✺✔✻✼✺✾✽☞✿❁❀❃❂❅❄❇❆✦✽
✽✒❂❅❈❉❈❉❊✴❄❇❆✦✽☞❋✾✿❁❆✦❂●❄■❍
❏▲❑◆▼
❀❃❂❅❖✑❋❃❖✑❆✟✻P❆◗✿❙❘
❚ ❯❲❱❨❳❬❩ ❱❭❯ ❱❫❪❴❪❛❵❝❜✫❚❞❳❞❡❢❱❭❯ ❳❞❡✹❣❤❩❞✐ ❱❭❥ ✐❅❳✫❚❞❳❞❡❢❱❴❯❦❚❫❧♥♠
❜❬❧♦❱❫❪❛❩❞✐✴✐✒❩❬✐
♣rq✔s✉t✇✈①q❉②④③❙⑤✹⑥⑦③✉⑧⑨q
⑩✑✄❷❶❸✣✶✙✲✞✭✎❅✄◗✕❹✞✚✌✇❺✾❻■✣✏✞❽❼❸✄◗✎✒✣✶✞✭✓▲✆✟❾✝❿➁➀✾✜✤✌☞➂✉➃✥✣✢➄✥✄✟✎✑✓❙➅❢✕➆✓✖➇➈✄◗✙✝❾✲✓✖✞✭➉➈❿
➊⑨➋✲➌✡➍➏➎✢➐✢➎✥➎✒➀✾✜✝✌➈❿⑦➊✍✓✖✕➆✁☎✣✏✕✹➄
➑◗➒❅➓✢➔☎→↔➣➆↕➆➙❸➛✥➜❸➝➞➛✢➟✩➠➏➡➞➠✶➢❸➛P➤✡➥➆➦
➧➨q①q❢⑧①③➫➩❸q①⑥➭♣rt✇✈➲➯P✈
⑩✑✄❷❶❸✣✶✙✲✞✭✎❅✄◗✕❹✞✚✌✇❺✾❻■✣✏✞❽❼❸✄◗✎✒✣✶✞✭✓▲✆✟❾✝❿➁➀✾✜✤✌☞➂✉➃✥✣✢➄✥✄✟✎✑✓❙➅❢✕➆✓✖➇➈✄◗✙✝❾✲✓✖✞✭➉➈❿
➊⑨➋✲➌✡➍➏➎✢➐✢➎✥➎✒➀✾✜✝✌➈❿⑦➊✍✓✖✕➆✁☎✣✏✕✹➄
➑◗➒❅➓✢➔☎→↔➣➨➳➏➵✹➸✶➠✩➝➺↕➭➦✦➻❹➡➞➠✶➢❸➛P➤✡➥➆➦
➼➾➽ ✜✟✎✑✓✖✞↔✞❷✄✝➄r✸➺✬✑✈➲➚✏➪
➑◗➒④➶➾➑◗➹
✬✢✮✢✮✶✵➆✱➘✣✢✆✤✆✝✄✡❶➾✞✡✄✤➄❃✓✖✕☞➴P✕➾✣✏✁✢❺✟✌✶✙✲✎➷✸✦➬✒q➲➮
➹✤➔➱→ ✬✢✮✢✮✥✯
q①♣✃➩✘✬✶✮✢✮✥✮✑➩➏❐
➶✩❒✇➑✦❮✟❰①❮◗→➱➓✥Ï✛Ï✛➔☎Ð➆❮◗➓✶❰✛➔ ➚✢Ñ
➣⑦Ò ✮✥Ó✑✸✦✮❹✱
Ò
✮✢Ô➏✯✢✯✩✱
Ò
✮✥Ô
Ò
✮
②
➑◗Õ➏Ö
➚
➹✤×❹Ï◗➣⑨Ø✼Õ✩❮✟→➱➔➱❮✦➓✶→➱→☎Õ④ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ
➓✢➹✛Õ
➮
➹
➚
❮◗➑✦Ï✤ÏÙ➑➺Ï ✱✶Ú
➔ÜÛ
❐
Ï✛➔ ➚✥Ñ✒➮
➹
➚
❮✟➑✦Ï✤Ï✛➑✦Ï ✱✶②
➹✤➑◗➔ ÑÞÝ
Ï❢❰✤ß❸➑
➚
➹✛Õ ➚✢à
ÏÙ❰✛➹✤➔ Ñ❸á
Ï
✱✶➧
➓✶→➱➒
➮
➹
➚
➶❹➓✢➶❸➔➱→☎➔☎❰✡Õ
➂✾✜✟❾✝✞↔✙✛✣✢✆◗✞
Ø
➚✢Ñ
ÏÙ➔â×✩➑◗➹✉➓✑➹✛➑✦➓✢→
➪
➓✢→
❐
➑➺×ãÏ✇❰✝➓✏❰✤➔ ➚✥Ñ
➓✶➹✤Õ
➮
➹
➚
❮◗➑✦Ï✛Ï
X
=
{
Xs
:
s
∈
R}
.
ä➚
➹➲➓④Ð❸å✩➑✦×
t
∈
R
➓Ñ
×ã➓✒ÏÙ➑◗❰
D
➔Ñ
❰✛ß❹➑æÏÙ❰✤➓✶❰✛➑æÏ
➮
➓✢❮✟➑ ➚✶à
X
✱ →➱➑✟❰gt
➓Ñ
×
dt
×✩➑ Ñ❸➚❰✤➑①❰✛ß❸➑✾ÏÙ❰✤➓✶➹✛❰✛➔ Ñ❸á
➓
Ñ
×❅❰✛ß❸➑æ➑
Ñ
×✩➔ Ñ❸á
❰✤➔☎➒✒➑
✱
➹✤➑✦Ï
➮
➑➺❮✲❰✤➔
➪
➑✦→☎Õ ✱➞➚✶à
➓
Ñ
➑✟å❸❮
❐
➹✤Ï✛➔ ➚✢Ñ✒à
➹
➚
➒ç➓
Ñ
×✒❰
➚
D
✪Ï✇❰✤➹✤➓✥×❸×✩→➱➔ Ñ❸á
t
✰✟è⑦t✡Ñ ❰✤➹➚
×
❐
❮✟➑✔➓✶→âÏ
➚
❰✛ß❸➑
➚
❮✦❮ ❐❸➮
➓✏❰✛➔ ➚✢Ñ
❰✛➔➱➒✒➑✦Ï
I
t
+
➓
Ñ
×
I
t
−
➓✶➶ ➚✏➪
➑
➓
Ñ
×✃➶➾➑◗→
➚
Ö
✱
➹✤➑✦Ï
➮
➑✦❮✟❰✛➔
➪
➑◗→➱Õ
✱
❰✤ß❸➑
➚
➶❹ÏÙ➑✦➹
➪
➑➺×■→☎➑
➪
➑◗→P➓✏❰✾❰✤➔➱➒✑➑
t
×❐
➹✤➔ Ñ❸á
Ï
❐
❮✝ß✫➓
Ñ
➑✟å❸❮
❐
➹✝Ï✛➔ ➚✥ÑÞè✒t✡Ñ
❰✤ß❸➔➱Ï Ñ❸➚
❰✤➑
Ö✼➑æÏ✛ß
➚
Ö◆❰✤ß❹➓✏❰✂❰✛ß❸➑
➮
➓✢➔☎➹✝Ï
(I
t
+
, I
t
−
)
➓
Ñ
×
(t
−
gt, dt
−
t)
➓✶➹✤➑①➔â×✩➑Ñ
❰✛➔â❮◗➓✶→➱→➱Õ❅×✩➔âÏÙ❰✛➹✤➔☎➶
❐
❰✛➑✦× è➁é
ß❸➔âÏ✂Ï
➚
➒✒➑✦Ö✉ß❹➓✏❰
❮
❐
➹✛➔ ➚✢❐
Ï
➮
➹
➚✥➮
➑◗➹✛❰✡Õ✾➔➱Ï
✱
➔
Ñ❁à
➓✢❮✟❰
✱
Ï✛➑◗➑
Ñ
❰
➚
➶➾➑✂➓
à
➓✢➔☎➹✤→☎Õ❙Ï✛➔☎➒
➮
→➱➑✂❮ ➚✢Ñ
Ï✛➑✦ê
❐
➑
Ñ
❮✟➑ ➚✢à
❰✛ß❸➑✂ë Ñ❸➚
Ö
Ñ④á
➑
Ñ
➑◗➹✝➓✶→ ❐❸Ñ
➔
à❽➚
➹✛➒
Ï
➚
❒
➚✥❐
➹
Ñ
→â➓➺Ö❛Ö✉ß❸➔â❮✤ß➘➔➱➒
➮
→➱➔☎➑➺Ïæ❰✛ß❹➓✶❰❁❮ ➚✥Ñ
×✩➔Ü❰✤➔ ➚✢Ñ
➓✢→☎→➱Õ ➚✢Ñ
I
t
+
+
I
−
t
=
v
❰✤ß❸➑
➪
➓✢➹✛➔â➓✢➶❸→☎➑
I
t
+
✪
➓
Ñ
×➘➓✢→➱Ï
➚
I
t
−
✰
➔âÏ ❐❹Ñ
➔
à❽➚
➹✛➒✒→➱Õ■×✩➔âÏ✇❰✤➹✛➔➱➶
❐
❰✤➑✦× ➚✢Ñ
(0, v)
è✑ì ➑●➓✶→âÏ➚✴➮
➓✶➹✛❰✛➔â❮
❐
→â➓✶➹✤➔➱í◗➑④❰
➚
❰✛ß❹➑❅Ï✇❰✝➓✏❰✤➔ ➚✥Ñ
➓✶➹✤Õ❇×✩➔☎Û
❐
Ï✛➔ ➚✢Ñ
❮◗➓✥ÏÙ➑✒➓
Ñ
×
Ï✛ß
➚
Ö
✱
➑
èá➆è➱✱
❰✛ß❹➓✶❰✉❰✛ß❸➑❁×✩➔âÏ✇❰✤➹✤➔☎➶
❐
❰✤➔ ➚✢Ñ✴➚✢à
I
t
−
+
I
+
t
➔âÏ✉➓✑➒✒➔☎å➏❰
❐
➹✤➑ ➚✢à⑨á
➓✶➒✒➒❅➓✑×✩➔âÏÙ❰✛➹✤➔☎➶
❐
❰✛➔ ➚✢Ñ
Ï
è
î
❡➾ï☞ð❢ñ⑨ò❅óãô✴õPð❢ö✟ò❁ï
⑥
➑◗❰
X
=
{
Xs
:
s
∈
R}
➶➾➑✾➓✒Ï✇❰✝➓✏❰✤➔➚✥Ñ
➓✶➹✤Õ✒➒✒➑✦➓✢Ï
❐
➹✝➓✶➶❸→➱➑
➮
➹
➚
❮✟➑➺Ï✛Ï✼Ö✉➔☎❰✛ßã❰✤ß❸➑✾➹✝➓ Ñ❸á
➑
E
⊂
R
èä
➚
➹✉➓
á
➔
➪
➑
Ñ
D
⊂
E
→➱➑✟❰M
:=
{
s
∈
R
:
Xs
∈
D
}
cl
,
✪Ù✸➺✰Ö✉ß❸➑✦➹✛➑
cl
➒✒➑✦➓Ñ
Ï✂❰✛ß❸➑❁❮◗→
➚
Ï
❐
➹✛➑ ➚✢à
❰✛ß❸➑❁Ï✛➑✟❰➲➔
Ñ
❰✛ß❸➑❙➶❹➹✤➓✥❮✟➑✦Ï è⑦✈
➑✟å➏❰①×✩➑✟Ð
Ñ
➑
à❽➚
➹✚Ð❸å✩➑✦×
t
∈
R
gt
:= sup
{
s
≤
t
:
s
∈
M
}
,
dt
:= inf
{
s > t
:
s
∈
M
}
,
✪✭✬✥✰➓
Ñ
×
I
t
+
:=
Z
d
t
g
t
1
{X
s
>X
t
}
ds,
I
−
t
:=
Z
d
t
g
t
1
{X
s
<X
t
}
ds.
