Nonparametric drift estimation for diffusions from noisy data

Nonparametric drift estimation for diffusions from noisy

data

Emeline Schmisser

To cite this version:

Emeline Schmisser. Nonparametric drift estimation for diffusions from noisy data. Statistics

& Decisions, 2011, 28 (2), pp.119-150.

<10.1524/stnd.2011.1063>.

<hal-00419954>

HAL Id: hal-00419954

https://hal.archives-ouvertes.fr/hal-00419954

Submitted on 25 Sep 2009

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◆♦♥♣❛r❛♠❡tr✐❝ ❞r✐❢t ❡st✐♠❛t✐♦♥ ❢♦r ❞✐✛✉s✐♦♥s

❢r♦♠ ♥♦✐s② ❞❛t❛

❊♠❡❧✐♥❡ ❙❝❤♠✐ss❡r

✶ ■♥tr♦❞✉❝t✐♦♥

❙❡✈❡r❛❧ ♣❛♣❡rs ❞❡❛❧ ✇✐t❤ ♥♦♥♣❛r❛♠❡tr✐❝ ❞r✐❢t ❡st✐♠❛t✐♦♥ ❢♦r ♥♦♥ ♥♦✐s② ❞❛t❛✱ ✐✳❡✳ ❢♦r t❤❡ ❞✐r❡❝t ♦❜s❡r✈❛t✐♦♥ (Xkδ)✳ ✭s❡❡ ❡✳❣ ❍♦✛♠❛♥♥ ✭✶✾✾✾✮✱ ❈♦♠t❡ ❡t ❛❧✳ ✭✷✵✵✼✮✱ ❙❝❤♠✐ss❡r ✭✷✵✵✾✮ ❢♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❞✐✛✉s✐♦♥s✮✳ ❍♦✇❡✈❡r✱ ✐♥ ♣r❛t✐❝❡✱ ✐t ✐s ♦❢t❡♥ t❤❡ ❝❛s❡ t❤❛t ♦♥❡ ❝❛♥♥♦t ♦❜s❡r✈❡ ❡①❛❝t❧② (Xkδ)✳ ❚❤✐s ♠❛② ❜❡ ❞✉❡ t♦ ❡✐t❤❡r ♠❡❛s✉r❡♠❡♥ts ❞❡✈✐❝❡s ♦r t♦ ✇❤❛t ✐s ❝❛❧❧❡❞ ♠✐❝r♦str✉❝t✉r❡ ♥♦✐s❡ ❢♦r ✜♥❛♥❝✐❛❧ ❞❛t❛✳ ❋♦r ✐♥st❛♥❝❡✱ ❩❤❛♥❣ ❡t ❛❧✳ ✭✷✵✵✺✮ ❛♥❞ ❏❛❝♦❞ ❡t ❛❧✳ ✭✷✵✵✾✮ st✉❞② t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛t❡❞ ✈♦❧❛t✐❧✐t② R1 0 σ 2 tdt ✇✐t❤✐♥ ❛ ✜①❡❞✲❧❡♥❣t❤ t✐♠❡ ✐♥t❡r✈❛❧ ✭✐✳❡✳ M δM = 1✮✳ ■♥ t❤❡ s❛♠❡ ❝♦♥t❡①t✱ ●❧♦t❡r ❛♥❞ ❏❛❝♦❞ ✭✷✵✵✶✮ st✉❞② t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥t ❢r♦♠ ♥♦✐s② ❤✐❣❤✲ ❢r❡q✉❡♥❝② ❞❛t❛✳ ❲✐t❤ ❞❛t❛ ✇✐t❤✐♥ ♦❢ ✜①❡❞ t✐♠❡ ✐♥t❡r✈❛❧✱ ✐t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t t❤❡ ❞r✐❢t ❝❛♥♥♦t ❜❡ ❡st✐♠❛t❡❞✳ ❚❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝② ❞❛t❛ ❝♦♥t❡①t ❢♦r ❞✐✛✉s✐♦♥s ♦❜s❡r✈❡❞ ✇✐t❤ ♥♦✐s❡ r❡q✉✐r❡s s♣❡❝✐✜❝ t♦♦❧s t♦ r❡❞✉❝❡ t❤❡ ♥♦✐s❡ ❡✛❡❝t ✇❤✐❧❡ ❦❡❡♣✐♥❣ ❡♥♦✉❣❤ ✐♥❢♦r♠❛t✐♦♥ t♦ ❡st✐♠❛t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❞✐✛✉s✐♦♥✳ ■♥ ♦r❞❡r t♦ r❡❞✉❝❡ t❤❡ ♥♦✐s❡ ❡✛❡❝t✱ ✇❡ s♣❧✐t ❞❛t❛ ✐♥t♦ ❣r♦✉♣s ♦❢ ❡q✉❛❧ s✐③❡ p ❛♥❞ ❜✉✐❧❞ ❡♠♣✐r✐❝❛❧ ♠❡❛♥s ❛s ❢♦❧❧♦✇s✳ ❆ss✉♠✐♥❣ t❤❛t M = (n+ 2)p✱ ✇❡ s❡t N = np✱ ∆ =pδ❛♥❞ ❢♦rk= 0,1, . . . , n+ 2, ¯ Yk∆= ¯Xk∆+ ¯εk, ✭✸✮ ✇✐t❤ ¯ Xk∆= 1 p p X j=1 Xk∆+jδ ❛♥❞ ε¯k= 1 p p X j=1 εkp+j. ✭✹✮ ❚❤❡ ❣r♦✉♣ s✐③❡ p ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t ∆ = pδ ✐s s♠❛❧❧ ✇❤❡r❡❛s n∆ = N δ ✐s ❧❛r❣❡✳ ❚❤❡♥✱ ❜❛s❡❞ ♦♥ t❤❡ ♠❡❛♥s s❛♠♣❧❡ _Y¯_{k∆, k= 0,1, . . . , n+ 2✱ ✇❡ ❛♣♣❧②} t❤❡ ♠❡t❤♦❞ ♦❢ ❈♦♠t❡ ❡t ❛❧✳ ✭✷✵✵✼✮✳ ❋✐rst✱ ✇❡ ✜♥❞ ❛♥ ❛❞❡q✉❛t❡ r❡❣r❡ss✐♦♥✲t②♣❡ ❡q✉❛t✐♦♥✿ ¯ Y(k+1)∆−Y¯k∆ ∆ =b Y¯(k−1)∆ +♥♦✐s❡+r❡♠❛✐♥❞❡r. ✭❚❤❡ ❧❛❣ ✐s ❤❡r❡ t♦ ❛✈♦✐❞ ❝✉♠❜❡rs♦♠❡ ❝♦rr❡❧❛t✐♦♥s✳✮ ❚❤❡♥✱ ❛ ♣❡♥❛❧✐s❡❞ ❧❡❛st✲ sq✉❛r❡ ❛♣♣r♦❛❝❤ ✐s ✉s❡❞ t♦ ❜✉✐❧❞ ❛ ♥♦♥ ♣❛r❛♠❡tr✐❝ ❛❞❛♣t✐✈❡ ❡st✐♠❛t♦r ♦❢ b(.) ✭✇✐t❤♦✉t ❦♥♦✇❧❡❞❣❡ ♦❢ σ(.)✮✳ ❲❡ ✐♥tr♦❞✉❝❡ ❛ ❢❛♠✐❧② ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜✲ s♣❛❝❡s (Sm)♦❢ L2(A)❛♥❞ ❞❡✜♥❡ ❛ ❝♦❧❧❡❝t✐♦♥ ˆ_bm ♦❢ ❡st✐♠❛t♦rs ♦❢ bA =b✶A✳ ❚❤❡♥✱ ✐♥tr♦❞✉❝✐♥❣ ❛ ♣❡♥❛❧t②✱ ✇❡ s❡❧❡❝t✱ ❜② ❛ ❞❛t❛✲❞r✐✈❡♥ ♣r♦❝❡❞✉r❡✱ ❛♥ ❛❞❛♣t✐✈❡ ❡st✐♠❛t♦rˆ_{bmˆ ❛♠♦♥❣ t❤❡ ❝♦❧❧❡❝t✐♦♥✳ ❲❡ ♣r♦✈❡ t❤❛t t❤❡ ❛❞❛♣t✐✈❡ ❡st✐♠❛t♦r r✐s❦} ❛❝❤✐❡✈❡s t❤❡ ✉s✉❛❧ ♦♣t✐♠❛❧ ♥♦♥ ♣❛r❛♠❡tr✐❝ r❛t❡ ♦❢ t❤❡ ♥♦♥ ♥♦✐s② ❞❛t❛ ❝❛s❡ ✉♥✲ ❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ s✉❜✲●❛✉ss✐❛♥ ♥♦✐s❡s✱ ✇❤✐❝❤ ✐♥❝❧✉❞❡ ●❛✉ss✐❛♥ ♦r ❜♦✉♥❞❡❞ ♥♦✐s❡s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ ✇❡ s♣❡❝✐❢② t❤❡ ♠♦❞❡❧ ❛♥❞ t❤❡ ❛ss✉♠♣t✐♦♥s✳ ❙❡❝t✐♦♥ ✸ ❞❡s❝r✐❜❡s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡s✳ ■♥ ❙❡❝t✐♦♥ ✹✱ ✇❡ ♣r❡❝✐s❡ t❤❡ r❡❣r❡ss✐♦♥✲t②♣❡ ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❝♦❧❧❡❝t✐♦♥ˆ_bm ♦❢ ❡st✐♠❛t♦rs✳ ❚❤❡♦r❡♠ ✶ ❣✐✈❡s t❤❡ r✐s❦ ❜♦✉♥❞ ♦❢ ❛♥ ❡st✐♠❛t♦rˆ_{bm❢♦r ✜①❡❞m✳ ❚❤❡♦r❡♠ ✷ ❛♥❞ ✸ ❣✐✈❡ t❤❡ r✐s❦ ❜♦✉♥❞} ♦❢ t❤❡ ❛❞❛♣t✐✈❡ ❡st✐♠❛t♦rˆ_{bmˆ✳ ■♥ ❙❡❝t✐♦♥ ✺✱ ✇❡ ♣r♦♣♦s❡ ❡①❛♠♣❧❡s ♦❢ ♠♦❞❡❧s ✇✐t❤} ❞✐✛❡r❡♥t ♥♦✐s❡s ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✐♠♣❧❡♠❡♥t t❤❡ ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞ ♦♥ s✐♠✉❧❛t❡❞ ❞❛t❛✳ Pr♦♦❢s ❛r❡ ❣❛t❤❡r❡❞ ✐♥ ❙❡❝t✐♦♥ ✻ ❛♥❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐①✳ ✷

