Markov switching multifractal

<h2 id="msm-specification">MSM specification</h2>
<p>The MSM model can be specified in both discrete time and continuous time.
</p>
<h3>Discrete time</h3>
<p>Let 
  
    
      
        
          P
          
            t
          
        
      
    
    {\displaystyle P_{t}}
  
 denote the price of a financial asset, and let 
  
    
      
        
          r
          
            t
          
        
        =
        ln
        ⁡
        (
        
          P
          
            t
          
        
        
          /
        
        
          P
          
            t
            −
            1
          
        
        )
      
    
    {\displaystyle r_{t}=\ln(P_{t}/P_{t-1})}
  
 denote the return over two consecutive periods. In MSM, returns are specified as
</p>

r
          
            t
          
        
        =
        μ
        +
        
          
            
              σ
              ¯
            
          
        
        (
        
          M
          
            1
            ,
            t
          
        
        
          M
          
            2
            ,
            t
          
        
        .
        .
        .
        
          M
          
            
              
                
                  k
                  ¯
                
              
            
            ,
            t
          
        
        
          )
          
            1
            
              /
            
            2
          
        
        
          ϵ
          
            t
          
        
        ,
      
    
    {\displaystyle r_{t}=\mu +{\bar {\sigma }}(M_{1,t}M_{2,t}...M_{{\bar {k}},t})^{1/2}\epsilon _{t},}

<p>where 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 and 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
  are constants and {
  
    
      
        
          ϵ
          
            t
          
        
      
    
    {\displaystyle \epsilon _{t}}
  
} are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector:
</p>

M
          
            t
          
        
        =
        (
        
          M
          
            1
            ,
            t
          
        
        
          M
          
            2
            ,
            t
          
        
        …
        
          M
          
            
              
                
                  k
                  ¯
                
              
            
            ,
            t
          
        
        )
        ∈
        
          R
          
            +
          
          
            
              
                k
                ¯
              
            
          
        
        .
      
    
    {\displaystyle M_{t}=(M_{1,t}M_{2,t}\dots M_{{\bar {k}},t})\in R_{+}^{\bar {k}}.}

<p>Given the volatility state 
  
    
      
        
          M
          
            t
          
        
      
    
    {\displaystyle M_{t}}
  
, the next-period multiplier 
  
    
      
        
          M
          
            k
            ,
            t
            +
            1
          
        
      
    
    {\displaystyle M_{k,t+1}}
  
 is drawn from a fixed distribution M with probability 
  
    
      
        
          γ
          
            k
          
        
      
    
    {\displaystyle \gamma _{k}}
  
, and is otherwise left unchanged.
</p>
<table><tbody><tr><td>                              M                      k            ,            t                                {\displaystyle M_{k,t}}   drawn from distribution M</td><td>with probability                               γ                      k                                {\displaystyle \gamma _{k}}  </td></tr><tr><td>                              M                      k            ,            t                          =                  M                      k            ,            t            −            1                                {\displaystyle M_{k,t}=M_{k,t-1}}  </td><td>with probability                     1        −                  γ                      k                                {\displaystyle 1-\gamma _{k}}  </td></tr></tbody></table>
<p>The transition probabilities are specified by
</p>

γ
          
            k
          
        
        =
        1
        −
        (
        1
        −
        
          γ
          
            1
          
        
        
          )
          
            (
            
              b
              
                k
                −
                1
              
            
            )
          
        
      
    
    {\displaystyle \gamma _{k}=1-(1-\gamma _{1})^{(b^{k-1})}}
  
.
<p>The sequence 
  
    
      
        
          γ
          
            k
          
        
      
    
    {\displaystyle \gamma _{k}}
  
 is approximately geometric 
  
    
      
        
          γ
          
            k
          
        
        ≈
        
          γ
          
            1
          
        
        
          b
          
            k
            −
            1
          
        
      
    
    {\displaystyle \gamma _{k}\approx \gamma _{1}b^{k-1}}
  
 at low frequency. The marginal distribution M has a unit mean, has a positive support, and is independent of k.
</p>
<h4>Binomial MSM</h4>
<p>In empirical applications, the distribution M is often a discrete distribution that can take the values 
  
