Table of Ito Integrals












3














Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of?



Did a search but didn't come up with anything and was wondering if anyone knew of anything of the like...



Thanks.






stochastic-calculus stochastic-integrals







share|cite|improve this question






















  • I have Oksendahl and Klebaner and obviously they both have all of them spread throughout the book but neither has a quick reference table of common stochastic integrals. If you happen to know of where such a thing exists I'd be happy to get it. That's why I am asking.
    – Dirk Calloway
    Dec 6 '12 at 21:04






  • 1




    You should alsolook for symbolic packages for this! I think there is one for Maple.
    – kjetil b halvorsen
    Dec 7 '12 at 18:42
















3














Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of?



Did a search but didn't come up with anything and was wondering if anyone knew of anything of the like...



Thanks.






stochastic-calculus stochastic-integrals







share|cite|improve this question






















  • I have Oksendahl and Klebaner and obviously they both have all of them spread throughout the book but neither has a quick reference table of common stochastic integrals. If you happen to know of where such a thing exists I'd be happy to get it. That's why I am asking.
    – Dirk Calloway
    Dec 6 '12 at 21:04






  • 1




    You should alsolook for symbolic packages for this! I think there is one for Maple.
    – kjetil b halvorsen
    Dec 7 '12 at 18:42














3












3








3


2





Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of?



Did a search but didn't come up with anything and was wondering if anyone knew of anything of the like...



Thanks.






stochastic-calculus stochastic-integrals







share|cite|improve this question













Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of?



Did a search but didn't come up with anything and was wondering if anyone knew of anything of the like...



Thanks.






stochastic-calculus stochastic-integrals




stochastic-calculus stochastic-integrals






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 6 '12 at 13:40









Dirk Calloway

1721111




1721111












  • I have Oksendahl and Klebaner and obviously they both have all of them spread throughout the book but neither has a quick reference table of common stochastic integrals. If you happen to know of where such a thing exists I'd be happy to get it. That's why I am asking.
    – Dirk Calloway
    Dec 6 '12 at 21:04






  • 1




    You should alsolook for symbolic packages for this! I think there is one for Maple.
    – kjetil b halvorsen
    Dec 7 '12 at 18:42


















  • I have Oksendahl and Klebaner and obviously they both have all of them spread throughout the book but neither has a quick reference table of common stochastic integrals. If you happen to know of where such a thing exists I'd be happy to get it. That's why I am asking.
    – Dirk Calloway
    Dec 6 '12 at 21:04






  • 1




    You should alsolook for symbolic packages for this! I think there is one for Maple.
    – kjetil b halvorsen
    Dec 7 '12 at 18:42
















I have Oksendahl and Klebaner and obviously they both have all of them spread throughout the book but neither has a quick reference table of common stochastic integrals. If you happen to know of where such a thing exists I'd be happy to get it. That's why I am asking.
– Dirk Calloway
Dec 6 '12 at 21:04




I have Oksendahl and Klebaner and obviously they both have all of them spread throughout the book but neither has a quick reference table of common stochastic integrals. If you happen to know of where such a thing exists I'd be happy to get it. That's why I am asking.
– Dirk Calloway
Dec 6 '12 at 21:04




1




1




You should alsolook for symbolic packages for this! I think there is one for Maple.
– kjetil b halvorsen
Dec 7 '12 at 18:42




You should alsolook for symbolic packages for this! I think there is one for Maple.
– kjetil b halvorsen
Dec 7 '12 at 18:42










2 Answers
2






active

oldest

votes


















2














This actually isn't a bad start anyway if anyone is looking for the same...



Stochastic Calculus Cheat Sheet






share|cite|improve this answer





























    0














    $newcommand{d}{mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - tfrac{1}{2}t}$:



    begin{array} {|r|r|}
    hline
    X_t & d X_t = u d t + v d B_t\ hline
    B_t & d B_t\
    B_t^2 & 2 B_t d B_t + d t\
    B_t^2 - t & 2 B_t d B_t\
    B_t^3 & 3 B_t^2 d B_t + 3 B_t d t\
    e^{B_t} & e^{B_t}d B_t + tfrac{1}{2}e^{B_t}d t \
    e^{B_t - tfrac{1}{2}t} & e^{B_t - tfrac{1}{2}t} d B_t\
    e^{tfrac{1}{2}t}sin B_t & e^{tfrac{1}{2}t} cos B_t d B_t\
    e^{tfrac{1}{2}t}cos B_t & -e^{tfrac{1}{2}t} sin B_t d B_t\
    (B_t + t) e^{-B_t - tfrac{1}{2}t} & (1 - B_t - t) e^{-B_t - tfrac{1}{2}t} d B_t\hline
    end{array}



    All the above can be verified using Itô's formula and alongside the Itô isometry and integration by parts can be used to evaluate several stochastic integrals, their expectations and variances.



