▼♦♥t❡ ❈❛r❧♦ Pr✐❝✐♥❣
❲❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞✿ ◮ ❆s ❡①❡r❝✐s❡s ✇❡ ❤❛✈❡ ✇r✐tt❡♥ ❛ ❧❛r❣❡ ❛♠♦✉♥t ♦❢ ▼❆❚▲❆❇ st②❧❡ ❢✉♥❝t✐♦♥❛❧✐t② ✐♥ t❤❡ ✜❧❡ ♠❛t❧✐❜✳❝♣♣ ✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❝❛♥✿ ◮ ●❡♥❡r❛t❡ r❛♥❞♦♠ ♥✉♠❜❡rs ✇✐t❤ r❛♥❞❯♥✐❢♦r♠ ❛♥❞ r❛♥❞♥ ✳ ◮ ●❡♥❡r❛t❡ ♣❧♦ts ✇✐t❤ ♣❧♦t ❛♥❞ ❤✐st ✳ ◮ ❈♦♠♣✉t❡ st❛t✐st✐❝s ♦❢ ✈❡❝t♦rs ✇✐t❤ ♠✐♥ ✱ ♠❡❛♥ ✱ ♣r❝t✐❧❡ ❡t❝✳ ◮ ❲❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❤♦✇ t♦ ✇r✐t❡ s✐♠♣❧❡ ❝❧❛ss❡s✳ ❲❤❛t ✇❡ ✇✐❧❧ ❞♦ ♥♦✇✿ ◮ ❆❞❞ ❛ ❢✉♥❝t✐♦♥ t♦ ❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧ t♦ ❣❡♥❡r❛t❡ ♣r✐❝❡ ♣❛t❤s✳ ◮ ❚❡st ♦✉r ♣r✐❝❡ ♣❛t❤s ✉s✐♥❣ ♠❡❛♥ ❡t❝✳ ❛♥❞ ♣❧♦t t❤❡♠ ✉s✐♥❣ ♣❧♦t ✳ ◮ ❲r✐t❡ ❛ ❝❧❛ss ▼♦♥t❡❈❛r❧♦Pr✐❝❡r t❤❛t ✉s❡s ❛ ❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧ t♦ ❣❡♥❡r❛t❡ ♣r✐❝❡ ♣❛t❤s ❛♥❞ t❤❡♥ ✉s❡s r✐s❦✲♥❡✉tr❛❧ ♣r✐❝✐♥❣ t♦ ♣r✐❝❡ ❛ ❈❛❧❧❖♣t✐♦♥ ❜② ▼♦♥t❡ ❈❛r❧♦✳
●❡♥❡r❛t❡ ♣r✐❝❡ ♣❛t❤ s♣❡❝✐✜❝❛t✐♦♥ ❲❡ ✇✐s❤ t♦ ✇r✐t❡ ❛ ❢✉♥❝t✐♦♥ ❣❡♥❡r❛t❡Pr✐❝❡P❛t❤ ✇❤✐❝❤ t❛❦❡s ❛ ✜♥❛❧ ❞❛t❡ t♦❉❛t❡ ❛♥❞ ❛ ♥✉♠❜❡r ♦❢ st❡♣s ♥❙t❡♣s ❛♥❞ ❣❡♥❡r❛t❡s ❛ r❛♥❞♦♠ ❇❧❛❝❦✕❙❝❤♦❧❡s Pr✐❝❡ ♣❛t❤ ✇✐t❤ t❤❡ ❣✐✈❡♥ ♥✉♠❜❡r ♦❢ st❡♣s✳ ❝❧❛ss✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥④ ♣✉❜❧✐❝✿ ✥✥✥✥✳✳✳✥♦t❤❡r✥♠❡♠❜❡rs✥♦❢✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥✳✳✳ ✥✥✥✥st❞✿✿✈❡❝t♦r❁❞♦✉❜❧❡❃✥❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✮✥❝♦♥st❀ ⑥❀ ◆♦t❡ t❤❛t t❤❡ ❝❧❛ss ❞❡❝❧❛r❛t✐♦♥ ❡✛❡❝t✐✈❡❧② ❝♦♥t❛✐♥s t❤❡ s♣❡❝✐✜❝❛t✐♦♥✳ ■❢ ②♦✉ ❝❤♦♦s❡ ❣♦♦❞ ❢✉♥❝t✐♦♥ ❛♥❞ ✈❛r✐❛❜❧❡ ♥❛♠❡s✱ ②♦✉ ✇♦♥✬t ♥❡❡❞ t♦♦ ♠❛♥② ❝♦♠♠❡♥ts✳
●❡♥❡r❛t❡ r✐s❦✲♥❡✉tr❛❧ ♣r✐❝❡ ♣❛t❤ s♣❡❝✐✜❝❛t✐♦♥ ❲❡ ❛❧s♦ ✇❛♥t ❛ ❢✉♥❝t✐♦♥ ❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤ ✇❤✐❝❤ ❜❡❤❛✈❡s t❤❡ s❛♠❡✱ ❡①❝❡♣t ✐t ✉s❡s t❤❡ Q ✲♠❡❛s✉r❡ t♦ ❝♦♠♣✉t❡ t❤❡ ♣❛t❤✳ ❭❜❡❣✐♥④❝♣♣⑥ ❝❧❛ss✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥④ ♣✉❜❧✐❝✿ ✥✥✥✥✳✳✳✥♦t❤❡r✥♠❡♠❜❡rs✥♦❢✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥✳✳✳ ✥✥✥✥st❞✿✿✈❡❝t♦r❁❞♦✉❜❧❡❃✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✮✥❝♦♥st❀ ⑥❀ ❭❡♥❞④❝♣♣⑥