    
      
        
          m
          
            0
          
        
      
    
    {\displaystyle m_{0}}
  
 or 
  
    
      
        2
        −
        
          m
          
            0
          
        
      
    
    {\displaystyle 2-m_{0}}
  
 with equal probability. The return process 
  
    
      
        
          r
          
            t
          
        
      
    
    {\displaystyle r_{t}}
  
 is then specified by the parameters 
  
    
      
        θ
        =
        (
        
          m
          
            0
          
        
        ,
        μ
        ,
        
          
            
              σ
              ¯
            
          
        
        ,
        b
        ,
        
          γ
          
            1
          
        
        )
      
    
    {\displaystyle \theta =(m_{0},\mu ,{\bar {\sigma }},b,\gamma _{1})}
  
. Note that the number of parameters is the same for all  
  
    
      
        
          
            
              k
              ¯
            
          
        
        >
        1
      
    
    {\displaystyle {\bar {k}}>1}
  
.
</p>
<h3>Continuous time</h3>
<p>MSM is similarly defined in continuous time. The price process follows the diffusion:
</p>

d
              
                P
                
                  t
                
              
            
            
              P
              
                t
              
            
          
        
        =
        μ
        d
        t
        +
        σ
        (
        
          M
          
            t
          
        
        )
        
        d
        
          W
          
            t
          
        
        ,
      
    
    {\displaystyle {\frac {dP_{t}}{P_{t}}}=\mu dt+\sigma (M_{t})\,dW_{t},}

<p>where 
  
    
      
        σ
        (
        
          M
          
            t
          
        
        )
        =
        
          
            
              σ
              ¯
            
          
        
        (
        
          M
          
            1
            ,
            t
          
        
        …
        
          M
          
            
              
                
                  k
                  ¯
                
              
            
            ,
            t
          
        
        
          )
          
            1
            
              /
            
            2
          
        
      
    
    {\displaystyle \sigma (M_{t})={\bar {\sigma }}(M_{1,t}\dots M_{{\bar {k}},t})^{1/2}}
  
, 
  
    
      
        
          W
          
            t
          
        
      
    
    {\displaystyle W_{t}}
  
 is a standard Brownian motion, and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 and 
  
    
      
        
          
            
              σ
              ¯
            
          
        
      
    
    {\displaystyle {\bar {\sigma }}}
  
 are constants. Each component follows the dynamics:
</p>
<table><tbody><tr><td>                              M                      k            ,            t                                {\displaystyle M_{k,t}}   drawn from distribution M</td><td>with probability                               γ                      k                          d        t              {\displaystyle \gamma _{k}dt}  </td></tr><tr><td>                              M                      k            ,            t            +            d            t                          =                  M                      k            ,            t                                {\displaystyle M_{k,t+dt}=M_{k,t}}  </td><td>with probability                     1        −                  γ                      k                          d        t              {\displaystyle 1-\gamma _{k}dt}  </td></tr></tbody></table>
<p>The intensities vary geometrically with k:
</p>

γ
          
            k
          
        
        =
        
          γ
          
            1
          
        
        
          b
          
            k
            −
            1
          
        
        .
      
    
    {\displaystyle \gamma _{k}=\gamma _{1}b^{k-1}.}

<p>When the number of components 
  
    
      
        
          
            
              k
              ¯
            
          
        
      
    
    {\displaystyle {\bar {k}}}
  
 goes to infinity, continuous-time MSM converges to a multifractal diffusion, whose sample paths take a continuum of local Hölder exponents on any finite time interval.
</p>
<h2 id="inference-and-closed-form-likelihood">Inference and closed-form likelihood</h2>
<p>When 
  
    
      
        M
      
    
    {\displaystyle M}
  
 has a <a href="/facts/Discrete_probability_distribution/EpsKKVRu">discrete distribution</a>, the Markov state vector 
  
    
      
        
          M
          
            t
          
        
      
