Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 33

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