Cauchy process

<h2 id="symmetric-cauchy-process">Symmetric Cauchy process</h2>

The symmetric Cauchy process can be described by a <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> or <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a> subject to a <a href="/facts/L%C3%A9vy_distribution/g3fVHgGz">Lévy</a> <a href="/facts/Subordinator_(mathematics)/ckSDdmcC">subordinator</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> The Lévy subordinator is a process associated with a <a href="/facts/L%C3%A9vy_distribution/g3fVHgGz">Lévy distribution</a> having location parameter of 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 and a scale parameter of 
 
 
 
 
 t
 
 2
 
 
 
 /
 
 2
 
 
 {\displaystyle t^{2}/2}
 
.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> The Lévy distribution is a special case of the <a href="/facts/Inverse-gamma_distribution/zpyktNyN">inverse-gamma distribution</a>. So, using 
 
 
 
 C
 
 
 {\displaystyle C}
 
 to represent the Cauchy process and 
 
 
 
 L
 
 
 {\displaystyle L}
 
 to represent the Lévy subordinator, the symmetric Cauchy process can be described as:

C
        (
        t
        ;
        0
        ,
        1
        )
        
        :=
        
        W
        (
        L
        (
        t
        ;
        0
        ,
        
          t
          
            2
          
        
        
          /
        
        2
        )
        )
        .
      
    
    {\displaystyle C(t;0,1)\;:=\;W(L(t;0,t^{2}/2)).}

The Lévy distribution is the probability of the first hitting time for a Brownian motion, and thus the Cauchy process is essentially the result of two <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> Brownian motion processes.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
The <a href="/facts/L%C3%A9vy_process/m9A3YQW8">Lévy–Khintchine representation</a> for the symmetric Cauchy process is a triplet with zero drift and zero diffusion, giving a Lévy–Khintchine triplet of 
 
 
 
 (
 0
 ,
 0
 ,
 W
 )
 
 
 {\displaystyle (0,0,W)}
 
, where 
 
 
 
 W
 (
 d
 x
 )
 =
 d
 x
 
 /
 
 (
 π
 
 x
 
 2
 
 
 )
 
 
 {\displaystyle W(dx)=dx/(\pi x^{2})}
 
.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
The marginal <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of the symmetric Cauchy process has the form:<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

E
        ⁡
        
          
            [
          
        
        
          e
          
            i
            θ
            
              X
              
                t
              
            
          
        
        
          
            ]
          
        
        =
        
          e
          
            −
            t
            
              |
            
            θ
            
              |
            
          
        
        .
      
    
    {\displaystyle \operatorname {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=e^{-t|\theta |}.}

The marginal <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> of the symmetric Cauchy process is the <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a> whose density is<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

f
        (
        x
        ;
        t
        )
        =
        
          
            1
            π
          
        
        
          [
          
            
              t
              
                
                  x
                  
                    2
                  
                
                +
                
                  t
                  
                    2
                  
                
              
            
          
          ]
        
        .
      
    
    {\displaystyle f(x;t)={1 \over \pi }\left[{t \over x^{2}+t^{2}}\right].}

<h2 id="asymmetric-cauchy-process">Asymmetric Cauchy process</h2>
The asymmetric Cauchy process is defined in terms of a parameter 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
. Here

β
 
 
 {\displaystyle \beta }
 
 is the <a href="/facts/Skewness/rdS8luFa">skewness</a> parameter, and its <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> must be less than or equal to 1.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> In the case where 
 
 
 
 
 |
 
 β
 
 |
 
 =
 1
 
 
 {\displaystyle |\beta |=1}
 
 the process is considered a completely asymmetric Cauchy process.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
The Lévy–Khintchine triplet has the form 
 
 
 
 (
 0
 ,
 0
 ,
 W
 )
 
 
 {\displaystyle (0,0,W)}
 
, where 
 
 
 
 W
 (
 d
 x
 )
 =
 
 
 {
 
 
 
 A
 
 x
 
 −
 2
 
 
 
 d
 x
 
 
 
 if 
 
 x
 >
 0
 
 
 