✷ ▼♦❞❡❧ ❛♥❞ ❛ss✉♠♣t✐♦♥s

❋♦r ✜①❡❞ p ❛♥❞ δ✱ t❤❡ ♣r♦❝❡ss❡s X¯k∆ ❛♥❞ Y¯k∆ ❞❡✜♥❡❞ ✐♥ ✭✸✮✲✭✹✮ ❛r❡ str✐❝t❧② st❛t✐♦♥❛r②✳ ❚❤❡✐r ✐♥✈❛r✐❛♥t ❞❡♥s✐t✐❡s ❛r❡ r❡s♣❡❝t✐✈❡❧② ❞❡♥♦t❡❞ π¯p,δ ❛♥❞ ˜ πp,δ✳ ❆ss✉♠♣t✐♦♥ ✻✳ ❚❤❡r❡ ❡①✐st ♣♦s✐t✐✈❡ ❝♦♥st❛♥tsπ¯0✱¯π1 ✐♥❞❡♣❡♥❞❡♥t ♦❢ p❛♥❞δ s✉❝❤ t❤❛t✿ ∀x∈R,π¯p,δ(x)≤π¯1 ❛♥❞ ∀x∈B,π¯p,δ(x)≥π¯0 ✇❤❡r❡B = [a0−1, a1+ 1]✳ ❲❡ ♣r♦✈❡ t❤✐s ❛ss✉♠♣t✐♦♥ ✉♥❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ♦♥b ❛♥❞σ✳ Pr♦♣♦s✐t✐♦♥ ✶✳ ■❢ ✭❛✮ t❤❡ ♣r♦❝❡ss(Xt)_t≥0 ✐s ❛♥ ❖r♥st❡✐♥✲❯❤❧❡♥❜❡❝❦ ♣r♦❝❡ss✳ ✭❜✮ t❤❡ ❢✉♥❝t✐♦♥sb❛♥❞σ❛r❡C3✱_|b|✱_|b′_|✱_|σ|✱_|σ′_|✱_|σ′′_|❛r❡ ❜♦✉♥❞❡❞ ❛♥❞ σ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❜② σ1>0✱ ❆ss✉♠♣t✐♦♥ ✻ ✐s s❛t✐s✜❡❞✳ ■♥ ❢❛❝t✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❜♦✉♥❞ t❤❡ ❢✉♥❝t✐♦♥ ˜π. ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ✇✐❧❧ ❜❡ ✈❡r② ✉s❡❢✉❧❧ ✐♥ ♣r♦✈❡s✿ Pr♦♣♦s✐t✐♦♥ ✷✳ ❯♥❞❡r ❆ss✉♠♣t✐♦♥s ✺ ❛♥❞ ✻✱ t❤❡r❡ ❡①✐st ♣♦s✐t✐✈❡ ❝♦♥st❛♥tsπ˜0 ❛♥❞π˜1 s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧δ✱ ✐❢p≥p0 ✭❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ t❤❡ ❧❛✇ ♦❢ε1✮✿ ∀x∈R,π˜p,δ(x)≤π˜1 ❛♥❞ ∀x∈A,π˜p,δ(x)≥˜π0.