Pr✐✈❛t❡ ❤❡❧♣❡r ❢✉♥❝t✐♦♥ ❚♦ ✐♠♣❧❡♠❡♥t t❤❡s❡ ❢✉♥❝t✐♦♥s✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♣r✐✈❛t❡ ❢✉♥❝t✐♦♥ t❤❛t ❛❧❧♦✇s ②♦✉ t♦ ❝❤♦♦s❡ t❤❡ ❞r✐❢t ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♣r✐❝❡ ♣❛t❤✳ ❝❧❛ss✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥④ ✥✥✥✥✳✳✳✥♦t❤❡r✥♠❡♠❜❡rs✥♦❢✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥✳✳✳ ♣r✐✈❛t❡✿ ✥✥✥✥st❞✿✿✈❡❝t♦r❁❞♦✉❜❧❡❃✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❞r✐❢t✮✥❝♦♥st❀ ⑥❀ ❚❤✐s ❢✉♥❝t✐♦♥ ✐s ♣r✐✈❛t❡ ❜❡❝❛✉s❡ ✇❡✬✈❡ ♦♥❧② ❝r❡❛t❡❞ ✐t t♦ ♠❛❦❡ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡❛s✐❡r✳ ❯s❡rs ♦❢ t❤❡ ❝❧❛ss ❞♦♥✬t ♥❡❡❞ ✭♦r ❡✈❡♥ ✇❛♥t✮ t♦ ❦♥♦✇ ❛❜♦✉t ✐t✳
❆❧❣♦r✐t❤♠ ❢♦r ❇❧❛❝❦✕❙❝❤♦❧❡s ♣r✐❝❡ ♣❛t❤s ❆❧❣♦r✐t❤♠ ◮ ❉❡✜♥❡ δ t i = t i − t i − ✶ ◮ ❈❤♦♦s❡ ✐♥❞❡♣❡♥❞❡♥t✱ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ǫ i ✱ ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ✶ ✳ ◮ ❉❡✜♥❡ � µ − ✶ � � ✷ σ ✷ s t i = s t i − ✶ + δ t i + σ δ t i ǫ i ◮ ❉❡✜♥❡ S t i = ❡①♣ ( s t i ) ✳ ◮ S t i s✐♠✉❧❛t❡ t❤❡ st♦❝❦ ♣r✐❝❡ ❛t t❤❡ ❞❡s✐r❡❞ t✐♠❡s✳
■♠♣❧❡♠❡♥t t❤❡ ❤❡❧♣❡r ❢✉♥❝t✐♦♥ ✈❡❝t♦r❁❞♦✉❜❧❡❃✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✿✿❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✱ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❞r✐❢t✥✮✥❝♦♥st✥④ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥♣❛t❤✭♥❙t❡♣s✱✵✳✵✮❀ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥❡♣s✐❧♦♥✥❂✥r❛♥❞♥✭✥♥❙t❡♣s✥✮❀ ✥✥✥✥❞♦✉❜❧❡✥❞t✥❂✥✭t♦❉❛t❡✲❞❛t❡✮✴♥❙t❡♣s❀ ✥✥✥✥❞♦✉❜❧❡✥❛✥❂✥✭❞r✐❢t✥✲✥✈♦❧❛t✐❧✐t②✯✈♦❧❛t✐❧✐t②✯✵✳✺✮✯❞t❀ ✥✥✥✥❞♦✉❜❧❡✥❜✥❂✥✈♦❧❛t✐❧✐t②✯sqrt✭❞t✮❀ ✥✥✥✥❞♦✉❜❧❡✥❝✉rr❡♥t▲♦❣❙✥❂✥❧♦❣✭✥st♦❝❦Pr✐❝❡✥✮❀ ✥✥✥✥❢♦r✥✭✐♥t✥✐❂✵❀✥✐❁♥❙t❡♣s❀✥✐✰✰✮✥④ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❞▲♦❣❙✥❂✥❛✥✰✥❜✯❡♣s✐❧♦♥❬✐❪❀ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❧♦❣❙✥❂✥❝✉rr❡♥t▲♦❣❙✥✰✥❞▲♦❣❙❀ ✥✥✥✥✥✥✥✥♣❛t❤❬✐❪✥❂✥❡①♣✭✥❧♦❣❙✥✮❀ ✥✥✥✥✥✥✥✥❝✉rr❡♥t▲♦❣❙✥❂✥❧♦❣❙❀ ✥✥✥✥⑥ ✥✥✥✥r❡t✉r♥✥♣❛t❤❀ ⑥
■♠♣❧❡♠❡♥t t❤❡ ♣✉❜❧✐❝ ❢✉♥❝t✐♦♥s ✈❡❝t♦r❁❞♦✉❜❧❡❃✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✿✿❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✥✮✥❝♦♥st✥④ ✥✥✥✥r❡t✉r♥✥❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭✥t♦❉❛t❡✱✥♥❙t❡♣s✱✥❞r✐❢t✥✮❀ ⑥ ✈❡❝t♦r❁❞♦✉❜❧❡❃✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✿✿ ✥✥✥✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✥✮✥❝♦♥st✥④ ✥✥✥✥r❡t✉r♥✥❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥t♦❉❛t❡✱✥♥❙t❡♣s✱✥r✐s❦❋r❡❡❘❛t❡✥✮❀ ⑥ ◆♦t✐❝❡ t❤❛t ✇✐t❤ t❤✐s ❞❡s✐❣♥ ✇❡✬✈❡ ❛✈♦✐❞❡❞ ✇r✐t✐♥❣ t❤❡ s❛♠❡ ❝♦♠♣❧❡① ❝♦❞❡ t✇✐❝❡✳
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