e
are assuming that the price of some underlying asset is given
by
and we are considering a derivative
that may jump in value depending on some Cox process
 The
drift
 ,
volatility
 ,
intensity of the Cox process
and the size of the jump
![$\left[ V\right] $](graphics/notesBK__0__2044.gif)
are all functions of
 .
So this is a Markovian situation with the
being the state.
We form a
portfolio
In reallity we would, of course, sell the derivative
and hedge it with the underlying
and more liquid derivative
 :
However, the result would be the same and our unnatural way keeps expressions
symmetrical.
We calculate the
differential
and choose the
and
to remove both sources of
randomness:
We solve for the
   
For such
we
have  
Note that the expression on the LHS only depends on the function
while the expression on the RHS only depends on
 .
The functions
and
are not related. Therefore, both parts of the equation do not, in fact, depend
on
or
 .
We conclude that there is a function
depending on the state, such
that
The probabilistic meaning of the above equation is provided in the section
( Backward_equation_for_jump_diffusion).
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