✸ ❆♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡s

❚❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ Sm ✐s t❤❡♥Dm= 2m+r✳ ❚❤❡ ✐♥✜♥✐t❡ ♥♦r♠ ❛♥❞ t❤❡L2 ♥♦r♠ ❛r❡ ❝♦♥♥❡❝t❡❞✿ ❛❝❝♦r❞✐♥❣ t♦ ❙❝❤♠✐ss❡r ✭✷✵✵✾✮✱ Pr♦♣♦s✐t✐♦♥ ✸ ♣✳✻✱ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥tφ1 s✉❝❤ t❤❛t✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥t∈Sm✿ ktk2∞≤φ 2 1Dmktk2L2 ✭✶✶✮ ✇❤❡r❡ktk2∞= supt∈[0,1]|t(x)|❛♥❞ktk 2 L2 = R1 0 t2(x)dx✳ Pr♦♣♦s✐t✐♦♥ ✹✳ ❲❡ ❝❛♥ ❝♦♥str✉❝t ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s(ψλ)♦❢Sm s✉❝❤ t❤❛t✿ ∃φ2, ∀λ, ❝❛r❞({λ′, kψλψλ′k ∞6= 0})≤φ2. ❚❤✐s ♣r♦♣♦s✐t✐♦♥ ✇✐❧❧ ❜❡ ❧❛t❡r ♣r♦✈❡❞✳ ❆❝❝♦r❞✐♥❣ t♦ ▼❡②❡r ✭✶✾✾✵✮✱ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ❣❡♥❡r❛t❡ ❛ ♠✉❧t✐r❡s♦❧✉t✐♦♥ ❛♥❛❧②s✐s ♦❢L2_{(R)✱ ❛♥❞✱ ❛❝❝♦r❞✐♥❣ t♦ Pr♦♣♦✲} s✐t✐♦♥ ✹✱ ♣✳✺✵✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥t✐♥ ❛ ❇❡s♦✈ s♣❛❝❡Bα2,∞([0,1])✱ ✇✐t❤r≥α✱ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C s✉❝❤ t❤❛t t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ✭L2_{✮ ♦❢t ♦✈❡r} Sm ❞❡♥♦t❡❞ ❜②tm s❛t✐s✜❡s✿ kt−tmkL2 ≤2 −mα_{C. ✭✶✷✮}

✹ ❊st✐♠❛t✐♦♥

❛♥❞ ❞❡✜♥❡ t❤❡ ❡st✐♠❛t♦rˆ_bm❛s✿ ˆ_{bm= arg min} t∈Sm γn(t). ❘❡♠❛r❦ ✶✳ ❲❡ ❝❛♥ ❛❧✇❛②s ✜♥❞ ❛ ❢✉♥❝t✐♦♥ ♠✐♥✐♠✐s✐♥❣ γn✱ ❜✉t ✐t ♠❛② ❜❡ ♥♦t ✉♥✐q✉❡✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② v∗ t❤❡ tr❛♥s♣♦s❡ ♦❢ t❤❡ ✈❡❝t♦r v✳ ❙❡tt✐♥❣ T = (T∆, . . . , Tn∆)∗✱ t❤❡ r❛♥❞♦♠ ✈❡❝t♦r(ˆbm( ¯Y0), . . . ,ˆbm( ¯Y(n−1)∆))∗= Πm(T)✱ ✇❤❡r❡ Πm ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ ♣r♦❥❡❝t✐♦♥ ♦✈❡r t❤❡ s✉❜s♣❛❝❡ (t( ¯Y0), . . . , t( ¯Y(n−1)∆), t∈Sm ✱ ✐s ❛❧✇❛②s ✉♥✐q✉❡❧② ❞❡✜♥❡❞✳ ❚❤✐s ✐s ✇❤②✱ ❛s ✐♥ ❈♦♠t❡ ❡t ❛❧✳ ✭✷✵✵✼✮✱ ✇❡ ❝❤♦♦s❡ t❤❡ r✐s❦ ❢✉♥❝t✐♦♥ ❛s t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ ❛♥ ❡♠♣✐r✐❝❛❧ ♥♦r♠✿ R_{(ˆbm) =E} ˆbm−bA 2 n ✇❤❡r❡ktk2n= 1 n Pn k=1t2( ¯Y(k−1)∆)❛♥❞bA=b✶A✳