    Here is a table of stochastic integrals...



    begin{array} {|r|r|r|}
    hline
    text{Stochastic Integral} & text{Result} & text{Variance}\ hline
    int_0^t d B_s & B_t & t \
    int_0^t s d B_s & tB_t - int_0^t B_s d s & tfrac{1}{3}t^3 \
    int_0^t B_s d B_s & tfrac{1}{2}B_t^2 - tfrac{1}{2}t & tfrac{1}{2}t^2\
    int_0^t B_s^2 d B_s & tfrac{1}{3}B_t^3 - int_0^t B_s d s& 3t^2\
    int_0^t e^{B_s - tfrac{1}{2}s}d B_s
    & e^{B_t - tfrac{1}{2}t} - 1
    & e^{t}-1\
    hline
    end{array}






    share|cite|improve this answer























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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2














      This actually isn't a bad start anyway if anyone is looking for the same...



      Stochastic Calculus Cheat Sheet






      share|cite|improve this answer


























        2














        This actually isn't a bad start anyway if anyone is looking for the same...



        Stochastic Calculus Cheat Sheet






        share|cite|improve this answer
























          2












          2








          2






          This actually isn't a bad start anyway if anyone is looking for the same...



          Stochastic Calculus Cheat Sheet






          share|cite|improve this answer












          This actually isn't a bad start anyway if anyone is looking for the same...



          Stochastic Calculus Cheat Sheet







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 7 '12 at 18:14









          Dirk Calloway

          1721111




          1721111























              0














              $newcommand{d}{mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - tfrac{1}{2}t}$:



              begin{array} {|r|r|}
              hline
              X_t & d X_t = u d t + v d B_t\ hline
              B_t & d B_t\
              B_t^2 & 2 B_t d B_t + d t\
              B_t^2 - t & 2 B_t d B_t\
              B_t^3 & 3 B_t^2 d B_t + 3 B_t d t\
              e^{B_t} & e^{B_t}d B_t + tfrac{1}{2}e^{B_t}d t \
              e^{B_t - tfrac{1}{2}t} & e^{B_t - tfrac{1}{2}t} d B_t\
              e^{tfrac{1}{2}t}sin B_t & e^{tfrac{1}{2}t} cos B_t d B_t\
              e^{tfrac{1}{2}t}cos B_t & -e^{tfrac{1}{2}t} sin B_t d B_t\
              (B_t + t) e^{-B_t - tfrac{1}{2}t} & (1 - B_t - t) e^{-B_t - tfrac{1}{2}t} d B_t\hline
              end{array}



              All the above can be verified using Itô's formula and alongside the Itô isometry and integration by parts can be used to evaluate several stochastic integrals, their expectations and variances.



              Here is a table of stochastic integrals...



              begin{array} {|r|r|r|}
              hline
              text{Stochastic Integral} & text{Result} & text{Variance}\ hline
              int_0^t d B_s & B_t & t \
              int_0^t s d B_s & tB_t - int_0^t B_s d s & tfrac{1}{3}t^3 \
              int_0^t B_s d B_s & tfrac{1}{2}B_t^2 - tfrac{1}{2}t & tfrac{1}{2}t^2\
              int_0^t B_s^2 d B_s & tfrac{1}{3}B_t^3 - int_0^t B_s d s& 3t^2\
              int_0^t e^{B_s - tfrac{1}{2}s}d B_s
              & e^{B_t - tfrac{1}{2}t} - 1
              & e^{t}-1\
              hline
              end{array}






              share|cite|improve this answer




























                0














                $newcommand{d}{mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - tfrac{1}{2}t}$:



                begin{array} {|r|r|}
                hline
                X_t & d X_t = u d t + v d B_t\ hline
                B_t & d B_t\
                B_t^2 & 2 B_t d B_t + d t\
                B_t^2 - t & 2 B_t d B_t\
                B_t^3 & 3 B_t^2 d B_t + 3 B_t d t\
                e^{B_t} & e^{B_t}d B_t + tfrac{1}{2}e^{B_t}d t \
                e^{B_t - tfrac{1}{2}t} & e^{B_t - tfrac{1}{2}t} d B_t\
                e^{tfrac{1}{2}t}sin B_t & e^{tfrac{1}{2}t} cos B_t d B_t\
                e^{tfrac{1}{2}t}cos B_t & -e^{tfrac{1}{2}t} sin B_t d B_t\
                (B_t + t) e^{-B_t - tfrac{1}{2}t} & (1 - B_t - t) e^{-B_t - tfrac{1}{2}t} d B_t\hline
                end{array}



                All the above can be verified using Itô's formula and alongside the Itô isometry and integration by parts can be used to evaluate several stochastic integrals, their expectations and variances.