    
    {\displaystyle M_{t}}
  
 takes finitely many values 
  
    
      
        
          m
          
            1
          
        
        ,
        .
        .
        .
        ,
        
          m
          
            d
          
        
        ∈
        
          R
          
            +
          
          
            
              
                k
                ¯
              
            
          
        
      
    
    {\displaystyle m^{1},...,m^{d}\in R_{+}^{\bar {k}}}
  
. For instance, there are 
  
    
      
        d
        =
        
          2
          
            
              
                k
                ¯
              
            
          
        
      
    
    {\displaystyle d=2^{\bar {k}}}
  
 possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix 
  
    
      
        A
        =
        (
        
          a
          
            i
            ,
            j
          
        
        
          )
          
            1
            ≤
            i
            ,
            j
            ≤
            d
          
        
      
    
    {\displaystyle A=(a_{i,j})_{1\leq i,j\leq d}}
  
 with components 
  
    
      
        
          a
          
            i
            ,
            j
          
        
        =
        P
        
          (
          
            
              M
              
                t
                +
                1
              
            
            =
            
              m
              
                j
              
            
            
              |
            
            
              M
              
                t
              
            
            =
            
              m
              
                i
              
            
          
          )
        
      
    
    {\displaystyle a_{i,j}=P\left(M_{t+1}=m^{j}|M_{t}=m^{i}\right)}
  
.
Conditional on the volatility state, the return 
  
    
      
        
          r
          
            t
          
        
      
    
    {\displaystyle r_{t}}
  
 has Gaussian density
</p>

f
        (
        
          r
          
            t
          
        
        
          |
        
        
          M
          
            t
          
        
        =
        
          m
          
            i
          
        
        )
        =
        
          
            1
            
              2
              π
              
                σ
                
                  2
                
              
              (
              
                m
                
                  i
                
              
              )
            
          
        
        exp
        ⁡
        
          [
          
            −
            
              
                
                  (
                  
                    r
                    
                      t
                    
                  
                  −
                  μ
                  
                    )
                    
                      2
                    
                  
                
                
                  2
                  
                    σ
                    
                      2
                    
                  
                  (
                  
                    m
                    
                      i
                    
                  
                  )
                
              
            
          
          ]
        
        .
      
    
    {\displaystyle f(r_{t}|M_{t}=m^{i})={\frac {1}{\sqrt {2\pi \sigma ^{2}(m^{i})}}}\exp \left[-{\frac {(r_{t}-\mu )^{2}}{2\sigma ^{2}(m^{i})}}\right].}

<h3>Conditional distribution</h3>
<h3>Closed-form Likelihood</h3>
<p>The log likelihood function has the following analytical expression:
</p>

ln
        ⁡
        L
        (
        
          r
          
            1
          
        
        ,
        …
        ,
        
          r
          
            T
          
        
        ;
        θ
        )
        =
        
          ∑
          
            t
            =
            1
          
          
            T
          
        
        ln
        ⁡
        [
        ω
        (
        
          r
          
            t
          
        
        )
        .
        (
        
          Π
          
            t
            −
            1
          
        
        A
        )
        ]
        .
      
    
    {\displaystyle \ln L(r_{1},\dots ,r_{T};\theta )=\sum _{t=1}^{T}\ln[\omega (r_{t}).(\Pi _{t-1}A)].}

<p><a href="/facts/Maximum_likelihood_estimation/0Yq2dpQD">Maximum likelihood</a> provides reasonably precise estimates in finite samples.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h3>Other estimation methods</h3>
<p>When 
  
    
      
        M
      
    
    {\displaystyle M}
  
 has a <a href="/facts/Continuous_probability_distribution/EpsKKVRu">continuous distribution</a>, estimation can proceed by simulated method of moments,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> or simulated likelihood via a particle filter.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="forecasting">Forecasting</h2>
<p>Given 
  
    
      
        
          r
          
            1
          
        
        ,
        …
        ,
        
          r
          
            t
          
        
      
    
    {\displaystyle r_{1},\dots ,r_{t}}
  
, the conditional distribution of the latent state vector at date 
  
    
      
        t
        +
        n
      
    
    {\displaystyle t+n}
  
 is given by:
</p>

Π
                ^
              
            
          
          
            t
            ,
            n
          
        
        =
        
          Π
          
            t
          
        
        
          A
          
            n
          
        
        .
        