 
 B
 
 x
 
 −
 2
 
 
 
 d
 x
 
 
 
 if 
 
 x
 <
 0
 
 
 
 
 
 
 
 
 {\displaystyle W(dx)={\begin{cases}Ax^{-2}\,dx&{\text{if }}x>0\\Bx^{-2}\,dx&{\text{if }}x<0\end{cases}}}
 
, where 
 
 
 
 A
 ≠
 B
 
 
 {\displaystyle A\neq B}
 
, 
 
 
 
 A
 >
 0
 
 
 {\displaystyle A>0}
 
 and 
 
 
 
 B
 >
 0
 
 
 {\displaystyle B>0}
 
.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
Given this, 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 is a function of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
.
The characteristic function of the asymmetric Cauchy distribution has the form:<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>

E
        ⁡
        
          
            [
          
        
        
          e
          
            i
            θ
            
              X
              
                t
              
            
          
        
        
          
            ]
          
        
        =
        
          e
          
            −
            t
            (
            
              |
            
            θ
            
              |
            
            +
            i
            β
            θ
            ln
            ⁡
            
              |
            
            θ
            
              |
            
            
              /
            
            (
            2
            π
            )
            )
          
        
        .
      
    
    {\displaystyle \operatorname {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=e^{-t(|\theta |+i\beta \theta \ln |\theta |/(2\pi ))}.}

The marginal probability distribution of the asymmetric Cauchy process is a <a href="/facts/Stable_distribution/rvtRmhlz">stable distribution</a> with index of stability (i.e., α parameter) equal to 1.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Engelbert, H.J., Kurenok, V.P. & Zalinescu, A. (2006). "On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes". In Kabanov, Y.; Liptser, R.; Stoyanov, J. (eds.). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="9783540307884" target="_blank">9783540307884</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Winkel, M. "Introduction to Levy processes" (PDF). pp. 15–16. Retrieved 2013-02-07. <a href="http://www.stats.ox.ac.uk/~winkel/lp1.pdf" target="_blank">http://www.stats.ox.ac.uk/~winkel/lp1.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Jacob, N. (2005). Pseudo Differential Operators & Markov Processes: Markov Processes And Applications, Volume 3. Imperial College Press. p. 135. ISBN 9781860945687. <a href="9781860945687" target="_blank">9781860945687</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Bertoin, J. (2001). "Some elements on Lévy processes". In Shanbhag, D.N. (ed.). Stochastic Processes: Theory and Methods. Gulf Professional Publishing. p. 122. ISBN 9780444500144. <a href="9780444500144" target="_blank">9780444500144</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Engelbert, H.J., Kurenok, V.P. & Zalinescu, A. (2006). "On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes". In Kabanov, Y.; Liptser, R.; Stoyanov, J. (eds.). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer. p. 228. ISBN 9783540307884.{{cite book}}: CS1 maint: multiple names: authors list (link) <a href="9783540307884" target="_blank">9783540307884</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kroese, D.P.; Taimre, T.; Botev, Z.I. (2011). Handbook of Monte Carlo Methods. John Wiley & Sons. p. 214. ISBN 9781118014950. <a href="9781118014950" target="_blank">9781118014950</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. <a href="http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf" target="_blank">http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. <a href="http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf" target="_blank">http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53. <a href="http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf" target="_blank">http://www.applebaum.staff.shef.ac.uk/Brauns2notes.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. <a href="9780387878591" target="_blank">9780387878591</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. <a href="9780387878591" target="_blank">9780387878591</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Cinlar, E. (2011). Probability and Stochastics. Springer. p. 332. ISBN 9780387878591. <a href="9780387878591" target="_blank">9780387878591</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Itô, K. (2006). Essentials of Stochastic Processes. American Mathematical Society. p. 54. ISBN 9780821838983. <a href="9780821838983" target="_blank">9780821838983</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Kovalenko, I.N.; et al. (1996). Models of Random Processes: A Handbook for Mathematicians and Engineers. CRC Press. pp. 210–211. ISBN 9780849328701. <a href="9780849328701" target="_blank">9780849328701</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
</ol>

Cauchy process open-in-new

Cauchy process