✹✳✶ ❘✐s❦ ♦❢ t❤❡ ♥♦♥✲❛❞❛♣t✐✈❡ ❡st✐♠❛t♦r

✹✳✷ ❖♣t✐♠✐s❛t✐♦♥ ♦❢ t❤❡ ❞✐♠❡♥s✐♦♥ s♣❛❝❡

✹✳✸ ❆❞❛♣t✐✈❡ ❡st✐♠❛t✐♦♥✿ s✉❜✲●❛✉ss✐❛♥ ❝❛s❡

❘❡♠❛r❦ ✸✳ ●❛✉ss✐❛♥ ❛♥❞ ❜♦✉♥❞❡❞ ✈❛r✐❛❜❧❡s s❛t✐s❢② t❤✐s ❛ss✉♠♣t✐♦♥✳ ❋♦r ✐♥✲ st❛♥❝❡✱ ❛ ✉♥✐❢♦r♠ ❧❛✇ ♦♥ [−υ, υ] ✐s s✉❜✲●❛✉ss✐❛♥ ♦❢ ♣❛r❛♠❡t❡r υ2_{✳ ▼♦r❡♦✈❡r✱} ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦❢ ❧❛✇ f(x)∝ x2k_e−x2 /(2σ2 )_{✱ ✇❤❡r❡k ✐s ❛♥ ✐♥t❡❣❡r✱ ✐s s✉❜✲} ●❛✉ss✐❛♥ ♦❢ ♣❛r❛♠❡t❡rυ2_{= (2k+ 1)σ}2_✳ ❚❤❡♦r❡♠ ✷✳ ❯♥❞❡r ❆ss✉♠♣t✐♦♥s ✶✲✽✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✈❡rs❛❧ ❝♦♥st❛♥t κs✉❝❤ t❤❛t✱ ✐❢ ✇❡ s❡t pen(m)≥ κ σ 2 0+υ2 Dm n∆ , ❢♦rp∼δ−1/2_{✱ t❤❡ r✐s❦ ♦❢ t❤❡ ❛❞❛♣t✐✈❡ ❡st✐♠❛t♦r ✐s ❜♦✉♥❞❡❞ ❜②} Ekˆbmˆ −bAk2n ≤C inf m∈Mn k bm−bAk2L2+pen(m) +C σ 2 0+τ2 N δ +C ′_δ1/2_. ❚❤❡ ♣❛r❛♠❡t❡rυ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❦♥♦✇♥✳ ❖♥ t❤❡ ❝♦♥tr❛r②✱σ2 0 ✐s ✉♥❦♥♦✇♥✱ ❜✉t ✇❡ ❝❛♥ r❡♣❧❛❝❡ ✐t ❜② ❛ r♦✉❣❤ ❡st✐♠❛t♦r ✭s❡❡ s❡❝t✐♦♥ ✺✳✷ ♦❢ ❈♦♠t❡ ❡t ❛❧✳ ✭✷✵✵✼✮✮✳ ❚❤❡ ❛❞❛♣t✐✈❡ ❡st✐♠❛t♦r ❛✉t♦♠❛t✐❝❛❧❧② r❡❛❧✐③❡s t❤❡ ❜✐❛s✲✈❛r✐❛♥❝❡ ❝♦♠✲ ♣r♦♠✐s❡✿ ✇❤❡♥❡✈❡r bA ❜❡❧♦♥❣s t♦ s♦♠❡ ❇❡s♦✈ ❜❛❧❧✱ ✐❢ r ≥ α✱ ˆbmˆ ❛❝❤✐❡✈❡s t❤❡ ♦♣t✐♠❛❧ ❝♦rr❡s♣♦♥❞✐♥❣ ♥♦♥♣❛r❛♠❡tr✐❝ r❛t❡✳