      
    
    {\displaystyle {\hat {\Pi }}_{t,n}=\Pi _{t}A^{n}.\,}

<p>MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample. Calvet and Fisher<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH,<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> and Fractionally Integrated GARCH.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> Lux<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> obtains similar results using linear predictions.
</p>
<h2 id="applications">Applications</h2>
<h3>Multiple assets and value-at-risk</h3>
<p>Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p>
<h3>Asset pricing</h3>
<p>In financial economics, MSM has been used to analyze the pricing implications of multifrequency risk. The models have had some success in explaining the excess volatility of stock returns compared to fundamentals and the negative skewness of equity returns. They have also been used to generate multifractal jump-diffusions.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p>
<h2 id="related-approaches">Related approaches</h2>
<p>MSM is a stochastic volatility model<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by <a href="/facts/James_D._Hamilton/68PjzdAt">James D. Hamilton</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> 
MSM is closely related to the Multifractal Model of Asset Returns.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process. 
MSM provides a pure regime-switching formulation of multifractal measures, which were pioneered by <a href="/facts/Beno%25C3%25AEt_Mandelbrot/fa8tKP93">Benoit Mandelbrot</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a></li>
<li><a href="/facts/Rogemar_Mamon/JWQtkoFY">Rogemar Mamon</a></li>
<li><a href="/facts/Markov_chain/5oP5hntk">Markov chain</a></li>
<li>Multifractal model of asset returns</li>
<li><a href="/facts/Multifractal/Fly3MCTb">Multifractal</a></li>
<li><a href="/facts/Stochastic_volatility/d5zBCcUP">Stochastic volatility</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20120402215902/http://wiki.wstein.org/2008/simuw">Financial Time Series, Multifractals and Hidden Markov Models</a></li></ul>

