Page 75 - 068

61

where y is assumed to be a nonnegative random variable drawn from lognormal distribution
t
 
which is ln y having independent and identical normal distribution with mean  and
t
2
variance  . Then, Merton introduced the new equation for describing the behavior of asset
price with jump which is driven by the Brownian motion B , the Poisson process N with
t t
constants drift  and volatility  in Eq. (3).

To receive the models of asset price from both cases, we need the following proposition

(Cont and Tankov, 2004).

Proposition 3.1. (Itô’s formula for jump diffusion processes)

Let X be a diffusion process with jumps, defined as the sum of a drift term, a Brownian

stochastic integral and a compound Poisson process

t t N t
s 
0 
s 
X  X  a ds  b dB   X
t s i
0 0 i 1
where a and b are continuous non anticipating processes with
t t
 T 
E  bdt   t   .
 0   


Then, for any C function, f  : 0,T      , the process Y  f  ,t X can be
1,2
t
t
represented as



s  
 ,f t X  f  0,X   t   f   ,s X   f   ,s X a ds
t 0 s  s  X s
0  
1  t 2 f   t  f  
  
    ,s X b ds    2   ,s X b dB
2 0   X 2 s s  0   X s s  s


  
    f X    X  f X i    . (a)
i
 1,i T i  t   T i T 
Proof. [ See Cont and Tankov, 2004]