                Here is a table of stochastic integrals...



                begin{array} {|r|r|r|}
                hline
                text{Stochastic Integral} & text{Result} & text{Variance}\ hline
                int_0^t d B_s & B_t & t \
                int_0^t s d B_s & tB_t - int_0^t B_s d s & tfrac{1}{3}t^3 \
                int_0^t B_s d B_s & tfrac{1}{2}B_t^2 - tfrac{1}{2}t & tfrac{1}{2}t^2\
                int_0^t B_s^2 d B_s & tfrac{1}{3}B_t^3 - int_0^t B_s d s& 3t^2\
                int_0^t e^{B_s - tfrac{1}{2}s}d B_s
                & e^{B_t - tfrac{1}{2}t} - 1
                & e^{t}-1\
                hline
                end{array}






                share|cite|improve this answer


























                  0












                  0








                  0






                  $newcommand{d}{mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - tfrac{1}{2}t}$:



                  begin{array} {|r|r|}
                  hline
                  X_t & d X_t = u d t + v d B_t\ hline
                  B_t & d B_t\
                  B_t^2 & 2 B_t d B_t + d t\
                  B_t^2 - t & 2 B_t d B_t\
                  B_t^3 & 3 B_t^2 d B_t + 3 B_t d t\
                  e^{B_t} & e^{B_t}d B_t + tfrac{1}{2}e^{B_t}d t \
                  e^{B_t - tfrac{1}{2}t} & e^{B_t - tfrac{1}{2}t} d B_t\
                  e^{tfrac{1}{2}t}sin B_t & e^{tfrac{1}{2}t} cos B_t d B_t\
                  e^{tfrac{1}{2}t}cos B_t & -e^{tfrac{1}{2}t} sin B_t d B_t\
                  (B_t + t) e^{-B_t - tfrac{1}{2}t} & (1 - B_t - t) e^{-B_t - tfrac{1}{2}t} d B_t\hline
                  end{array}



                  All the above can be verified using Itô's formula and alongside the Itô isometry and integration by parts can be used to evaluate several stochastic integrals, their expectations and variances.



                  Here is a table of stochastic integrals...



                  begin{array} {|r|r|r|}
                  hline
                  text{Stochastic Integral} & text{Result} & text{Variance}\ hline
                  int_0^t d B_s & B_t & t \
                  int_0^t s d B_s & tB_t - int_0^t B_s d s & tfrac{1}{3}t^3 \
                  int_0^t B_s d B_s & tfrac{1}{2}B_t^2 - tfrac{1}{2}t & tfrac{1}{2}t^2\
                  int_0^t B_s^2 d B_s & tfrac{1}{3}B_t^3 - int_0^t B_s d s& 3t^2\
                  int_0^t e^{B_s - tfrac{1}{2}s}d B_s
                  & e^{B_t - tfrac{1}{2}t} - 1
                  & e^{t}-1\
                  hline
                  end{array}






                  share|cite|improve this answer














                  $newcommand{d}{mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - tfrac{1}{2}t}$:



                  begin{array} {|r|r|}
                  hline
                  X_t & d X_t = u d t + v d B_t\ hline
                  B_t & d B_t\
                  B_t^2 & 2 B_t d B_t + d t\
                  B_t^2 - t & 2 B_t d B_t\
                  B_t^3 & 3 B_t^2 d B_t + 3 B_t d t\
                  e^{B_t} & e^{B_t}d B_t + tfrac{1}{2}e^{B_t}d t \
                  e^{B_t - tfrac{1}{2}t} & e^{B_t - tfrac{1}{2}t} d B_t\
                  e^{tfrac{1}{2}t}sin B_t & e^{tfrac{1}{2}t} cos B_t d B_t\
                  e^{tfrac{1}{2}t}cos B_t & -e^{tfrac{1}{2}t} sin B_t d B_t\
                  (B_t + t) e^{-B_t - tfrac{1}{2}t} & (1 - B_t - t) e^{-B_t - tfrac{1}{2}t} d B_t\hline
                  end{array}



                  All the above can be verified using Itô's formula and alongside the Itô isometry and integration by parts can be used to evaluate several stochastic integrals, their expectations and variances.



                  Here is a table of stochastic integrals...



                  begin{array} {|r|r|r|}
                  hline
                  text{Stochastic Integral} & text{Result} & text{Variance}\ hline
                  int_0^t d B_s & B_t & t \
                  int_0^t s d B_s & tB_t - int_0^t B_s d s & tfrac{1}{3}t^3 \
                  int_0^t B_s d B_s & tfrac{1}{2}B_t^2 - tfrac{1}{2}t & tfrac{1}{2}t^2\
                  int_0^t B_s^2 d B_s & tfrac{1}{3}B_t^3 - int_0^t B_s d s& 3t^2\
                  int_0^t e^{B_s - tfrac{1}{2}s}d B_s
                  & e^{B_t - tfrac{1}{2}t} - 1
                  & e^{t}-1\
                  hline
                  end{array}







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 9 hours ago

























                  answered 2 days ago









                  Pantelis Sopasakis

                  2,007832




                  2,007832






























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