Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Stochastic Integration
The function can be extended with integration rules that allow explicit calculation of many Ito integrals.
Ito stochastic differential of semimartingale expression f | |
Ito integral of stochastic differential expression sd | |
reset all structures | |
create independent Brownian motions in list names init | |
introduce semimartingale X X 0 dX==sd | |
computes the drift differential of expression sd | |
computes the value of the expression f t 0 |
Functions of Itovsn3 paclet that are useful for stochastic integration.
Load package
In[1]:=
Needs["FernandoDuarte`Itovsn3`"]
Set up two Brownian motions and a semimartingale that depends on them:
Z
In[2]:=
 | ItoReset |
| BrownBasis |
| Itosde |
In[5]:=
 | ItoD |
| ItoIntegral |
Out[5]=
dXY
In[6]:=
| ItoIntegral |
Out[6]=
-1+X
In[7]:=
| ItoD |
 | ItoIntegral |
 | ItoD |
2
X
2
X
Out[7]=
5dt+2dX+dtItoIntegral[dt+2dXX]+2dXXItoIntegral[dt+2dXX]
2
X
3
X
In[8]:=
| Drift |
| ItoD |
| ItoIntegral |
 | ItoD |
2
X
2
X
Out[8]=
dt(5+ItoIntegral[dt+2dXX])
2
X
can be programmed with further Ito integral properties. The identity:
d()=(n+1)dX+d()dX
n+1
X
n
X
n+1
2
n
X
leads to:
(n+1)∫dX=-Subscript[,0]-∫d()dX
n
X
n+1
X
n+1
X
n+1
2
n
X
We implement this identity as follows:
In[9]:=
| ItoIntegral |
| ItoD |
1
2
2
X
| InitialValue |
2
X
| ItoIntegral |
 | ItoExpand |
2
dX
In[10]:=
| ItoIntegral |
n_Integer
X_
| ItoD |
1
n+1
n+1
X
| InitialValue |
n+1
X
1
2
| ItoIntegral |
n-1
X
| ItoExpand |
2
dX
We can now solve some Ito integrals:
In[11]:=
| ItoIntegral |
Out[11]=
1
2
2
X
In[12]:=
| ItoD |
| ItoIntegral |
Out[12]=
dXX
In[13]:=
| ItoIntegral |
3
X
Out[13]=
1
4
4
X
2
X
In[14]:=
| ItoD |
| ItoIntegral |
3
X
Out[14]=
dX
3
X
We now call on the well-known relationship between Ito differentials and Hermite polynomials:
In[15]:=
H[n_,x_,t_]:=HermiteHn,
x
2t
n
2t
In[16]:=
Simplify
[H[n+1,X,t]]/.n{0,1,2,3,4,5,6}
| ItoD |
2(n+1)H[n,X,t]
Out[16]=
{dX,dX,dX,dX,dX,dX,dX}
We need some linearity for the Ito integrals for the next part:
In[17]:=
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
| ItoIntegral |
sda_
n_Integer
| ItoIntegral |
n
For and , we have:
n=0
n=1
In[23]:=
0Simplify
[Expand[H[0,X,t]dX]]-
| ItoIntegral |
H[1,X,t]-
[0,Simplify[H[1,X,t]]]
| InitialValue |
2
Out[23]=
True
In[24]:=
0Simplify
[Expand[H[1,X,t]dX]]-
| ItoIntegral |
H[2,X,t]-
[0,Simplify[H[2,X,t]]]
| InitialValue |
4
Out[24]=
True
For , our rules are not enough to compute the stochastic integral to :
n=2
0
In[25]:=
0Simplify
[Expand[H[2,X,t]dX]]-
| ItoIntegral |
H[3,X,t]-
[0,Simplify[H[3,X,t]]]
| InitialValue |
6
Out[25]=
04tX-4ItoIntegral[dXt]-4ItoIntegral[dtX]
We can compute these integrals explicitly adding some new rules. The expression:
In[26]:=
SetAttributes[p,Constant];SetAttributes[q,Constant];
[]
| ItoD |
p
t
q
X
Out[28]=
-dtq+dt+dXq+dtp
1
2
p
t
-2+q
X
1
2
2
q
p
t
-2+q
X
p
t
-1+q
X
-1+p
t
q
X
leads to:
d()==mdt+(n+1)dX+(n+1)ndt
m
t
n+1
X
m-1
t
n+1
X
m
t
n
X
1
2
m
t
n-1
X
from which we derive the following rules:
In[29]:=
| ItoIntegral |
| InitialValue |
| ItoIntegral |
| ItoD |
| ItoIntegral |
1
2
2
X
| InitialValue |
2
X
| ItoIntegral |
2
X
| ItoIntegral |
| ItoIntegral |
m_Integer
t
m
t
| InitialValue |
m
t
| ItoIntegral |
| ItoD |
m
t
| ItoIntegral |
m_Integer
t
1
2
m
t
2
X
| InitialValue |
m
t
2
X
| ItoIntegral |
m-1
t
2
X
| ItoIntegral |
m
t
| ItoIntegral |
n_Integer
X
1
n+1
n+1
X
| InitialValue |
n+1
X
| ItoIntegral |
n+1
X
| ItoIntegral |
1
2
n-1
X
| ItoIntegral |
m_Integer
t
n_Integer
X
1
n+1
m
t
n+1
X
| InitialValue |
m
t
n+1
X
| ItoIntegral |
m-1
t
n+1
X
| ItoIntegral |
1
2
m
t
n-1
X
Now we can compute the integral with :
n=2
In[35]:=
0Simplify
[Expand[H[2,X,t]dX]]-
| ItoIntegral |
H[3,X,t]-
[0,Simplify[H[3,X,t]]]
| InitialValue |
6
Out[35]=
True
And, indeed, for any other :
n
In[36]:=
Table0Simplify
[Expand[H[n,X,t]dX]]-,{n,3,10}
| ItoIntegral |
H[n+1,X,t]-
[0,Simplify[H[n+1,X,t]]]
| InitialValue |
2(n+1)
Out[36]=
{True,True,True,True,True,True,True,True}
Finally, a couple of random checks:
In[37]:=
| ItoIntegral |
5
t