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

<ol>
<li id="fn:1"><p>Calvet, L.; Fisher, A. (2001). "Forecasting multifractal volatility" (PDF). Journal of Econometrics. 105: 27–58. doi:10.1016/S0304-4076(01)00069-0. S2CID 119394176. <a href="http://archive.nyu.edu/bitstream/2451/26894/2/wpa99017.pdf" target="_blank">http://archive.nyu.edu/bitstream/2451/26894/2/wpa99017.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Calvet, L. E. (2004). "How to Forecast Long-Run Volatility: Regime Switching and the Estimation of Multifractal Processes". Journal of Financial Econometrics. 2: 49–83. CiteSeerX 10.1.1.536.8334. doi:10.1093/jjfinec/nbh003. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Calvet, L. E. (2004). "How to Forecast Long-Run Volatility: Regime Switching and the Estimation of Multifractal Processes". Journal of Financial Econometrics. 2: 49–83. CiteSeerX 10.1.1.536.8334. doi:10.1093/jjfinec/nbh003. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Calvet, Laurent; Fisher, Adlai (July 2003). "Regime-switching and the estimation of multifractal processes". NBER Working Paper No. 9839. doi:10.3386/w9839. <a href="https://doi.org/10.3386%2Fw9839" target="_blank">https://doi.org/10.3386%2Fw9839</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Lux, T. (2008). "The Markov-Switching Multifractal Model of Asset Returns". Journal of Business & Economic Statistics. 26 (2): 194–210. doi:10.1198/073500107000000403. S2CID 55648360. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Calvet, L. E.; Fisher, A. J.; Thompson, S. B. (2006). "Volatility comovement: A multifrequency approach". Journal of Econometrics. 131 (1–2): 179–215. CiteSeerX 10.1.1.331.152. doi:10.1016/j.jeconom.2005.01.008. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Calvet, L. E. (2004). "How to Forecast Long-Run Volatility: Regime Switching and the Estimation of Multifractal Processes". Journal of Financial Econometrics. 2: 49–83. CiteSeerX 10.1.1.536.8334. doi:10.1093/jjfinec/nbh003. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Gray, S. F. (1996). "Modeling the conditional distribution of interest rates as a regime-switching process". Journal of Financial Economics. 42: 27–77. doi:10.1016/0304-405X(96)00875-6. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Klaassen, F. (2002). "Improving GARCH volatility forecasts with regime-switching GARCH" (PDF). Empirical Economics. 27 (2): 363–394. doi:10.1007/s001810100100. S2CID 29571612. <a href="https://pure.uva.nl/ws/files/3524906/21100_v60.pdf" target="_blank">https://pure.uva.nl/ws/files/3524906/21100_v60.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Bollerslev, T.; Ole Mikkelsen, H. (1996). "Modeling and pricing long memory in stock market volatility". Journal of Econometrics. 73: 151–184. doi:10.1016/0304-4076(95)01736-4. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Lux, T. (2008). "The Markov-Switching Multifractal Model of Asset Returns". Journal of Business & Economic Statistics. 26 (2): 194–210. doi:10.1198/073500107000000403. S2CID 55648360. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Calvet, L. E.; Fisher, A. J.; Thompson, S. B. (2006). "Volatility comovement: A multifrequency approach". Journal of Econometrics. 131 (1–2): 179–215. CiteSeerX 10.1.1.331.152. doi:10.1016/j.jeconom.2005.01.008. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Calvet, Laurent E.; Fisher, Adlai J. (2008). Multifractal volatility theory, forecasting, and pricing. Burlington, MA: Academic Press. ISBN 9780080559964. <a href="9780080559964" target="_blank">9780080559964</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Taylor, Stephen J (2008). Modelling financial time series (2nd ed.). New Jersey: World Scientific. ISBN 9789812770844. <a href="9789812770844" target="_blank">9789812770844</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>Wiggins, J. B. (1987). "Option values under stochastic volatility: Theory and empirical estimates" (PDF). Journal of Financial Economics. 19 (2): 351–372. doi:10.1016/0304-405X(87)90009-2. <a href="http://dml.cz/bitstream/handle/10338.dmlcz/135564/Kybernetika_39-2003-6_1.pdf" target="_blank">http://dml.cz/bitstream/handle/10338.dmlcz/135564/Kybernetika_39-2003-6_1.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>Hamilton, J. D. (1989). "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle". Econometrica. 57 (2): 357–384. CiteSeerX 10.1.1.397.3582. doi:10.2307/1912559. JSTOR 1912559. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
<li id="fn:17"><p>Hamilton, James (2008). "Regime-Switching Models". New Palgrave Dictionary of Economics (2nd ed.). Palgrave Macmillan Ltd. ISBN 9780333786765. <a href="9780333786765" target="_blank">9780333786765</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li>
<li id="fn:18"><p>Mandelbrot, Benoit; Fisher, Adlai; Calvet, Laurent (September 1997). "A multifractal model of asset returns". Cowles Foundation Discussion Paper No. 1164. SSRN 78588. <a href="/wiki/SSRN_(identifier)" target="_blank">/wiki/SSRN_(identifier)</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li>
<li id="fn:19"><p>Mandelbrot, B. B. (2006). "Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier". Journal of Fluid Mechanics. 62 (2): 331–358. doi:10.1017/S0022112074000711. S2CID 222375985. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li>
<li id="fn:20"><p>Mandelbrot, Benoit B. (1983). The fractal geometry of nature (Updated and augm. ed.). New York: Freeman. ISBN 9780716711865. <a href="9780716711865" target="_blank">9780716711865</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li>
<li id="fn:21"><p>Mandelbrot, Benoit B.; J.M. Berger; et al. (1999). Multifractals and 1/f noise : wild self-affinity in physics (1963 - 1976) (Repr. ed.). New York, NY [u.a.]: Springer. ISBN 9780387985398. <a href="9780387985398" target="_blank">9780387985398</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li>
</ol>

Markov switching multifractal open-in-new

Markov switching multifractal