❰✛➑ ❮◗❮ ➓✏❰✤➔ ❰✛➔➱➒✒➑✦Ï Ï✇❰✝➓✏❰✛➔ ➓✶➹✤Õ ➹ ❮◗➑✦Ï✤ÏÙ➑➺Ï é ß❸➑❙➒❅➓✶➔ Ñ ➹✛➑➺Ï ❐ →Ü❰ ➚✢à ❰✛ß❸➔âÏ Ñ❸➚ ❰✛➑✾➔âÏ ✿✁✄✂✆☎✞✝✟✂✆✠ ✧☛✡✌☞✎✍✑✏
X
✒ ✍✔✓✖✕✗✓ ✒✙✘✛✚ ✍✁✜✣✢✤✏✦✥✧✏★✥☛✍✪✩✬✫ ✘ ✩✎✍✭✫✮✏✰✯Leb
{
s
:
Xs
=
X
0}
= 0 a.s.
✜✣✥☛✍✱✫✱✍✔⑥ ➑✦➶ ✕✲✏✤✓✖✳✵✴✶✕✪✷ ✘ ✫✸✏✦✥☛✍✌☞✄✍ ✒ ✍✱✕✺✹✼✻✽✍✿✾❀✍❁✓✖✕✲✻✖✫❁✍✟❂❄❃✄✥☛✍✭✳
(I
t
+
, I
−
t
)
d
= (t
−
gt, dt
−
t),
✪▲✵➞✰✜✣✥☛✍✱✫✱✍
d
=
✾❀✍❁✓✟✳✼✕❆❅❇✢❇✕✎✢★✴✆✍✭✳✬✏✦✢★❈✭✓✖❉❊✢✰✳❋❉●✓✶✜❍✜✣✢✤✏✦✥❏■✤❂✄❑ ✘ ✫❁✍ ✘✛✚ ✍✭✫▼▲✙❈ ✘ ✳✵✴✛✢✤✏✦✢ ✘ ✳☛✍✮✴ ✘ ✳V
:=
I
t
+
+
I
−
t
=
dt
−
gt
✏✦✥☛✍ ✫✱✓✟✳✵✴ ✘ ✾ ✚ ✓✖✫◆✢●✓ ✒ ❉❖✍✱✕
I
t
+
, I
−
t
, t
−
gt,
✓✟✳✵✴
dt
−
t
✓✟✫❁✍❆✢★✴✆✍✱✳☛✏✦✢★❈✭✓✖❉P❉◗✯❘✴✛✢❇✕✲✏✦✫❙✢✒ ✻✞✏❚✍✮✴❯✏★✥☛✍✔❈ ✘ ✾❋✾ ✘ ✳❘✴✛✢❇✕▼✏★✫◆✢ ✒ ✻❊✏★✢ ✘ ✳ ✒ ✍✭✢✰✳✙✹❘✏★✥☛✍❱✻✖✳❊✢✰✷ ✘ ✫❙✾❲✴✛✢❇✕▼✏★✫◆✢ ✒ ✻❊✏★✢ ✘ ✳ ✘ ✳
(0, V
).
é ß❸➑ ➮ ➹ ➚✥➮ ➑◗➹✛❰✡Õ ✪❽✵➏✰ Ö✂➓✥Ï ➚ ➶❹Ï✛➑◗➹ ➪ ➑✦×✘➔ Ñ❨❳☎✸ Ò✑❩⑨❰ ➚ ➶➾➑ ➪ ➓✢→☎➔â× à❽➚ ➹✔➹✤➑❁❬❹➑✦❮✟❰✛➑➺×❪❭✼➹ ➚ Ö Ñ ➔➱➓ Ñ ➒ ➚ ❰✛➔ ➚✢Ñr➚✢Ñ
R+
Ö✉➔☎❰✛ß Ñ ➑ á ➓✏❰✤➔ ➪ ➑❙×✩➹✤➔ à ❰ ✱RBM
↓
,
à✖➚ ➹①ÏÙß ➚ ➹✛❰ ✱ ➓ Ñ × à❽➚ ➹①Ï✇❰✝➓✏❰✛➔ ➚✢Ñ ➓✶➹✤Õ❅➑◗å❸❮ ❐ ➹✝ÏÙ➔ ➚✢Ñ Ï à ➹ ➚ ➒ ✮ ❰ ➚●✮❸è✼⑥ ➓✶❰✛➑✦➹✚❰✤ß❸➑❁➓ ❐ ❰✛ß ➚ ➹✝Ï ➚✶à ❰✛ß❸➔âÏ Ñ❸➚ ❰✤➑ à❽➚✥❐❸Ñ ×(4)
❰ ➚ ➶➾➑ ➪ ➓✢→☎➔â× à❽➚ ➹P➓✶→➱→ ➮➾➚ Ï✛➔Ü❰✤➔ ➪ ➑◗→➱Õ④➹✤➑✦❮ ❐ ➹✤➹✤➑ Ñ ❰✍→➱➔ Ñ ➑➺➓✶➹P×❸➔ÜÛ ❐ Ï✛➔ ➚✥Ñ Ï ❐❸Ñ ×✩➑✦➹✼ÏÙ➒ ➚➏➚ ❰✛ß Ñ ➑✦Ï✤Ï ➓✢Ï✤Ï ❐ ➒ ➮ ❰✛➔ ➚✢Ñ Ï ➚✢Ñ ❰✤ß❸➑æÏ✤❮◗➓✢→☎➑ à❽❐❸Ñ ❮✲❰✛➔ ➚✢Ñ ➓ Ñ ×❅❰✛ß❸➑✾Ï ➮ ➑✦➑✦×●➒✒➑✦➓✢Ï ❐ ➹✤➑ è⑦Ô ➔➱➒ ➧ ➔☎❰✛➒❅➓ Ñ❃➮➾➚ ➔ Ñ ❰✛➑✦× ➚✥❐ ❰✼❰ ➚☞❐ ÏP❰✤ß❸➑ Ñ ❰✤ß❸➑ à❽❐ →➱→ á ➑ Ñ ➑✦➹✤➓✢→☎➔☎❰✇Õ ✪ ➓✢Ï✍ÏÙ❰✤➓✶❰✛➑➺×✑➔ Ñ❅é ß❹➑ ➚ ➹✤➑◗➒ ✸➈✰⑨➚✢à ❰✛ß❹➑➲➹✛➑➺Ï ❐ →Ü❰✼➓ Ñ ×✑➹✤➑◗➒❅➓✢➹✛ë✢➑➺×❁❰✤ß❹➓✏❰P➔☎❰P➔âÏP➓❙❮ ➚✥Ñ ÏÙ➑➺ê ❐ ➑ Ñ ❮◗➑ ➚✶à ❰✛ß❹➑✴➹✤➑✦Ï ❐ →☎❰✤Ï✑➔ Ñ❫❳➱✸➺✬ ❩①➓ Ñ × ❳✵ ❩ è❍❴➲➚ Ö✼➑ ➪ ➑✦➹ ✱ ➶➾➑✦❮✦➓ ❐ Ï✛➑ãÖ✼➑✴ß➆➓ ➪ ➑ Ñ❸➚ ❰ à❽➚✢❐❹Ñ × ❰✤ß❸➑✴➔â×✩➑ Ñ ❰✛➔☎❰✇Õ ✪❽✵➏✰ ➔ Ñ ❰✛ß❸➑ →➱➔Ü❰✤➑◗➹✝➓✏❰ ❐ ➹✛➑ ✱ Ö✼➑ à ➑✦➑◗→➆❰✛ß❹➓✶❰✼➔Ü❰✂➔➱ÏPÖ ➚ ➹Ù❰✤ß➞Ö✉ß❸➔➱→➱➑✉❰ ➚ ×✩➔âÏ✛❮ ❐ Ï✤Ï✍➶❸➹✤➔➱➑❁❬❹Õ☞❰✛ß❸➔âÏP➔ Ñ ❰✤➑◗➹✤➑✦ÏÙ❰✛➔ Ñ❸á ➶ ❐ ❰ Ñ❸➚ ❰✼Ö✉➔➱×✩➑✦→☎Õ✒ë Ñ❹➚ Ö Ñ ➑✦ê ❐ ➓✢→☎➔☎❰✇Õ●➔ Ñ →â➓➈Ö è➁é ß❸➑❁×✩➔ÜÛ ❐ Ï✛➔ ➚✥Ñ ❮✦➓✢Ï✛➑æ➔âÏ✉➓✢→➱Ï ➚✑➪ ➑◗➹✤Õ❃➓ ➮❸➮ ➑➺➓✶→➱➔ Ñ❹á Ö✉➔☎❰✛ß Ñ ➔â❮✟➑❙➑✟å ➮ →➱➔➱❮◗➔Ü❰ à❽➚ ➹✤➒ ❐ →â➓✶➑ è Ø✼→➱➑✦➓✶➹✤→➱Õ ✱➏à ➹ ➚ ➒ é ß❸➑ ➚ ➹✤➑◗➒ ✸✢✱ ➔☎❰ à❽➚ →➱→➱Ö➲Ï✂❰✤ß❹➓✏❰I
t
+
d
=
I
t
−
.
