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/