✹✳✹ ❆❞❛♣t✐✈❡ ❡st✐♠❛t✐♦♥✿ ❣❡♥❡r❛❧ ❝❛s❡

✺ ◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ♦♥ ❡①❛♠♣❧❡s

♥♦✐s❡ ❛r❡ ✈❡r② s✐♠✐❧❛r t♦ t❤❡ ♦t❤❡r r❡s✉❧ts✳ ❆❝t✉❛❧❧②✱ ❡✈❡♥ ✐❢ t❤❡ ✈❛r✐❛❜❧❡s (εk) ❛r❡ ♥♦t s✉❜✲●❛✉ss✐❛♥✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✱ ✈❛r✐❛❜❧❡s(¯εk)❛r❡ ♥❡❛r❧② ●❛✉ss✐❛♥✳ ■❢ t❤❡ ❞r✐❢t ✐s ❧✐♥❡❛r✱ ♠♦st ♦❢ t❤❡ t✐♠❡✱ t❤❡ ❡st✐♠❛t♦rs ❛r❡ ❧✐♥❡❛r✳ ❚❤❡ ❝♦♥tr❛r② ✐s ♥♦t tr✉❡ ✭s❡❡ ▼♦❞❡❧s ✷ ❛♥❞ ✸✮✳ ▼♦r❡♦✈❡r✱ t❤❡ s♠❛❧❧❡r ∆ =pδ ❛♥❞ (N δ)−1_✱ t❤❡ s♠❛❧❧❡r t❤❡ ❡rr♦r✱ ♥❡✈❡rt❤❡❧❡ss✱ t❤❡ r✐s❦ ✐s ♥♦t ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤✐s ✈❛r✐❛t✐♦♥✳ ❚❤✐s ❝❛♥ ❜❡ ❞✉❡ t♦ t❤❡ ❝♦♥st❛♥ts ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ r✐s❦✳ ❚❤❡ ❜❡st ♣♦ss✐❜❧❡ ❡st✐♠❛t♦rs ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♦❜t❛✐♥❡❞ ❢♦rN= 10−6_{✱δ= 10}−4_{✱ ❡✐t❤❡r} ❢♦rp= 100✱ ❡✐t❤❡r ❢♦rp= 103_✳

✻ Pr♦♦❢s

✻✳✶ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶

▲❡t ✉s ❞❡♥♦t❡ ❜② bm t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ✭L2✮ ♦❢bA ♦✈❡rSm ❛♥❞ s❡t✱ ❢♦r t∈Sm✱ νn(t) = 1 n n X k=1 Zk∆t( ¯Y(k−1)∆), ρn(t) = 1 n n X k=1 Rk∆t( ¯Y(k−1)∆), En(t) = 1 n n X k=1 (Ak∆+Bk∆+Ik∆)t( ¯Y(k−1)∆), ✭✶✽✮ ✭s❡❡ ✭✶✺✮✲ ✭✶✻✮✮✳ ❇② ❞❡✜♥✐t✐♦♥ ✱γn(ˆbm)−γn(b)≤γn(bm)−γn(b)✱ s♦✿ ˆbm−b 2 n≤ kbm−bk 2 n+ 2νn(ˆbm−bm) + 2ρn ˆ bm−bm + 2En ˆ bm−bm ❛♥❞✱ ❛sˆ_{bm❛♥❞ bm❛r❡A✲s✉♣♣♦rt❡❞✿} ˆbm−bA 2 n≤ kbm−bAk 2 n+ 2νn(ˆbm−bm) + 2ρn ˆ_bm−bm + 2En ˆ bm−bm . ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ♥♦r♠ktk2π˜= R At2(x)˜π(x)dx✇❤❡r❡π˜ = ˜πp,δ✐s t❤❡ ❞❡♥s✐t② ♦❢ ¯ Y∆✳ ❆s t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡sY¯k∆❤❛✈❡ ❞❡♥s✐t②π˜✱ ✇❡ ❤❛✈❡ t❤❛tE ktk2n =ktk2˜π✳ ▲❡t ✉s ✐♥tr♦❞✉❝❡ t❤❡ s❡t Ωn =    ω,∀t∈ [ m,m′ Sm+Sm′, ktk2n ktk2π˜ −1 ≤ 1₂    ✐♥ ✇❤✐❝❤ ♥♦r♠sk.kn ❛♥❞k.k˜π ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ♦♥Ωn✱ ✇❡ ❤❛✈❡ ktk2π˜ ≤2ktk 2 n≤3ktk 2 ˜ π. ✭✶✾✮ ■♥ ❛ ✜rst st❡♣✱ ✇❡ ