Stochastic calculus

<h2 id="it-integral">Itô integral</h2>
<p class="note">Main article: <a href="/facts/It%25C3%25B4_calculus/2uSR02fY">Itô calculus</a></p>
<p>The <a href="/facts/It%25C3%25B4_integral/2uSR02fY">Itô integral</a> is central to the study of stochastic calculus. The integral 
  
    
      
        ∫
        H
        
        d
        X
      
    
    {\displaystyle \int H\,dX}
  
 is defined for a <a href="/facts/Semimartingale/DUU5kWtC">semimartingale</a> <i>X</i> and locally bounded predictable process <i>H</i>. 
</p>
<h2 id="stratonovich-integral">Stratonovich integral</h2>
<p class="note">Main article: <a href="/facts/Stratonovich_integral/sK9p7005">Stratonovich integral</a></p>
<p>The Stratonovich integral or Fisk–Stratonovich integral of a <a href="/facts/Semimartingale/DUU5kWtC">semimartingale</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 against another <a href="/facts/Semimartingale/DUU5kWtC">semimartingale</a> <i>Y</i> can be defined in terms of the Itô integral as
</p>

∫
          
            0
          
          
            t
          
        
        
          X
          
            s
            −
          
        
        ∘
        d
        
          Y
          
            s
          
        
        :=
        
          ∫
          
            0
          
          
            t
          
        
        
          X
          
            s
            −
          
        
        d
        
          Y
          
            s
          
        
        +
        
          
            1
            2
          
        
        
          
            [
            
              X
              ,
              Y
            
            ]
          
          
            t
          
          
            c
          
        
        ,
      
    
    {\displaystyle \int _{0}^{t}X_{s-}\circ dY_{s}:=\int _{0}^{t}X_{s-}dY_{s}+{\frac {1}{2}}\left[X,Y\right]_{t}^{c},}

<p>where [<i>X</i>, <i>Y</i>]<i>t</i><i>c</i> denotes the optional <a href="/facts/Quadratic_variation/15jHz4xU">quadratic covariation</a> of the continuous parts of <i>X</i>
and <i>Y</i>, which is the optional quadratic covariation minus the jumps of the processes 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
, i.e.
</p>

[
            
              X
              ,
              Y
            
            ]
          
          
            t
          
          
            c
          
        
        :=
        [
        X
        ,
        Y
        
          ]
          
            t
          
        
        −
        
          ∑
          
            s
            ≤
            t
          
        
        Δ
        
          X
          
            s
          
        
        Δ
        
          Y
          
            s
          
        
      
    
    {\displaystyle \left[X,Y\right]_{t}^{c}:=[X,Y]_{t}-\sum \limits _{s\leq t}\Delta X_{s}\Delta Y_{s}}
  
.
<p>The alternative notation
</p>

∫
          
            0
          
          
            t
          
        
        
          X
          
            s
          
        
        
        ∂
        
          Y
          
            s
          
        
      
    
    {\displaystyle \int _{0}^{t}X_{s}\,\partial Y_{s}}

<p>is also used to denote the Stratonovich integral.
</p>
<h2 id="applications">Applications</h2>
<p>An important application of stochastic calculus is in <a href="/facts/Mathematical_finance/P9otEltQ">mathematical finance</a>, in which asset prices are often assumed to follow <a href="/facts/Stochastic_differential_equation/ol9a9WNq">stochastic differential equations</a>. For example, the <a href="/facts/Black%25E2%2580%2593Scholes_model/TPPG5YWQ">Black–Scholes model</a> prices options as if they follow a <a href="/facts/Geometric_Brownian_motion/0Ve9dW6c">geometric Brownian motion</a>, illustrating the opportunities and risks from applying stochastic calculus.
</p><p>In communication theory, one might consider a random data stream—say, a binary sequence of 100 million bits. After encoding and modulation, the resulting power spectrum may exhibit distinct features such as peaks or unexpected interference. Analyzing these characteristics requires the precise methods of stochastic processes.
</p><p>Before applying these tools, however, it is essential to review the underlying mathematical foundations. Mathematical methods are intended to rigorously prove or disprove hypotheses using logical reasoning. Misapplication of these tools can lead to misleading or incorrect conclusions without any clear indication of where the error occurred. Therefore, mastering well-formed mathematical inferences is a necessary prerequisite.
</p><p>Transitioning from classical calculus to stochastic processes also involves learning new methods, including rigorous proof techniques tailored to randomness. In this context, measure theory plays a central role by ensuring that statements in probability—such as definitions of probability density functions, total probability, and conditional probability—are mathematically sound and well-defined.
</p>
<h2 id="stochastic-integrals">Stochastic integrals</h2>
<p>Besides the classical Itô and Fisk–Stratonovich integrals, many other notions of stochastic integrals exist, such as the <a href="/facts/Skorokhod_integral/2ghwWfbR">Hitsuda–Skorokhod integral</a>, the Marcus integral, and the <a href="/facts/Ogawa_integral/yvEbKjGP">Ogawa integral</a>.
</p>
<h2 id="see-also">See also</h2>
<ul>
<li>Mathematics portal</li></ul>

<ul><li><a href="/facts/It%25C3%25B4_calculus/2uSR02fY">Itô calculus</a></li>
<li><a href="/facts/It%25C3%25B4%2527s_lemma/NF7tTcgd">Itô's lemma</a></li>
<li><a href="/facts/Stratonovich_integral/sK9p7005">Stratonovich integral</a></li>
<li><a href="/facts/Semimartingale/DUU5kWtC">Semimartingale</a></li>
<li><a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a></li></ul>

<ul><li>Thomas Mikosch, 1998, Elementary Stochastic Calculus, World Scientific, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 981-023543-7</li>
<li>Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd Edition). World Scientific Publishing, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781848168312</li>
<li>Szabados, T.S.; Székely, B.Z. (2008). "Stochastic Integration Based on Simple, Symmetric Random Walks". <i>Journal of Theoretical Probability</i>. 22: 203–219. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0712.3908">0712.3908</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs10959-007-0140-8">10.1007/s10959-007-0140-8</a>. <a href="https://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3908v2.pdf">Preprint</a></li></ul>

Stochastic calculus open-in-new

Stochastic calculus