csymbol cd="ambiguous" id="p1.1.m1.1.11.2.3.cmml" xref="p1.1.m1.1.11.2">superscript</csymbol>
                      <ci id="p1.1.m1.1.11.2.1.cmml" xref="p1.1.m1.1.11.2.1">𝜎</ci>
                      <cn type="integer" id="p1.1.m1.1.11.2.2.1.cmml" xref="p1.1.m1.1.11.2.2.1">2</cn>
                    </apply>
                    <cn type="integer" id="p1.1.m1.1.11.3.cmml" xref="p1.1.m1.1.11.3">2</cn>
                  </apply>
                </apply>
                <ci id="p1.1.m1.1.13.cmml" xref="p1.1.m1.1.13">𝑡</ci>
              </apply>
              <apply id="p1.1.m1.1.20.2.2.3.2.1.2.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.2">
                <times id="p1.1.m1.1.20.2.2.3.2.1.2.1.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.2.1"/>
                <ci id="p1.1.m1.1.15.cmml" xref="p1.1.m1.1.15">𝜎</ci>
                <apply id="p1.1.m1.1.20.2.2.3.2.1.2.2.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.2.2">
                  <csymbol cd="ambiguous" id="p1.1.m1.1.20.2.2.3.2.1.2.2.1.cmml" xref="p1.1.m1.1.20.2.2.3.2.1.2.2">subscript</csymbol>
                  <ci id="p1.1.m1.1.16.cmml" xref="p1.1.m1.1.16">𝑊</ci>
                  <ci id="p1.1.m1.1.17.1.cmml" xref="p1.1.m1.1.17.1">𝑡</ci>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle S_{t}=S_{0}\exp\left(\left(\mu-{\frac{\sigma^{2}}{2}}\right)t+%
\sigma W_{t}\right).}</annotation>
  </semantics>
</math>

SVG (12.814 KB / 4.761 KB) :

upper S Subscript t Baseline equals upper S 0 times exp left-parenthesis left-parenthesis mu minus StartFraction sigma squared Over 2 EndFraction right-parenthesis times t plus sigma times upper W Subscript t Baseline right-parenthesis period

MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools) rendering

MathML (0 B / 8 B) :

SVG image empty. Force Re-Rendering

SVG (0 B / 8 B) :


PNG (0 B / 8 B) :


Translations to Computer Algebra Systems

Translation to Maple

In Maple: S[t]= S[0]*exp((mu -((sigma)^(2))/(2))*t + sigma*W[t])

Information about the conversion process:

\exp: Exponential function; Example: \exp@@{z}

Will be translated to: exp($0)

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.2#E19

Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Trace



Translation to Mathematica

In Mathematica: Subscript[S, t]= Subscript[S, 0]*Exp[(\[Mu]-Divide[\[Sigma]^(2),2])*t + \[Sigma]*Subscript[W, t]]

Information about the conversion process:

\exp: Exponential function; Example: \exp@@{z}

Will be translated to: Exp[$0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.2#E19

Mathematica: https://



Similar pages

Calculated based on the variables occurring on the entire Geometric Brownian motion page

Identifiers

MathML observations

0results

0results

no statistics present please run the maintenance script ExtractFeatures.php

0 results

0 results