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Re: Stochastic calculus in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg34399] Re: [mg34363] Stochastic calculus in Mathematica
  • From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
  • Date: Sat, 18 May 2002 03:51:02 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Presumably Ln  means the natural logarithm? Also, you should never use 
capital I for anything but Sqrt[-1].
If I have understood you correctly the following should be what you are 
looking for:

<< "ItosLemma`"

In[2]:=
{TimeSymbol, TimeIncrement, BrownianIncrement,
    CorrelationSymbol} = {t, dt, dB, rho}

Out[2]=
{t, dt, dB, rho}

In[3]:=
ItoMake[X[t], a*mu1[t], {sigma1, 0}]

Out[3]=
a dt mu1[t] + sigma1 dB
                        1

In[4]:=
ItoMake[Y[t], b*mu2[t], {0, sigma2}]

Out[4]=
b dt mu2[t] + sigma2 dB
                        2

Now assuming you have uncorollated Brownians:

In[5]:=
Simplify[ItoD[(X[t] - Log[X[t]])*(Y[t] - Log[Y[t]])]]

Out[5]=
             2           2
1     sigma2  X   sigma1  Y
- dt (--------- + --------- +
2         2           2
          Y           X

                                            2
      2 b mu2 (-1 + Y) (X - Log[X])   sigma2  Log[X]
      ----------------------------- - -------------- +
                    Y                        2
                                            Y

                                            2
      2 a mu1 (-1 + X) (Y - Log[Y])   sigma1  Log[Y]
      ----------------------------- - --------------) +
                    X                        2
                                            X

   sigma1 (-1 + X) (Y - Log[Y]) dB
                                  1
   -------------------------------- +
                  X

   sigma2 (-1 + Y) (X - Log[X]) dB
                                  2
   --------------------------------
                  Y

with corollated ones:

In[6]:=
Simplify[ItoD[(X[t] - Log[X[t]])*(Y[t] - Log[Y[t]]),
    OrthogonalBrownians -> False]]

Out[6]=
    1                          2
------- (2 sigma1 (-1 + X) X Y  (Y - Log[Y]) dB  +
    2  2                                        1
2 X  Y

               2
     2 sigma2 X  (-1 + Y) Y (X - Log[X]) dB  +
                                           2

               2  3            3              3  2
     dt (sigma2  X  - 2 b mu2 X  Y + 2 b mu2 X  Y  +

              2  3              3            2  3
        sigma1  Y  - 2 a mu1 X Y  + 2 a mu1 X  Y  -

         2        2
        X  (sigma2  + 2 b mu2 (-1 + Y) Y) Log[X] -

              2  2                     2
        sigma1  Y  Log[Y] + 2 a mu1 X Y  Log[Y] -

                 2  2
        2 a mu1 X  Y  Log[Y] +

        2 sigma1 sigma2 (-1 + X) X (-1 + Y) Y rho   ))
                                                 1,2
Andrzej Kozlowski
Toyama International University
JAPA


On Friday, May 17, 2002, at 07:30  PM, Narve wrote:

> Hi!
>
> I have two stochastic processes defined
>
> as
>
> dI = a*(mu1(t)-Ln(I))+sigma1*dZ1
> dP=b*(mu2(t)-Ln(P))+sigma2*dZ2
>
> where mu1(t) denotes a deterministic function of time, a and b are
> constants, sigma1 and sigma2 denotes two constant standard deviations
> and dZ1 and dZ2 are two brownian motions. Thus, I have two
> mean-reverting processes of the Ln-values of two variables.
>
> The problem is that I want to multiply the two processes (one is a
> volume process, the other a price process) and compute the stochastic
> derivative of the resulting expression. I got the ItosLemma notebook off
> the web, but cannot figure out how (if) this can be done. Anyone ?
>
> Cheers,
> Narve
>
>
>
>
N
http://platon.c.u-tokyo.ac.jp/andrzej/



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