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where  is again the drift,  is again the volatility, N is the counting process which is a
t
N t
Poisson process with intensity . The  Y represents the jump part of the process.
i
i 1
Note that   means that there are no any jumps occur in the process and the
0
solution in Eq. (4) is reduced to be the same as the solution of Eq. (2)

   2 
S  S exp   t    B  (5)
t 0  2   t 
   
Eq. (5) is known as the Geometric Brownian Motion (GBM) and wildly used in financial

mathematics.

4 Simulations

In order to discretize eq. (4) and (5), assume that we have a fixed set of date

0  t  t  t    ...  t  t  t   T with step time t . Let us first consider the
t
0 1 0 n n 1
continuous model (5) with constants  and  . For the today price at t  , the discretization
t
i
version of (5) is

    2  
S S exp     t    B 
t i 0   2   i  t i  

and for the price of the next day t  t   is
t
i
    2  
S  S exp      t    t    B  .
t  t 0   2    i t  t  
i
i
Then, the return can be obtained by

    2  
S exp      t    t    B 
S 0  2   i t  t  
i
t  t  .
i
S     2  
t i S exp     t    B 
0 i t
 2   i  
Then we have

    2  
 
S S exp      t    B  (6)
t  t t i   2    t  t  
i
i