Economic Foundations of Energy Investments - Green Energy and Efficiency: An Economic Perspective - page 87

EðS t Þ ¼ S 0 e k S ðtt 0 Þ þ S m ð e k S ðtt 0 Þ Þ ; 1 ð 52 Þ

which amounts to

EðS tþ D t Þ ¼ S t e k S D t e k S D t þ S m ð 1 Þ : ð 53 Þ

Also, the conditional variance is

r S
2 k S ½

e 2 k S ðtt 0 Þ : Var ðS t Þ ¼ 1 ð 54 Þ

Since both mean and variance remain nite as t !1 , this process is stationary.

Equation ( 11 ) is the continuous-time version of a rst-order autoregressive

process AR ( 1 ) in discrete time:

e k D t ÞþS t e k D t S tþ D t ¼ S m ð 1 þ e tþ D t ¼ a þ bS t þ e tþ D t ; ð 55 Þ

where e t : Nð 0

; r e Þ

, and the following notation holds:

a

a S m ð 1 bÞ)S m ¼ ð 56 Þ

b ;

1

) k ¼ ln b

b e kDt

ð 57 Þ

D t :

Also

r 2
2 k ½

2

e 2 k D t

ðr e Þ ¼ 1 ) ð 58 Þ

2 2

2 kðr e Þ

ðr e Þ

ln b 2

) r 2

¼

e 2 kDt ¼

:

1

D t ½ b 2

1

58 ) enable the continuous-time process parameters ( k , S m , r )to
be recovered on estimating the regression coef

Equations ( 56 -

cients ( a , b ) and the standard

deviation of the regression residuals ( r e ).

References

1. Abadie LM, Chamorro JM (2013) Investment in energy assets under uncertainty. Springer,
London
2. Abadie LM, Ortiz RA, Galarraga I (2012) Determinants of energy ef ciency investments in
the U.S. Energy Policy 45:551 566

3. Abadie LM (2009) Valuation of long-term investments in energy assets under uncertainty.
Energies 2(3):738 - 768

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