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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

0

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