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and the value
φ
1 after the jump needs to be
φ 1
φ
d
φ ) =
h
,
(2.60)
g (
φ
0
where h is the size of the jump [the contribution of the first term on the right-hand side
of Eq. ( 2.57 ) is negligible in the time frame in which a jump occurs]. If the left-hand
side of Eq. ( 2.60 ) can be analytically integrated, the size of the jump in the trajectory
of
φ 1 φ 0 , is directly found; otherwise the solution should be found numerically.
We can obtain an interesting representation of
φ
,
φ 1 φ 0 by considering the trans-
= φ
φ )d
φ ,using( 2.60 ) to express
z 1 [ z (
formed variable z (
φ
)
1
/
g (
φ 1 as
φ 1 =
φ 0 )
+
h ] (where the superscript
1 represents the inverse function) and expanding
φ 1 in a
Taylor series around h
=
0. We find
1
i ! g ( i ) (
φ 0 ) h i
φ 1 = φ 0 +
,
(2.61)
i
=
1
φ = φ 0 . An analogous representa-
tion is obtained by Caddemi and Di Paola ( 1996 )and Pirrotta ( 2005 ) following a
different reasoning. The first three terms of the expansion are
φ 0 ) d g ( i ) ( φ )
where g (1) (
φ 0 )
= g (
φ 0 )and g ( i + 1) (
φ 0 )
= g (
d
φ
6 g (
φ 0 ) h 3
1
1
2 g (
φ 0 ) h 2
φ 0 ) g 2 (
[ g (
φ 0 )] 2 g (
.
(2.62)
We recognize that the first term corresponds to Ito's rule and the higher-order terms
in general do not vanish.
Consider now the rule frequently used for the Stratonovich convention to calculate
φ 1 φ 0 =
g (
φ 0 ) h
+
φ 0 ) g (
+
φ 0 )
+
φ
in g (
φ
) as half the sum of the values before and after the jump:
φ 1 φ 0 = g φ
h .
+ φ
0
1
(2.63)
2
Setting
φ 1 = φ 0 +
g [(
φ 0 + φ 1 )
/
2] h on the right-hand side of Eq. ( 2.63 ) and expand-
] in a Taylor series in h until O ( h 3 ), we obtain
ing g [
·
0 ] g φ 0 + φ 1
2
h 2
0 ] g 2 φ 0 + φ 1
2
h 3
1
1
2 g [
8 g [
φ
φ
=
g [
φ
0 ] h
+
φ
+
φ
.
(2.64)
1
0
We repeat the substitution and the expansion with the second and third terms on the
right-hand side of Eq. ( 2.64 )byusing( 2.61 ); we neglect the terms of orders higher
than 3 to obtain
1
0 ) h 3
1
1
2 g (
0 ) h 2
8 g (
0 ) g 2 (
4 [ g (
0 )] 2 g (
,
(2.65)
which is clearly different from Eq. ( 2.62 ) for the terms of orders higher than 2 in h .
As a consequence, convention ( 2.63 ) is in general inconsistent with relation ( 2.61 );
φ
φ
=
g ( x 0 ) h
+
φ
0 ) g (
φ
+
φ
φ
0 )
+
φ
φ
1
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