✪✭✯✢✰
t✡Ñ
❰✛ß❸➑✘❮✦➓✢Ï✛➑
X
➔âÏ✑➓RBM
↓
➓Ñ × Ï✇❰✝➓✏❰✛➔ ➚✢Ñ ➓✶➹✤Õ➘➑✟å❸❮ ❐ ➹✝Ï✛➔ ➚✥Ñ Ï à ➹ ➚ ➒ ✮ ❰ ➚➘✮ ➓✢➹✛➑✴❮ ➚✥Ñ ÏÙ➔â×✩➑✦➹✛➑➺× ➚✢Ñ ➑✴Ö ➚✢❐ →â× ➑✟å ➮ ➑➺❮✲❰æ❰✤ß❹➓✏❰æ❰✤ß❸➑ ➚ ❮◗❮ ❐❸➮ ➓✏❰✤➔ ➚✥Ñ ❰✛➔➱➒✑➑✒➶➾➑◗→ ➚ Ö❬❰✛ß❸➑ ➚ ➶➆ÏÙ➑◗➹ ➪ ➑✦×r→☎➑ ➪ ➑◗→⑦➔âÏæ➶❸➔ á✢á ➑◗➹ ✪ ➔ Ñ Ï ➚ ➒✒➑✑Ï✛➑ Ñ ÏÙ➑ ✰ ❰✤ß❹➓ Ñ ❰✛ß❸➑ ❰✤➔☎➒✒➑✔➓✶➶ ➚✏➪ ➑ ✱ ➶ ❐ ❰P❰✤ß❸➑①➹✤➓ Ñ × ➚ ➒ Ñ ➑➺Ï✛Ï ➚✢à ❰✛ß❹➑➲→➱➑ ➪ ➑✦→✽❵Ù➶❹➓✢→➱➓ Ñ ❮✟➑✦Ï❜❛✂❰✛ß❹➑①➹✤➓ Ñ × ➚ ➒ ➪ ➓✢➹✛➔â➓✶➶❹→☎➑➺Ï➁Ï ➚ ❰✤ß❹➓✏❰ ✪↔✯✢✰ ß ➚ →â×❸Ï è ì ➑●➹✤➑ à ➑◗➹④➓✢→➱Ï ➚ ❰ ➚❝❳✳ ❩ ✱ Ö✉ß❹➑◗➹✤➑
(5)
à❽➚ ➹④➓RBM
↓
➔âÏ❁Ï✛ß➚ Ö Ñ ❰ ➚ ➶➾➑❃➓❇❮ ➚✥Ñ Ï✛➑✦ê ❐ ➑ Ñ ❮◗➑ ➚✢à ➹✤➑ ➪ ➑✦➹✤Ï✛➔☎➶❹➔☎→➱➔Ü❰✇Õ■➔ Ñ Ï ➮ ➓✢❮◗➑ ➚✢à ❰✛ß❸➑❙➑◗å✩❮ ❐ ➹✝ÏÙ➔ ➚✢Ñ Ï è é ß❸➑ ➮ ➓ ➮ ➑✦➹❙➔âÏ ➚ ➹ á ➓ Ñ ➔âÏÙ➑➺×✃Ï ➚ ❰✛ß❹➓✶❰④➔ Ñ ❰✛ß❸➑ Ñ ➑◗å➞❰☞Ï✛➑✦❮✲❰✤➔ ➚✥Ñ Ö✼➑ ➮ ➹ ➚➈➪ ➑ é ß❸➑ ➚ ➹✤➑◗➒
1
è✴é ß❸➑ ➮ ➹ ➚➏➚✶à Ö ➚ ➹✤ë✢➑✦× ➚✢❐ ❰ à ➹ ➚ ➒ ➧ ➔Ü❰✤➒❅➓ ÑÞÝ Ï✑➹✤➑◗➒❅➓✢➹✛ë✫➹✛➑✦→☎➔➱➑✦Ï ➚✥Ñ Ï ➚ ➒✒➑ã➹✤➑✦Ï ❐ →☎❰✤Ï à ➹ ➚ ➒ ❳➱✸➺✬ ❩➲➓ Ñ × ❳✵ ❩➲Ö✉ß❸➔â❮✤ß ➓✢➹✛➑❃Ð❹➹✝Ï✇❰✒➹✛➑➺❮◗➓✢→☎→➱➑✦× è t✡Ñ◆➩ ➑➺❮✲❰✤➔ ➚✥Ñ3
Ö✼➑ ➮ ➹✛➑➺ÏÙ➑ Ñ ❰❅➓ Ñ ➓✶→☎❰✛➑✦➹ Ñ ➓✶❰✛➔ ➪ ➑ ➮ ➹ ➚✩➚✶à①➚✶àæé ß❸➑ ➚ ➹✛➑✦➒1
➔ Ñ ❰✛ß❸➑❇❮◗➓✥ÏÙ➑✴Ö✉ß❸➑ ÑX
➔âÏ✒➓✃→➱➔ Ñ ➑➺➓✶➹ ×✩➔☎Û ❐ ÏÙ➔ ➚✢ÑÞè✍é ß❸➑✔➒❅➓✢➔ Ñ ❰ ➚✩➚ →➾➔ Ñ ❰✛ß❸➔âÏ ➮ ➹ ➚✩➚✶à ➔➱Ï✼❰✛ß❸➑ ä ➑◗Õ Ñ ➒❅➓ Ñ✙❞✡② ➓✢❮ à❽➚ ➹✤➒ ❐ →â➓ è⑦é ß❸➑✾❮ ➚ ➒✒➒ ➚✢Ñ ×✩➔âÏ✇❰✤➹✛➔➱➶ ❐ ❰✤➔ ➚✥Ñ ➚✶à(I
t
+
, I
−
t
)
➓
Ñ
×
(t
−
gt, dt
−
t)
➔➱Ï✔➓✶→âÏ➚ ❮✤ß❹➓✢➹✤➓✥❮✲❰✤➑◗➹✤➔☎í✦➑✦× ➪ ➔â➓✒❰✛ß❸➑ ⑥❏❡◗➪ Õã➒✒➑✦➓✥Ï ❐ ➹✤➑ ➚✢à ❰✛ß❹➑④➔ Ñ➏➪ ➑◗➹✝ÏÙ➑❁→ ➚ ❮◗➓✶→ ❰✤➔☎➒✒➑✴➓✏❰④❰✤ß❸➑ ➮➾➚ ➔ Ñ ❰✑Ö✉ß❸➑◗➹✤➑●❰✛ß❸➑ã➑◗å✩❮ ❐ ➹✝ÏÙ➔ ➚✢Ñ Ï④ÏÙ❰✤➓✢➹Ù❰✑➓ Ñ × ➑ Ñ × è➘q①➮❸➮ →➱Õ➞➔ Ñ❸á➘② ➹✛➑✦➔ Ñ➭Ý Ï☞Ï ➮ ➑✦❮✟❰✛➹✝➓✶→P❰✛ß❸➑ ➚ ➹✤Õ ➚✶à ÏÙ❰✛➹✤➔ Ñ❹á Ï✼❰✛ß❸➑❁×✩➔➱ÏÙ❰✛➹✤➔➱➶ ❐ ❰✛➔ ➚✥Ñã➚✢à
V
✪ Ö✉ß❸➔â❮✝ß✘×✩➑✟❰✤➑◗➹✤➒✒➔ Ñ ➑➺Ï✂❰✛ß❸➑✂❒ ➚ ➔ Ñ ❰➲×✩➔âÏÙ❰✛➹✤➔☎➶ ❐ ❰✛➔ ➚✢Ñ✴➚✶à(I
t
+
, I
−
t
)
✰ ➔âÏ✉ÏÙß ➚ Ö Ñ ❰ ➚ ➶➾➑❁➓✑➒✑➔☎å➏❰ ❐ ➹✤➑ ➚✢à⑨á ➓✢➒✑➒❅➓✑×❸➔➱ÏÙ❰✛➹✤➔☎➶ ❐ ❰✛➔ ➚✢Ñ Ï è ❢ ❣✐❤ ï ❤ ñ✄❥❧❦ õ❆❥✁♠ ❤ ♥✪♦✺♣ q❨rts❊✉✗✈①✇✿②◆③✶s✙④✞②❜⑤⑦⑥✗s❊②⑨⑧⑩r⑦③✧⑧❷❶−
g
0
❸ r✿✇d
0
⑥
➑◗❰
X
=
{
Xs
:
s
∈
R}
➶➾➑➲➓Ñ ➓✢➹✛➶❸➔☎❰✛➹✝➓✶➹✤Õ❙ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ ➓✢➹✛Õ ➮ ➹ ➚ ❮◗➑✦Ï✤Ï❢❰✤➓✢ë➞➔ Ñ❸á✾➪ ➓✢→ ❐ ➑➺Ï⑦➔ Ñ
E
⊂
R
.
t ❰✍➔➱Ï✍➓✢Ï✤Ï ❐ ➒✒➑✦× ❰✤ß❹➓✏❰❙❰✛ß❹➑●Ï✛➓✢➒ ➮ →➱➑ ➮ ➓✶❰✛ß❹Ï ➚✶àX
➓✶➹✤➑✑➹✤➔ á ß✥❰❁❮ ➚✢Ñ ❰✤➔ Ñ➞❐❸➚✢❐ Ï❙➓ Ñ ×✃ß❹➓ ➪ ➑✒→➱➑ à ❰④→☎➔➱➒✒➔Ü❰✝Ï ✪ ❮✦➓✢×✩→â➓ á➏✰✲è✒ì ➑●❮ ➚✥Ñ ÏÙ➔â×✩➑✦➹X
➔ Ñ ❰✛ß❹➑✑❮✦➓ Ñ❸➚✢Ñ ➔➱❮✦➓✶→⑨Ï ➮ ➓✢❮✟➑(Ω,
F
)
➚✶à ❮◗➓✢×❸→➱➓ á❅à❽❐❸Ñ ❮✟❰✛➔ ➚✢Ñ Ï è❙⑥ ➑◗❰{
θs
:
s
∈
R}
×✩➑ Ñ❸➚ ❰✤➑④❰✛ß❸➑ ❐ Ï ❐ ➓✶→⑦Ï✛ß❸➔ à ❰ ➚✢➮ ➑✦➹✤➓✶❰ ➚ ➹✼➔ Ñ ❰✤ß❸➔➱Ï à ➹✝➓✶➒✒➑✦Ö ➚ ➹✤ë è ä ➚ ➹✉➓✑Ï✛➑✟❰D
⊂
E
➓Ñ
×
t
= 0
×✩➑✟ÐÑ
➑
M, d
0
➓ Ñ ×
g
0
➓✢Ï✂➔ Ñ✫✪✇✸➺✰ ➓ Ñ × ✪✭✬✥✰✲è ♣❇➚ ➹✛➑ ➚➈➪ ➑✦➹ ✱ Ï✛➑✟❰Ò →☎➑➺❮✲❰✤➹ ➔➱❮④Ø ➒✒➒ ➔â❮◗➓✶❰✛➔ Ï✚➔ ➹ ➶➆➓✶➶❸➔➱→☎➔☎❰✇Õ
ì
➑
Ñ❸➚
Ö❭❮
➚
→➱→☎➑➺❮✲❰ ✱➁à❽➚
→➱→
➚
Ö✉➔ Ñ❸á❨❳☎✸➈✬
❩
✪
Ö✉ß❸➑✦➹✛➑
✱
➔
Ñ à
➓✢❮✟❰
✱
➑
➪
➑
Ñ
➒
➚
➹✤➑
á
➑
Ñ
➑✦➹✤➓✢→✼❮◗➓✥ÏÙ➑❃➔âÏ✑❮ ➚✢Ñ
Ï✛➔➱×❸➑◗➹✤➑✦× ✰✲✱
Ï
➚
➒✒➑
à❽➚
➹✤➒
❐
→â➓✶➑✑❮ ➚✢Ñ
❮◗➑◗➹
Ñ
➔
Ñ❸á
❰✤ß❸➑❅×✩➔âÏ✇❰✤➹✛➔➱➶
❐
❰✤➔ ➚✥Ñ
Ï
➚✶à
g
0
➓
Ñ
×
V
:=
d
0
−
g
0
.
é
ß❹➑✒❮✟➹
❐
❮◗➔➱➓✢→⑦❮ ➚✥Ñ
❮✟➑
➮
❰✾ß❸➑◗➹✤➑◗➶➏Õ❇➔➱Ï
❰✛ß❹➑
➧
➓✶→➱➒❨➒✒➑✦➓✥Ï
❐
➹✛➑
è
✂✂✁ ❑⑨❏☎✄✦❏
☎
❑✝✆
✡✟✞✹❼❸✄✾✗➲✣✏✁✎ ✎❅✄✝✣➈❾
➽
✙✛✄
Q
✣➈❾✝❾◗✌✦✆◗✓❽✣✶✞❷✄✝➄✡✠⑦✓✖✞❽❼X
✓â❾④➄✢✄✭➴✍✕✹✄✤➄ã✜✟➉Q
(B) :=
E
(
| {
s
: 0
< s <
1, s
∈
L, θs
∈
B
} |
)
,
B
∈ F
,
✠❢❼❸✄◗✙✛✄
| · |
➄✥✄✟✕✹✌✏✞✡✄✲❾❙✞❽❼❸✄❁✕ ➽ ✎❅✜✤✄◗✙❁✌✇❺✂❶❸✌✏✓✖✕➆✞▲❾④✌Ù❺✾✞❽❼❸✄❁❾◗✄✟✞✼✓✖✕■✞❽❼❸✄✑✜✟✙✛✣✥✆✤✄✟❾◗✠▼ ✝✟☎☞☛⑩☎✍✌ ❏✎✄✦❏☎ ❑✑✏ ✡❙➊➭✌✶✙④✣●✎✒✄✝✣➈❾ ➽ ✙✛✣✢✜◗✁☎✄✼❺ ➽ ✕✹✆✟✞↔✓▲✌✏✕
f
:
R
×
Ω
→
[0,
∞
)
E
¡
f
(θg
0
,
−
g
0
)
1
{−∞<g
0
<
0
}
¢
=
Z
Ω
Q
(dω)
Z
d
0
0
f
(t, ω)
dt.
✪Ò
✰
➋✲✕❅❶❸✣✶✙✲✞✭✓▲✆ ➽ ✁☎✣✶✙✝❿
P
(
−∞
< g
0
<
0, θg
0
∈
dω) =
Q
(dω)
d
0
(ω),
✪✓✒✶✰
P
(
−
g
0
∈
da) =
Q
(d
0
> a)
da,
a >
0.
✪▲➬➞✰
❻■✌✏✙✛✄✝✌✏➇✏✄✟✙✝❿
P
(V
∈
dv) =
v
Q
(d
0
∈
dv),
✪✭✳✥✰
✣✏✕✹➄❅✆✤✌✶✕➾➄✏✓✖✞↔✓▲✌✏✕✹✣✏✁❽✁➉❃✌✏✕✘✞❽❼❸✄✼❶❸✣✶✞✖❼➏❾
{
Xg
0
+
s
:
s
≥
0
}
✞✖❼❹✄❙➄✶✓â❾✲✞✭✙✲✓▲✜
➽
✞✭✓▲✌✶✕■✌✇❺
−
g
0
➄✥✄❷❶❹✄✟✕✹➄➈❾❙✌✶✕❹✁➉❅✌✏✕
V
✣✏✕✹➄✓â❾❙✞✖❼❹✄
➽
✕➆✓❺✟✌✏✙✲✎ ➄✏✓â❾✲✞↔✙✲✓▲✜
➽
✞↔✓▲✌✏✕✫✌✶✕
(0, V
).
✗✂✙✛✌✦✌Ù❺✟✠æ➩
➑✦➑ ❳➱✸➺✬
❩
é
ß❹➑
➚
➹✤➑◗➒ ➮ÞèP✬✢✳✢✮
➓
Ñ
×❇Ø
➚
➹
➚
→➱→➱➓✢➹✛Õ
➮ÞèP✬✢✳✢➬❹è
❀❀✂✆✠✕✔✙✝✗✖✙✘❏✡æ✪
➔
✰❢t✡Ñ❯❳➱✸➺✬
❩❸➔Ü❰➁➔➱Ï➁➓✶→âÏ ➚✔➮
➹
➚➈➪
➑➺×✾❰✤ß❹➓✏❰⑦❰✛ß❸➑
➧
➓✢→☎➒❉➒✒➑✦➓✥Ï
❐
➹✛➑✼➔➱Ï➁➓①➒
❐
→☎❰✛➔
➮
→➱➑ ➚✶à
❰✛ß❸➑
t
❰✛✚æ➑◗å✩❮
❐
➹✝ÏÙ➔ ➚✢Ñ
→â➓➺Ö
è
Ø
➚
➒
➮
➓✶➹✤➔ Ñ❹á☞à❽➚
➹✛➒
❐
→➱➓✢➑ ✪▲➬➞✰
➓
Ñ
×
✪✭✳✥✰
Ö✉➔Ü❰✤ß ✪Ù✸
Ò
✰
➓
Ñ
×
✪Ù✸✗✒✢✰
➔
Ñ❇➧
➹
➚✢➮➾➚
Ï✛➔☎❰✛➔ ➚✢Ñ✘➬✒á
➔
➪
➑✦Ï➲➓
Ñ
➔
Ñ
×✩➔â❮◗➓✏❰✤➔ ➚✢Ñ
à❽➚
➹✚❰✤ß❸➔âÏ
à
➓✢❮✲❰
✪
➔
Ñ
❰✛ß❹➑❁×✩➔ÜÛ
❐
Ï✛➔ ➚✥Ñ
❮✦➓✢Ï✛➑ ✰✟è
✪
➔➱➔ ✰✂➧
➹
➚✢➮➾➚
ÏÙ➔☎❰✛➔ ➚✢Ñ✴÷
Õ➏➔➱➑◗→â×❸Ï✉➓✢→➱Ï
➚
➑✦➓✥ÏÙ➔➱→➱Õ
✪
❮
à✇è①✪✇✸➺➬✥✰Ù✰
P
(
−
g
0
∈
da , d
0
∈
db) =
da π(a, db),
Ö✉ß❸➑✦➹✛➑æ❰✛ß❹➑❙➒✑➑➺➓✢Ï
❐
➹✛➑
π
➔âÏ✉❮✝ß❹➓✢➹✤➓✥❮✲❰✛➑✦➹✛➔➱í◗➑➺×➪
➔➱➓
π(a, B) =
µ(a
+
B),
a
+
B
:=
{
a
+
b
:
b
∈
B
}
Ö✉➔☎❰✛ß
B
➓❋❭➚
➹✛➑✦→➨Ï✛➑✟❰ ➚✢Ñ
R+
➓Ñ
×
µ(dv) :=
P
(V
∈
dv).
♥✪♦★♥ q✢✜✣✜✵⑥✥✤
❸
s❊②❜⑧⑩r s❊②✧✦ ✈❏③✧❶✭⑧❷④✑✜✍★✩✜✣✪✺②✧✜
❸
✪✧✪✓★ ③✶s
❸
s❊②❜⑧ r
❸
④✫★✬✤✿④✞⑧✭✜✬✈❏③✆③✽✈❏③
ì
➑r❮ ➚✥Ñ
ÏÙ➔â×✩➑✦➹ Ñ❸➚
Ö❫➓✫❮✟Õ✩❮◗→☎➔â❮◗➓✢→☎→➱Õ ÏÙ❰✤➓✏❰✤➔ ➚✢Ñ
➓✢➹✛Õ ➒✒➑➺➓✢Ï
❐
➹✤➓✢➶❸→➱➑
➮
➹
➚
❮✟➑➺Ï✛Ï ➚✥Ñ
Ð
Ñ
➔Ü❰✤➑✘❰✤➔☎➒✒➑r➔
Ñ
❰✛➑◗➹
➪
➓✢→➲➓
Ñ
× ➔☎❰✤Ï
Ï
➚
❒
➚✥❐
➹
Ñ
❰✤➔☎➒✒➑✦Ï①➓✶➶ ➚✏➪
➑✾➓
Ñ
×ã➶➾➑✦→
➚
Ö ❰✛ß❹➑❙➔
Ñ
➔Ü❰✤➔➱➓✢→Þ→➱➑
➪
➑◗→
è
Ø✼Õ✩❮✟→➱➔➱❮✦➓✶→➱→☎Õ❃ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ
➓✢➹✛➔☎❰✇Õ❅ß❸➑✦➹✛➑✦➶➞Õ❃➒✒➑✦➓
Ñ
Ï✚➹ ➚✢❐❸á
ß❸→➱Õ
❰✛ß➆➓✏❰✉❰✛ß❸➑
➮
➑✦➹✛➔
➚
×✩➔â❮✔➑◗å➏❰✛➑
Ñ
ÏÙ➔ ➚✢Ñ✴➚✶à
❰✤ß❸➑
➮
➹
➚
❮✟➑➺Ï✛Ï✂➔âÏ✉ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ
➓✢➹✛Õ
è
✂✂✁ ❑⑨❏☎✄✦❏☎ ❑✯✮ ✡✰✞➾❼❹✄①✎❅✄✤✣➈❾ ➽ ✙✛✣✢✜◗✁☎✄➁❶✹✙✛✌✦✆✤✄✟❾✝❾
{
Xt
: 0
≤
t < l
}
,
✠❢❼❹✄✟✙✛✄l >
0
✓â❾➨➴✲✱➞✄✝➄➈❿➭✓â❾①✆✤✣✶✁✖✁☎✄✝➄☞✆◗➉✢✆◗✁✓▲✆✤✣✶✁✖✁Ü➉ ❾✲✞❷✣✶✞✭✓▲✌✶✕➾✣✶✙✲➉●✓❺❁✞✖❼❹✄✉❶➾✙✛✌✦✆✝✄✟❾✤❾{
Yt
:=
Xt|l
:
t
∈
R}
,
✠❢❼❸✄◗✙✛✄t
|
l
✎✒✄✝✣✏✕❸❾t
✎❅✌✦➄➽
✁☎✌
l,
✓â❾✾❾✲✞✡✣✏✞↔✓▲✌✏✕➾✣✶✙✲➉●✓✖✕➘✞✖❼❸✄➽
❾
➽
✣✏✁➨❾✟✄◗✕❸❾◗✄✟❿✍✓▲✠✖✄➺✠Ü❿✹❺✲✌✶✙☞✣✏✕❹➉
s
∈
R
✞❽❼❸✄✉❶➾✙✛✌✦✆✝✄✲❾✝❾◗✄✟❾{
Yt
}
✣✏✕✹➄{
Ys
+
t
}
✣✶✙✛✄④✓▲➄✢✄◗✕❹✞↔✓▲✆✤✣✶✁➭✓✖✕■✁➱✣✳✠P✠
é
ß❸➑✔➔➱➒ ➮➾➚
➹✛❰✤➓
Ñ
❰
➮
➹
➚✢➮
➑◗➹✛❰✇Õ ➚✶à
❮✟Õ✩❮✟→➱➔â❮◗➓✢→☎→➱Õ✒ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ
➓✢➹✛Õ
➮
➹
➚
❮✟➑➺Ï✛Ï✛➑✦Ï
Ñ
➑◗➑✦×✩➑➺×●➔
Ñ
❰✤ß❸➑
➮
➹
➚➏➚✢à➨➚✢à❢é
ß❸➑
➚
➹✤➑◗➒
✸
➔➱Ï
á
➔
➪
➑
Ñ
➔
Ñ❝❳✵
❩
é
ß❹➑
➚
➹✤➑◗➒ ÷❹è☎✸✥è
ä
➚
➹➲❰✛ß❹➑☞❮ ➚✢Ñ➏➪
➑
Ñ
➔➱➑
Ñ
❮◗➑ ➚✶à
❰✛ß❹➑❁➹✤➑✦➓✥×✩➑◗➹➲Ö✼➑☞Ï✇❰✝➓✏❰✤➑④➓
Ñ
×
➮
➹
➚✏➪
➑✾❰✤ß❸➔➱Ï①➹✤➑✦Ï
❐
→☎❰①➔
Ñ
❰✛ß❹➑ à❽➚
➹✤➒❨×✩➔➱➹✛➑➺❮✲❰✤→☎Õã➓ ➮❸➮
→➱➔â❮◗➓✶➶❹→☎➑ à❽➚
➹
➚✥❐
➹
➮❸❐
➹
➮➾➚
Ï✛➑✵✴✩ß
➚
Ö✼➑
➪
➑✦➹ ✱➏à❽➚
→➱→
➚
Ö✉➔ Ñ❹á
❮◗→
➚
ÏÙ➑✦→☎Õ ❳✵
❩
❰✛➑ ❮◗❮ ➓✏❰✤➔ ❰✛➔➱➒✒➑✦Ï Ï✇❰✝➓✏❰✛➔ ➓✶➹✤Õ ➹ ❮◗➑✦Ï✤ÏÙ➑➺Ï
▼
✝✟☎ ☛❷☎✍✌ ❏☎✄✦❏
☎
❑
✁ ✡✄✂➭✄✟✞
X
=
{
Xt
: 0
≤
t <
1
}
✜✤✄✒✣❃✎✒✄✝✣➈❾➽
✙✛✣✢✜◗✁☎✄❅✆✟➉✢✆◗✁✓▲✆✝✣✏✁❽✁➉ã❾✝✞✡✣✏✞↔✓▲✌✏✕✹✣✏✙✲➉❙❶➾✙✛✌➺✆✤✄✟❾✝❾❁❾
➽
✆✝❼
✞❽❼❸✣✏✞
Leb
{
t
:
Xt
=
X
0}
= 0
a.s.
✪Ù✸✦✮➞✰✞✹❼❸✄◗✕✃✞✖❼❹✄☞✌✦✆✝✆ ➽ ❶❸✣✏✞↔✓▲✌✏✕➘✞✭✓✖✎❅✄✟❾
Z
1
0
1
{X
t
≤X
0
}
dt
✣✶✕➾➄
Z
1
0
1
{X
t
≥X
0
}
dt
✣✏✙✛✄ ➽ ✕➆✓❺✟✌✏✙✲✎✑✁➉ã✌✏✕
(0,
1)
➄✏✓â❾✲✞↔✙✲✓▲✜ ➽ ✞✡✄✤➄●✙✛✣✶✕➾➄✥✌✏✎ ➇✏✣✏✙✲✓▲✣✥✜✟✁☎✄✟❾◗✠ ✗✂✙✛✌✦✌Ù❺✟✠✔é❢➚ÏÙ❰✤➓✢➹Ù❰❅Ö✉➔Ü❰✤ß
✱
➹✛➑➺❮◗➓✶→➱→ é❢❐
❮✤ë✥➑◗➹
Ý
Ï✑➑◗å➞❰✤➑
Ñ
Ï✛➔ ➚✢Ñ ➚✢à
❰✤ß❸➑
Ó
→➱➔
➪
➑
Ñ
ë
➚✛❞
Ø✂➓
Ñ
❰✛➑✦→☎→➱➔✂❰✛ß❹➑
➚
➹✤➑◗➒
✪
ÏÙ➑✦➑ ❳☎✸➺➬
❩
✰
➣
➔
à
Z
=
{
Zn
}
➔âÏ●➓ Ï✇❰✝➓✏❰✤➔➚✥Ñ
➓✶➹✤Õ Ï✛➑✦ê
❐
➑
Ñ
❮◗➑ ➚✶à
➹✤➓
Ñ
×
➚
➒
➪
➓✶➹✤➔➱➓✢➶❸→➱➑✦Ï❅➓
Ñ
×
I
Z
➔âÏ✒❰✤ß❸➑r➔ Ñ➞➪➓✶➹✤➔➱➓
Ñ
❰
σ
❞ Ð❹➑◗→â××✩➑◗❰✛➑◗➹✤➒✒➔
Ñ
➑✦×ã➶➏Õ
Z
✪✖à❽➚ ➹✚❰✤ß❸➔➱Ï➲❮➚✢Ñ
❮◗➑
➮
❰①ÏÙ➑✦➑
✱
➑
èá➆è☎✱ ❳✯
❩
✰
❰✛ß❸➑
Ñ
➓
è
Ï
è
sup
x
¯
¯
¯
¯
¯
1
n
n
X
i
=1
1
{Z
i
≤x}
−
P
(Z
1
≤
x
| I
Z
)
¯
¯
¯
¯
¯
→
0 as
n
→ ∞
.
✪Ù✸✢✸➈✰
⑥
➑◗❰
Y
➶➾➑❅❰✤ß❸➑ãÏ✇❰✝➓✏❰✤➔ ➚✥Ñ➓✶➹✤Õ
➮
➹
➚
❮◗➑✦Ï✤Ï
➚
➶✩❰✤➓✢➔
Ñ
➑➺×➘➶➏Õ➘➓
➮
➑✦➹✛➔
➚
×✩➔â❮❅❮ ➚✥Ñ
❰✤➔ Ñ➏❐
➓✏❰✤➔ ➚✥Ñ✫➚✢à
X
➓✢Ï❁➔Ñ
❰✛➹
➚
×
❐
❮◗➑✦×✫➔
Ñ
Ú
➑✟Ð
Ñ
➔☎❰✛➔ ➚✢Ñ●✯
➓
Ñ
×☞→☎➑◗❰
I
Y
➶➾➑✂❰✤ß❸➑✉➔ Ñ➏➪➓✶➹✤➔â➓
Ñ
❰
σ
❞ Ð❹➑◗→â×➚✶à
Y.
é ß❸➑Ñ☞à❽➚
➹✍➓✢→☎→
n
❰✤ß❸➑➲ÏÙ➑➺ê❐
➑
Ñ
❮◗➑
{
Z
k
(
n
)
:=
Y
k
2
n
}
➔âÏ✉ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ
➓✢➹✛Õ●➓
Ñ
×ãÖ✼➑✾ß➆➓
➪
➑
à❽➚
➹①➓✶→➱→
x
➓Ñ
×
➮➾➚
Ï✛➔Ü❰✤➔
➪
➑✾➔
Ñ
❰✤➑
á
➑✦➹✤Ï
m
➓è
Ï
è
Z
1
0
1
{X
s
≤x}
ds
= lim
n→∞
1
2
n
2
n
−
1
X
k
=0
1
{Z
(n)
k
≤x}
= lim
n→∞
1
m
2
n
(2
n
−
1)
m
X
k
=0
1
{Z
(n)
k
≤x}
.
✈①➚
❰✛➔â❮✟➑❁❰✤ß❹➓✏❰æ➶➞Õã❰✤ß❸➑④➒✒➑➺➓✢Ï
❐
➹✤➓✢➶❸➔➱→☎➔☎❰✡Õ✘➓✢Ï✤Ï
❐
➒
➮
❰✤➔ ➚✥Ñ
❰✛ß❸➑☞➔
Ñ
❰✛➑
á
➹✝➓✶→❢➓✢➶ ➚➈➪
➑❁➔➱Ï✔Ö✼➑◗→➱→⑨×✩➑◗Ð
Ñ
➑✦×
è
ä
➹
➚
➒
✪✇✸✢✸➈✰
➔☎❰
à❽➚
→➱→
➚
Ö➲Ï✂❰✛ß➆➓✏❰①➓
è
Ï
è
1
2
n
2
n
−
1
X
k
=0
1
{Z
(n)
k
≤x}
= lim
m→∞
1
m
2
n
(2
n
−
1)
m
X
k
=0
1
{Z
k
(n)
≤x}
=
P
(Y
0
≤
x
| I
n
),
Ö✉ß❸➑✦➹✛➑
I
n
➔âÏ❃❰✤ß❸➑➘➔ Ñ➞➪➓✶➹✤➔➱➓
Ñ
❰
σ
❞ ➓✶→á
➑✦➶❸➹✤➓ ×✩➑✟❰✤➑◗➹✤➒✑➔
Ñ
➑➺× ➶➏Õ
Z
(
n
)
è❭✂Õ◆❰✛ß❸➑✫➒✒➓✢➹Ù❰✤➔ Ñ❹á
➓✶→➱➑✃❮ ➚✢Ñ➞➪
➑✦➹
á
➑
Ñ
❮◗➑
❰✤ß❸➑
➚
➹✤➑◗➒
✱
Ï✛➔
Ñ
❮✟➑
I
Y
=
σ
{I
1
,
I
2
, . . .
}
,
Ö✼➑❙ß❹➓➪
➑✾➓
è
Ï
è
Z
1
0
1
{X
s
≤x}
ds
=
n→∞
lim
P
(Y
0
≤
x
| I
n
)
=
lim
n→∞
E
(
P
(Y
0
≤
x
| I
Y
)
| I
n
)
=
P
(Y
0
≤
x
| I
Y
).
✪✇✸➺✬✥✰
✈
➑✟å➏❰①×✩➑✟Ð
Ñ
➑
à❽➚
➹➲➓
Ñ
Õ❀❭
➚
➹✤➑◗→➨Ï✛➑✟❰
B
η(B) :=
P
(Y
0
∈
B
| I
Y
).
ä
➹
➚
➒
✪✇✸➈✬✢✰
➔☎❰ à❽➚
→➱→
➚
Ö➲Ï✂❰✛ß❹➓✶❰➲➓
è
Ï
è
Z
1
0
1
{X
s
≤X
0
}
ds
=
η((
−∞
, Y
0
]),
➓
Ñ
×
✱➾à
➹
➚
➒❫❰✛ß❸➑✒➓✥Ï✛Ï
❐
➒
➮
❰✛➔
➚✢Ñ ✪✇✸➺✮✥✰✟✱
x
7→
η((
−∞
, x])
➔➱Ï✾❮ ➚✥Ñ❰✛➔ Ñ➞❐❹➚✢❐
Ï
è✆☎
Ï✛➔ Ñ❸á
❰✛ß❸➑☞❰
➚
Ö✂➑◗➹
➮
➹
➚✥➮
➑◗➹✛❰✡Õ✘Ö✼➑
ß❹➓
➪
➑❙➓
è
Ï
è
P
(Y
0
∈
B
|
η) =
E
(
P
(Y
0
∈
B
| F
Y
)
|
η) =
E
(η(B)
|
η) =
η(B)
Ï✛ß
➚
Ö✉➔ Ñ❸á
❰✛ß➆➓✏❰
η
➔âÏ➁❰✛ß❸➑①➹✛➑ á✢❐→â➓✶➹
➪
➑◗➹✝Ï✛➔ ➚✥Ñ✑➚✶à
P
(Y
0
∈ · |
η).
é
ß❸➑◗➹✤➑ à❽➚
➹✛➑
✱
➶➏Õ④❰✛ß❹➑①❮ ➚✥Ñ
❰✛➔ Ñ➞❐
➔Ü❰✇Õ ➚✢à
η,
➔☎❰Pß➚
→➱×❸Ï
❰✤ß❹➓✏❰
η((
−∞
, Y
0
])
➔âÏ ❐❸Ñ
➔
à❽➚
➹✤➒✒→☎Õã×✩➔âÏ✇❰✤➹✤➔☎➶
❐
❰✤➑✦× ➚✢Ñ
(0,
1),
➓✥Ï✉❮✟→â➓✶➔➱➒✑➑➺×→☎➑➺❮✲❰✤➹ ➔➱❮④Ø ➒✒➒ ➔â❮◗➓✶❰✛➔ Ï✚➔ ➹ ➶➆➓✶➶❸➔➱→☎➔☎❰✇Õ
ì
➑✘ß❹➓
➪
➑✴❰✛ß❸➑ à❽➚
→➱→
➚
Ö✉➔ Ñ❸á
Ï
❐
➹
➮
➹✤➔âÏÙ➔ Ñ❸á
→☎Õ
á
➑
Ñ
➑◗➹✝➓✶→✉❮
➚
➹
➚
→☎→â➓✶➹✛Õ ❮ ➚➈➪
➑✦➹✛➔ Ñ❸á❹✱
➑
èá❹è➱✱
➓✶→➱→✉➑✟å❸❮
❐
➹✤Ï✛➔ ➚✥Ñ
➓
Ñ
×
➚
❰✛ß❹➑◗➹
➶❸➹✤➔➱×
á
➑✦Ï
è
✽❱☎✞✝✟☎✁✂✔✙✝☎✄✝✆ ✡ ✂➭✄◗✞
Z
=
{
Zt
: 0
≤
t < l
}
✜✝✄✴✣r✎❅✄✤✣✏❾ ➽ ✙✛✣✥✜✟✁☎✄❁❶✹✙Ù✌➺✆✤✄✟❾✝❾ã✣✏✕✹➄U
➽ ✕➆✓❺✟✌✏✙✲✎✑✁➉✃✌✶✕(0, l)
➄✏✓â❾✲✞↔✙✲✓▲✜ ➽ ✞✡✄✤➄●✙✛✣✶✕➾➄✥✌✏✎ ➇✏✣✏✙✲✓▲✣✥✜✟✁☎✄✑✓✖✕➾➄✥✄✡❶❸✄✟✕✹➄✢✄◗✕❹✞✉✌Ù❺
Z
✠✚➂æ❾✝❾ ➽ ✎✒✄❁✞✖❼❹✣✏✞Leb
{
t
:
Zt
=
ZU
}
= 0
a.s.
✞✹❼❸✄◗✕✃✞✖❼❹✄☞✌✦✆✝✆ ➽ ❶❹✣✏✞↔✓▲✌✏✕➘✞✭✓✖✎❅✄✟❾
Z
l
0
1
{Z
t
<Z
U
}
dt
✣✶✕➾➄
Z
l
0
1
{Z
t
>Z
U
}
dt
✣✏✙✛✄ ➽ ✕➆✓❺✟✌✏✙✲✎✑✁➉ã➄✏✓â❾✲✞✭✙✲✓▲✜ ➽ ✞✡✄✤➄ã✌✏✕
(0, l).
✗✂✙✛✌✦✌Ù❺✟✠
ä
➚
➹✴➓✢→☎→
s
∈
[0, l]
✱ ❰✛ß❸➑➘➹✤➓Ñ
×
➚
➒
➪
➓✢➹✛➔â➓✶➶❸→➱➑
U
′
(s) :=
U
+
s
➒➚
×
❐
→
➚
l
➔âÏ✴➓✶→âÏ ➚ ❐❹Ñ➔
à❽➚
➹✛➒✒→➱Õ
×✩➔âÏ✇❰✤➹✛➔➱➶
❐
❰✤➑✦× ➚✢Ñ
(0, l),
➓Ñ
×
✱
❰✛ß
❐
Ï
✱
Y
=
{
Yt
: 0
≤
t < l
}
✱ Ö✉ß❸➑◗➹✤➑Yt
:=
ZU
′
(
t
)
✱
➔âÏã❮◗Õ✩❮✟→➱➔➱❮✦➓✶→➱→☎Õ
ÏÙ❰✤➓✏❰✤➔ ➚✥Ñ
➓✶➹✤Õ è❢ì
➑❙ß❹➓
➪
➑
Z
l
0
1
{Z
t
<Z
U
}
dt
=
Z
l
0
1
{Z
U
′
(t)
<Z
U
}
dt
=
Z
l
0
1
{Y
t
<Y
0
}
dt.
Ø
➚✢Ñ
ÏÙ➑➺ê
❐
➑
Ñ
❰✤→☎Õ
✱
❰✛ß❹➑❙❮◗→➱➓✢➔☎➒ à❽➚
→➱→
➚
Ö➲Ï
à
➹
➚
➒
➧
➹
➚✥➮➾➚
Ï✛➔Ü❰✤➔ ➚✥Ñ
6
è♥✪♦✟✞ ✠ ④❊⑧✪⑧❷❶ ⑧❷❶☛✡ ✉✗✈❏⑧❷④❊✈ ✦ ♣
⑥
➑◗❰
{
Xs
:
s
∈
R}
➶➾➑✒➓ã➒✑➑➺➓✢Ï❐
➹✝➓✶➶❸→➱➑☞ÏÙ❰✤➓✶❰✛➔ ➚✢Ñ
➓✢➹✛Õ
➮
➹
➚
❮✟➑✦Ï✤Ïæ➓✢Ïæ×❸➑✟Ð
Ñ
➑➺×■➔ Ñ✫➩
➑✦❮✟❰✛➔
➚✢Ñ ✸➺✯❸è❙ì
➑✒❮ ➚✢Ñ
Ï✛➔â×✩➑◗➹
❰✛ß❹➑❁❮◗➓✢Ï✛➑
t
= 0.
❭✼➑➺❮◗➓❐
ÏÙ➑
I
0
+
+
I
0
−
=
d
0
−
g
0
=:
V
➔☎❰❙➔âÏ✾➑ Ñ❸➚✥❐❸á
ß✃❰
➚
Ï✛ß
➚
Ö❉❰✤ß❹➓✏❰
✱
➑
èá❹è➱✱
❰✛ß❹➑❅❮ ➚✢Ñ
×✩➔☎❰✛➔ ➚✢Ñ
➓✶→➁×❸➔➱ÏÙ❰✛➹✤➔➱➶
❐
❰✛➔ ➚✢Ñ
Ï
➚✢à
I
0
−
➓
Ñ
×
d
0
á
➔
➪
➑
Ñ
V
❮➚
➔
Ñ
❮✟➔â×✩➑
è
ä
➹
➚
➒
➧
➹
➚✥➮➆➚
ÏÙ➔☎❰✤➔ ➚✥Ñ❅÷
Ö✼➑①ë Ñ❸➚
Ö ❰✤ß❹➓✏❰
d
0
á
➔
➪
➑
Ñ
V
➔➱Ï ❐❹Ñ➔
à❽➚
➹✛➒✒→➱Õ✒×✩➔âÏ✇❰✤➹✛➔➱➶
❐
❰✤➑✦× ➚✢Ñ
(0, V
).
é❢➚④➮➹
➚✏➪
➑✉❰✛ß❹➓✶❰
❰✛ß❹➔➱Ï✉➔âÏ✉➓✢→➱Ï
➚
❰✤ß❸➑❁❮◗➓✢Ï✛➑ à✖➚
➹
I
0
−
×✩➑◗Ð
Ñ
➑
à❽➚
➹
0
≤
t < V
Zt
:=
Xg
0
+
t
➓
Ñ
×✴❮ ➚✢Ñ
ÏÙ➔â×✩➑◗➹
I
0
−
:=
Z
d
0
g
0
1
{X
s
<X
0
}
ds
=
Z
V
0
1
{Z
t
<Z
−
g0
}
dt.
❭✂Õ
➧
➹
➚✥➮➾➚
Ï✛➔Ü❰✤➔ ➚✥Ñ✒÷❸✱✢á
➔
➪
➑
Ñ
V
❰✛ß❹➑➲➹✤➓Ñ
×
➚
➒
➪
➓✶➹✤➔➱➓✢➶❸→☎➑
−
g
0
➔âÏ ❐❹Ñ
➔
à❽➚
➹✛➒✒→➱Õ☞×❸➔➱ÏÙ❰✛➹✤➔☎➶
❐
❰✛➑➺× ➚✥Ñ
(0, V
)
➶❐
❰
➚
❰✛ß
❞
➑◗➹✤Ö✉➔âÏÙ➑✂➔
Ñ
×✩➑
➮
➑
Ñ
×✩➑
Ñ
❰
➚✶à
Z.
Ø ➚✢ÑÏ✛➑✦ê
❐
➑
Ñ
❰✛→➱Õ
✱
❮
➚
➒④➶❸➔
Ñ
➔
Ñ❸á
❰✤ß❸➔➱Ï➁Ö✉➔☎❰✛ß☞❰✤ß❸➑✚➹✤➑✦Ï
❐
→☎❰⑦➔
Ñ
Ø
➚
➹
➚
→➱→➱➓✢➹✛Õ
✒
❮
➚✥Ñ
❮✟→
❐
×✩➑➺Ï
❰✛ß❹➑
➮
➹
❰✛➑ ❮◗❮ ➓✏❰✤➔ ❰✛➔➱➒✒➑✦Ï Ï✇❰✝➓✏❰✛➔ ➓✶➹✤Õ ➹ ❮◗➑✦Ï✤ÏÙ➑➺Ï
✁
ö✄✂➘ô❯♠➭ö◗ò❁ï õ❆❥✁♠
❤
✞✪♦✺♣ ✠ ④❊⑧✪⑧❷❶ ⑧❷❶☛✡ ✉✗✈❏⑧❷④❊✈ ✦ ♣✆☎✗②
❸
s❊✉✗✈✞✝❄✈✣★✗r✥✦
❸
r✠✟☛✡
❸
✜ ❶❙⑧⑩④☞✦ ⑥✥✪
❸
ì
➑
➮
➹
➚✏➪
➑
é
ß❸➑
➚
➹✤➑◗➒ ✸✔à❽➚
➹✉➓✑Ï✇❰✝➓✏❰✤➔ ➚✥Ñ
➓✶➹✤Õ❅×✩➔ÜÛ
❐
Ï✛➔ ➚✥Ñ
X
=
{
Xs
:
s
∈
R}
→☎➔➪
➔
Ñ❹á
➔
Ñ
➓
Ñ
➔
Ñ
❰✤➑◗➹
➪
➓✶→
[0, r)
➚➹
[0, r]
Ö✉ß❸➑✦➹✛➑✮
➔âÏ⑨➓①➹✛➑▼❬❹➑✦❮✟❰✛➔ Ñ❸á
➶
➚✢❐❸Ñ
×❹➓✶➹✤Õ✾➓
Ñ
×❁➔
Ñ
❰✛ß❸➑✚❮✦➓✢Ï✛➑ ➚✶à
❰✤ß❸➑✂ß❹➓✶→ à❹➚✥➮
➑
Ñ
➔
Ñ
❰✛➑✦➹
➪
➓✶→
r
➔âÏ❢➑✦➔Ü❰✤ß❸➑◗➹Ñ
➓✏❰
❐
➹✝➓✶→
➚
➹✔➑
Ñ
❰✤➹✝➓
Ñ
❮✟➑ ❞↔Ñ❹➚
❰
❞
➑✟å✩➔☎❰①➓
Ñ
×❇➔
Ñ
❰✛ß❸➑
➚
❰✤ß❸➑◗➹æ❮◗➓✥ÏÙ➑
r
➔➱Ï✔➹✛➑❁❬➆➑✦❮✲❰✤➔ Ñ❹á❹è➲t❰æ➔➱Ï✔➓✢→➱Ï
➚
➓✢Ï✤Ï
❐
➒✑➑➺×✴❰✛ß➆➓✏❰
D
=
{
0
}
➔Ñ
✪Ù✸➺✰✲✱
➔
è
➑
è☎✱
M
=
{
t
:
Xt
= 0
}
.
é
ß❸➑➲❮◗➓✢Ï✛➑✦Ï⑦Ö✉ß❸➑
Ñ
❰✛ß❸➑➲ÏÙ❰✤➓✶❰✛➑✉Ï
➮
➓✢❮◗➑ ➚✶à
X
➔➱Ï⑦❰✛ß❹➑✚Ö✉ß➚
→➱➑
R
➚
➹
D
➔➱Ï✍➓Ñ
➔
Ñ
❰✛➑✦➹
➪
➓✶→❸❮◗➓
Ñ
➶➾➑✚❰✛➹✤➑✦➓✏❰✤➑✦×✑Ï✛➔☎➒✒➔➱→➱➓✢➹✛→➱Õ
è
é
ß❸➑
á
➑
Ñ
➑◗➹✝➓✏❰
➚
➹
➚✶à
X
➔âÏ✉×✩➑Ñ❸➚
❰✤➑✦×ã➶➏Õ
G
=
d
dm
d
dS
,
Ö✉ß❸➑✦➹✛➑
S
➔âÏ✂❰✤ß❸➑❁Ï✛❮✦➓✶→➱➑ à❽❐❸Ñ❮✲❰✛➔ ➚✢Ñ
➓
Ñ
×
m
➔➱Ï✚❰✛ß❸➑❁Ï➮
➑◗➑➺×ã➒✒➑✦➓✢Ï
❐
➹✤➑ è⑦ì
➑❁➓✥Ï✛Ï
❐
➒✒➑✔❰✤ß❹➓✏❰
m(dx) =
m(x)
dx
and
S(x) =
Z
x
0
S
′
(y)
dy
Ö✉➔☎❰✛ß❇❮ ➚✥Ñ
❰✤➔ Ñ➏❐❸➚✢❐
Ï
m(x)
➓Ñ
×
S
′
(x).
s➑➺❮◗➓✶→➱→✹❰✤ß❹➓✏❰✉❰✤ß❸➑❁Ï✇❰✝➓✏❰✤➔ ➚✥Ñ
➓✶➹✤Õ●×✩➔➱ÏÙ❰✛➹✤➔➱➶
❐
❰✛➔ ➚✢Ñ✘➚✶à
X
➔➱Ïá
➔
➪
➑
Ñ
➶➞Õ
µ(dx) :=
m(dx)/m(E)
Ö✉➔☎❰✛ß
m(E)
<
∞
.
ä ➔☎åy
∈
E
➓Ñ
×ã➔
Ñ
❰✛➹
➚
×
❐
❮✟➑
u(x) :=
E
x
Ã
exp
³
−
α
Z
H
0
0
1
{
0
≤X
s
≤y}ds
−
β
Z
H
0
0
1
{X
s
>y}ds
´
!
,
Ö✉ß❸➑✦➹✛➑
E
x
×✩➑ Ñ❸➚
❰✛➑✾❰✛ß❸➑❙➑◗å
➮
➑➺❮✲❰✝➓✏❰✛➔ ➚✢Ñ
➓✢Ï✤Ï
➚
❮✟➔â➓✏❰✛➑➺×●Ö✉➔☎❰✛ß
X
á➔
➪
➑
Ñ
❰✛ß❹➓✶❰
X
0
=
x
➓
Ñ
×
H
0
:= inf
{
t >
0 :
Xt
= 0
}
.
ä
➹
➚
➒❝❰✤ß❸➑
ä
➑◗Õ
Ñ
➒❅➓ Ñ❊❞❷②
➓✥❮ à❽➚
➹✤➒
❐
→â➓✘➔Ü❰ à❽➚
→➱→
➚
Ö➲Ï✾❰✤ß❹➓✏❰
u(x), x >
0,
➔âÏ❙❰✤ß❸➑ ❐❸Ñ➔âê
❐
➑●➶ ➚✥❐❸Ñ
×✩➑➺× ÏÙ➒ ➚➏➚
❰✛ß
Ï
➚
→
❐
❰✛➔ ➚✢Ñ✴➚✶à
❰✤ß❸➑
á
➑
Ñ
➑◗➹✝➓✶→➱➔☎í✦➑✦×ã×✩➔☎Û➾➑✦➹✛➑
Ñ
❰✛➔â➓✶→Þ➑✦ê
❐
➓✶❰✛➔ ➚✢Ñ
G
u(x) =
½
α u(x),
0
< x < y,
β u(x),
x > y
Ï✤➓✏❰✤➔➱Ï
à
Õ➏➔ Ñ❸á
❰✛ß❸➑❁❮ ➚✢Ñ
×❸➔Ü❰✤➔ ➚✥Ñ
u(0) = 1
èä
➚
➹
x
=
y
Ö✼➑❙ß❹➓➪
➑
u(y) =
ψ
+
α
(0)
ϕβ(y)
ψ
α
+
(y)
ϕβ(y)
−
ψα(y)
ϕ
+
β
(y)
,
Ö✉ß❸➑✦➹✛➑
ψα
➓Ñ
×
ϕα
➓✶➹✤➑✔❰✛ß❸➑❙➔Ñ
❮✟➹✤➑✦➓✥ÏÙ➔ Ñ❸á
➓
Ñ
×❃❰✛ß❸➑❁×✩➑➺❮✟➹✤➑✦➓✢Ï✛➔ Ñ❸á④à❽❐❸Ñ
×❸➓✶➒✒➑
Ñ
❰✤➓✶→❢Ï
➚
→
❐
❰✤➔ ➚✢ÑÞ✱
➹✤➑✦Ï
➮
➑✦❮✲❰✤➔
➪
➑◗→➱Õ ✱✩➚✶à
❰✤ß❸➑❙➑✦ê
❐
➓✏❰✤➔ ➚✥Ñ
G
u(x) =
αu(x),
x >
0.
✪✇✸➺÷✥✰ä
➚
➹
ψα
❰✛ß❸➑❃ë➞➔➱→➱→☎➔ Ñ❸á❮
➚✢Ñ
×✩➔Ü❰✤➔ ➚✥Ñ
ψα(0+) = 0
➒❐
Ï✇❰☞➶➾➑●➔☎➒ ➮➾➚
Ï✛➑✦× è✴é
ß❹➑ Ñ❸➚
❰✝➓✏❰✤➔ ➚✥Ñ
ϕ
+
β
,
à❽➚
➹❁➔
Ñ
ÏÙ❰✤➓
Ñ
❮✟➑
✱
➒✒➑✦➓
Ñ
Ï✂❰✛ß❹➑❁×✩➑◗➹✤➔
➪
➓✏❰✛➔
➪
➑æÖ✉➔☎❰✛ß✘➹✤➑✦Ï
➮
➑✦❮✲❰✚❰
➚
❰✛ß❹➑❁Ï✛❮✦➓✶→➱➑ à❽❐❸Ñ
❮✟❰✛➔ ➚✢ÑÞèP✈
➑◗å➏❰ Ñ❹➚
❰✛➔ Ñ❸á
❰✤ß❹➓✏❰
d
dm
r(y) :=
d
dm
¡
ψ
+
α
(y)
ϕβ(y)
−
ψα(y)
ϕ
+
β
(y)
¢
→☎➑➺❮✲❰✤➹ ➔➱❮④Ø ➒✒➒ ➔â❮◗➓✶❰✛➔ Ï✚➔ ➹ ➶➆➓✶➶❸➔➱→☎➔☎❰✇Õ
➓
Ñ
×
❐
ÏÙ➔ Ñ❸á
❰✤ß❸➑✾❰✛➔➱➒✒➑❙➹✛➑
➪
➑◗➹✝ÏÙ➔➱➶❸➔➱→☎➔☎❰✇Õ ➚✶à
Ï✇❰✝➓✏❰✛➔ ➚✢Ñ
➓✶➹✤Õ●×✩➔ÜÛ
❐
Ï✛➔ ➚✥Ñ
Ï✉Ö✼➑✾ß❹➓
➪
➑
E
¡
exp(
−
αI
t
−
−
βI
+
t
)
¢
=
Z
E
E
¡
exp(
−
αI
t
−
−
βI
t
+
)
|
Xt
=
y
¢
P
(Xt
∈
dy)
=
Z
E
¡
u(y)
¢
2
µ(dy)
=
¡
ψ
+
α
(0)
¢
2
α
−
β
Z
E
µ(dy)
ϕβ(y)
ψα(y)
d
dm
µ
−
1
r(y)
¶
=
1
m(E) (α
−
β
)
Ã
ϕ
+
β
(0)
ϕβ(0)
−
ϕ
+
α
(0)
ϕα(0)
!
=
1
m(E) (α
−
β
)
µ
1
Gα(0,
0)
−
1
Gβ(0,
0)
¶
,
✪✇✸✦✵➞✰Ö✉ß❸➑✦➹✛➑
✱
➔
Ñ
❰✛ß❸➑
Ñ
➑◗å➞❰➁❰
➚
❰✛ß❸➑✉→➱➓✥Ï✇❰✍ÏÙ❰✛➑ ➮Þ✱
Ö✼➑✚ß➆➓
➪
➑✚➔
Ñ
❰✛➑
á
➹✝➓✏❰✛➑➺×④➶➞Õ
➮
➓✢➹Ù❰✝Ï⑦➓
Ñ
×
Gα(0,
0)
×✩➑ Ñ❸➚❰✤➑✦Ï⑨❰✤ß❸➑
Ó
➹✛➑✦➑
Ñ
ë✢➑◗➹
Ñ
➑✦→Þ➓✏❰
(0,
0)
à❽➚➹
X
✪✖à❽➚ ➹➲➒➚
➹✤➑æ➔ Ñ✩à❽➚
➹✤➒❅➓✏❰✛➔ ➚✥Ñ
➓✶➶ ➚✥❐
❰
Ó
➹✛➑✦➑
Ñ
ë✢➑◗➹
Ñ
➑✦→➱Ï✉ÏÙ➑✦➑ ❳➱✸
❩
✰✲è
t
❰⑨➔âÏ⑨Ï✛➑◗➑
Ñ
➔
Ñ
❰✛ß❸➑✚ÏÙ➔➱➒✒➔☎→â➓✶➹⑨Ö✂➓➺Õ
➚
➹❢➶➞Õ
❐
ÏÙ➔ Ñ❸á
❰✛ß❸➑➲Ø✼ß❹➓
➮
➒✒➓ Ñ✙❞❷②✔➚
→➱➒ ➚✢á✢➚
➹
➚➈➪
➑➺ê
❐
➓✏❰✤➔ ➚✥Ñ❃✪
Ï✛➑◗➑ ❳➬
❩
➧
➹
➚✢➮➾➚
Ï✛➔☎❰✛➔ ➚✢Ñ
÷❸è✵➞✰
❰✛ß❹➓✶❰➲❰✤ß❸➑✉❒
➚
➔
Ñ
❰
⑥
➓
➮
→â➓✢❮◗➑✾❰✛➹✝➓
Ñ
Ï
à❽➚
➹✤➒ ➚✶à
(t
−
gt, dt
−
t)
➔➱Ï①➓✢→➱Ï ➚❅á➔
➪
➑
Ñ
➶➏Õ❃❰✤ß❸➑❁➹✤➔
á
ß✥❰
❞
ß➆➓
Ñ
×✘Ï✛➔➱×✩➑ ➚✢à
(14)
èä
➹
➚
➒❴❰✛ß❸➑❁Ï
➮
➑✦❮◗➔➱➓✢→ à✖➚
➹✛➒
✪✇✸✦✵➞✰✼➚✢à
❰✛ß❸➑
⑥
➓
➮
→â➓✢❮◗➑✔❰✛➹✝➓
Ñ
Ï
à❽➚
➹✤➒ ➚✢à
(I
t
+
, I
−
t
)
➔☎❰ à❽➚
→➱→
➚
Ö➲Ï✂❰✛ß➆➓✏❰
(I
t
+
, I
t
−
)
d
= (U V,
(1
−
U
)V
),
Ö✉ß❸➑✦➹✛➑
V
=
I
t
+
+
I
t
−
➓
Ñ
×
U
➔âÏæ➓❐❸Ñ
➔
à❽➚
➹✛➒✒→➱Õ ➚✢Ñ
(0,
1)
×✩➔âÏ✇❰✤➹✛➔➱➶❐
❰✤➑✦×r➹✝➓
Ñ
×
➚
➒
➪
➓✢➹✛➔â➓✶➶❹→☎➑❁➔
Ñ
×✩➑
➮
➑
Ñ
×✩➑
Ñ
❰
➚✶à
V
✪ Ï✛➑◗➑❳➬
❩
➧
➹
➚✢➮➾➚
ÏÙ➔☎❰✛➔
➚✢Ñ✘✯✩è✒✢✰P➮
➹
➚✏➪
➔
Ñ❸á
❰✤ß❸➑❙→➱➓✶❰Ù❰✤➑◗➹➲ÏÙ❰✤➓✶❰✛➑◗➒✒➑
Ñ
❰
➚✶à
❰✤ß❸➑
é
ß❸➑
➚
➹✛➑✦➒
è
✞✪♦★♥ ✈ r✿③✆②⑨s ★ ⑧❷❶
(
−
g
0
, d
0
)
②✺r s✙✈ ④☞✦ ③✧⑧⑩❶
❸✂✁☎✄
☎ ★ ✦ ✈
❸
③✆⑥✿④❊✈
⑥
➑◗❰
X
➓Ñ
×
M
➶➾➑æ➓✢ÏP➔Ñã➩
➑➺❮✲❰✤➔ ➚✥Ñ❃÷❸è➱✸✢è
Ø✼→➱➑✦➓✶➹✤→➱Õ
✱
❰✛ß❹➑æ×✩➔➱ÏÙ❰✛➹✤➔➱➶
❐
❰✛➔ ➚✢Ñ❃➚✶à
(t
−
gt, dt
−
t)
✪➓
Ñ
×
➚✶à
(I
t
+
, I
−
t
)
✰
×
➚
➑➺Ï Ñ❸➚
❰✚×✩➑
➮
➑
Ñ
×
➚✢Ñ
t
✴➞❰✛ß❸➑✦➹✛➑ à❽➚➹✤➑
✱
❰
➚
ÏÙ➔➱➒
➮
→➱➔
à
Õ✒❰✤ß❸➑ Ñ❸➚
❰✤➓✏❰✤➔ ➚✢ÑÞ✱
Ö✼➑①❰✝➓✶ë✢➑
t
= 0.
⑥ ➑✟❰A
=
{
As
:
s
≥
0
}
➶➾➑④❰✛ß❸➑✑➹✛➔
á
ß➞❰
❞
❮
➚✢Ñ
❰✛➔ Ñ➞❐❸➚✥❐
Ï➲➔ Ñ➞➪
➑✦➹✤Ï✛➑ ➚✶à
❰✤ß❸➑☞→
➚
❮◗➓✢→Þ❰✤➔☎➒✒➑ ➚✢à
{
Xs
:
s
≥
0
}
➓✶❰ ✮■✪❰✤➓✢ë✢➑
Ñ
Ö✉➔Ü❰✤ß■➹✛➑➺Ï
➮
➑✦❮✟❰①❰
➚
❰✛ß❹➑❙Ï
➮
➑◗➑✦×ã➒✒➑✦➓✥Ï
❐
➹✛➑ ✰✲è➁q
Ï✉➔➱Ï✚Ö✂➑◗→➱→➨ë Ñ❸➚
Ö
ÑÞ✱
A
➔âÏ✉➓✒Ï❐
➶
➚
➹✤×❸➔
Ñ
➓✶❰
➚
➹✚➓
Ñ
×
❐❸Ñ
×❸➑◗➹✚❰✤ß❸➑❙➓✥Ï✛Ï
❐
➒
➮
❰✤➔ ➚✥Ñ
X
0
= 0
E
0
¡
exp(
−
αAs)
¢
= exp
µ
−
s
Z
∞
0
¡
1
−
e
−αt
¢
n
+
(dt)
¶
= exp
µ
−
s
Z
∞
0
α e
−αt
n
+
(t,
∞
)
dt
¶
,
✪✇✸➈✯✢✰Ö✉ß❸➑✦➹✛➑æ❰✛ß❹➑ ⑥ ❡◗➪
Õ●➒✒➑➺➓✢Ï
❐
➹✛➑
n
+
➔➱Ïá
➔
➪
➑
Ñ
➶➞Õ
✪
ÏÙ➑✦➑ ❳✬
❩
➮➭èP✬✩✸◗✵➏✰
n
+
(dt) =
d
dS(x)
P
x(H
0
∈
dt)
¯
¯
x
=0+
▼ ✝✟☎☞☛⑩☎✍✌ ❏✎✄✦❏☎ ❑✝✆ ✡✟✞✴✓✖✞✖❼❇✞✖❼❹✄❁✕✹✌✏✞✡✣✏✞↔✓▲✌✏✕➘✣✏❾④✣✢✜✝✌✏➇✏✄✲❿
P
(
−
g
0
∈
dt) =
P
(d
0
∈
dt) =
n
+
(t,
∞
)
m(E)
dt,
✪✇✸
Ò
❰✛➑ ❮◗❮ ➓✏❰✤➔ ❰✛➔➱➒✒➑✦Ï Ï✇❰✝➓✏❰✛➔ ➓✶➹✤Õ ➹ ❮◗➑✦Ï✤ÏÙ➑➺Ï
P
(V
∈
dv) =
v
m(E)
n
+
(dv)
✠➁✓✖✞❽❼V
:=
d
0
−
g
0
;
✪✇✸✳✒✶✰
P
(d
0
∈
dt,
−
g
0
∈
ds)/dt ds
=
−
1
m(E)
d
dv
n
+
(v,
∞
)
¯
¯
¯
v
=
t
+
s
.
✪✇✸➺➬✥✰
☛✍✌✏✕❹❾◗✄✁
➽
✄✟✕➆✞✭✁➉➈❿✄✂✢✓✖➇➈✄◗✕
V
✞❽❼❸✄✃✙✛✣✶✕➾➄✥✌✏✎ ➇➈✣✶✙✲✓▲✣✢✜◗✁☎✄−
g
0
☎
✣✏✕➾➄◆✣✏✁❾◗✌
d
0
✆ ✓â❾➽
✕❹✓❺✟✌✏✙✲✎✑✁➉◆➄✏✓â❾✲✞✭✙✲✓▲✜
➽
✞✡✄✤➄◆✌✏✕
(0, V
).
✗✂✙✛✌✦✌Ù❺✞✝
ä
➚
➹✤➒
❐
→â➓ ✪Ù✸
Ò
✰
➔âÏ
➚
➶✩❰✝➓✶➔
Ñ
➑✦×ã➶➞Õ❃➔ Ñ➞➪
➑✦➹Ù❰✤➔ Ñ❹á
❰✤ß❸➑❁❮
➚
➹✛➹✤➑✦Ï ➮➾➚✢Ñ
×✩➔ Ñ❸á✑⑥
➓
➮
→â➓✢❮◗➑✔❰✛➹✝➓
Ñ
Ï
à❽➚
➹✤➒ è⑨t✡Ñ
×❸➑◗➑✦×
✱
E
(exp(
−
αd
0
)) =
−
ϕ
+
α
(0)
m(E)
α ϕα(0)
,
✪✇✸➺✳✥✰
➓
Ñ
×
à
➹
➚
➒❭ß❸➑✦➹✛➑æ❰✛ß❹➑✾➔ Ñ➞➪
➑◗➹✝Ï✛➔ ➚✥Ñ
❮✦➓
Ñ
➶➾➑❙× ➚✢Ñ
➑❙➓✢Ï✚➔ Ñ ❳✬
❩
➮ÞèP✬❸✸➺✯❸✱
Ï✛➑◗➑❙➓✶→âÏ
➚❯❳➱✸✦÷❸✱➨✸◗✵
❩
è✍✈➲➚
❰✛➔â❮✟➑æ❰✤ß❹➓✏❰ à✖➚
➹✚❰✛ß❸➑
➹✤➔
á
ß✥❰✉ß❹➓
Ñ
×ã➑
Ñ
×
➮➾➚
➔
Ñ
❰
r
➚✶àI
➔☎❰➲ß➚
→â×❸Ï
lim
x→r
ϕ
+
α(x) = 0
Ï✛➔
Ñ
❮◗➑
r
➔âÏ✂➑◗➔☎❰✛ß❹➑◗➹Ñ
➓✏❰
❐
➹✤➓✢→
➚
➹✂➑
Ñ
❰✛➹✝➓
Ñ
❮◗➑ ❞❷Ñ❸➚
❰
❞
➑◗å✩➔Ü❰
➚
➹✂➹✛➑ á✢❐
→â➓✶➹✂➓
Ñ
×●➹✤➑❁❬❹➑✦❮✟❰✛➔ Ñ❸á❹è➁✈
➑◗å➏❰✉❮ ➚✥Ñ
ÏÙ➔â×✩➑◗➹ à❽➚
➹✛➒
❐
→â➓✶➑
✪Ù✸✗✒✶✰
➓
Ñ
×
✪✇✸➺➬✥✰✟è
❭✂➑✦❮✦➓
❐
Ï✛➑
(
−
g
0
, d
0
)
d
= (U V,
(1
−
U
)V
),
Ö✉ß❸➑✦➹✛➑
V
=
d
0
−
g
0
➓
Ñ
×
U
➔âÏ✉➓ ❐❸Ñ➔
à❽➚
➹✤➒✒→☎Õ ➚✥Ñ
(0,
1)
×✩➔âÏ✇❰✤➹✛➔➱➶❐
❰✤➑✦×ã➹✝➓
Ñ
×
➚
➒
➪
➓✶➹✤➔➱➓✢➶❸→☎➑✔➔
Ñ
×✩➑
➮
➑
Ñ
×✩➑
Ñ
❰
➚✢à
V
➔☎❰ à❽➚→➱→
➚
Ö➲Ï
✪
Ï✛➑◗➑ ❳➱✸✗✒
❩
➧
➹
➚✢➮➾➚
ÏÙ➔☎❰✛➔ ➚✢Ñ✃✬❸è✵➏✰
❰✤ß❹➓✏❰æ❰✛ß❹➑✑×✩➑
Ñ
Ï✛➔Ü❰✇Õ
fV
➚✶àV
➔âÏ➚
➶✩❰✤➓✢➔
Ñ
➑✦×
à
➹
➚
➒ ❰✛ß❹➑✑×❸➑
Ñ
Ï✛➔Ü❰✇Õ
fg
0
➚✶à
−
g
0
➶➏Õ❅❰✤ß❸➑❙➹
❐
→☎➑
fV
(v) =
v
d
dv
fg
0
(v)
Õ➏➔➱➑◗→â×✩➔
Ñ❹á❇✪✇✸✗✒✢✰✲è✍♣❇➚
➹✤➑ ➚✏➪
➑◗➹
✱
❰✛ß❹➑✂❒
➚
➔
Ñ
❰①×✩➑
Ñ
ÏÙ➔☎❰✇Õ
fg
0
,d
0
➚✢à
(
−
g
0
, d
0
)
➔➱Ï
á
➔
➪
➑
Ñ
➶➞Õ
fg
0
,d
0
(u, v) =
fV
(u
+
v)/(u
+
v)
➓
Ñ
×❃❰✛ß❸➔âÏ✉➔âÏ✚➑➺ê
❐
➔
➪
➓✢→➱➑
Ñ
❰✉Ö✉➔☎❰✛ß ✪Ù✸✦➬✥✰✟è
❀❀✂✆✠✕✔✙✝✗✖✠✟❏✡☞⑥
➑◗❰
b
X
×✩➑ Ñ❸➚❰✤➑❅❰✛ß❸➑ã×✩➔☎Û
❐
ÏÙ➔ ➚✢Ñ ➚
➶✩❰✤➓✢➔
Ñ
➑✦×
à
➹
➚
➒
{
Xs
:
s
≥
0
}
➶➏Õ✃ë➏➔➱→☎→➱➔Ñ❸á
➓✶❰❁❰✛ß❹➑❅Ð❹➹✝ÏÙ❰
ß❸➔☎❰Ù❰✤➔ Ñ❹á
❰✛➔➱➒✒➑ ➚✶àP✮❹✱
➓
Ñ
×
b
p(t;
x, y)
❰✛ß❹➑④❰✛➹✝➓Ñ
Ï✛➔Ü❰✤➔ ➚✢Ñ
×✩➑
Ñ
Ï✛➔Ü❰✇Õ
✪
Ö✉➔Ü❰✤ß■➹✛➑➺Ï
➮
➑➺❮✲❰①❰
➚
❰✛ß❸➑✒Ï
➮
➑✦➑✦×❇➒✒➑✦➓✥Ï
❐
➹✤➑ ✰✉➚✢à
b
X
è✍éß❸➑
Ñ ✪
Ï✛➑◗➑ ❳✬
❩
✱❸➮➭è✉✸➺✯✏✵➏✰
P
x(H
0
∈
dt)/dt
=
d
dS(y)
p(t;
b
x, y)
¯
¯
¯
y
=0+
=:
p
b
+
(t;
x,
0).
❴
➑
Ñ
❮✟➑
✱
Ö✼➑✍➒❅➓➈Õ✔×✩➑✦➹✛➔
➪
➑⑦❰✤ß❸➑P×✩➑
Ñ
ÏÙ➔☎❰✇Õ
fg
0
,d
0
➶➞Õ
➮
➹
➚
❮✟➑◗➑➺×✩➔ Ñ❹á
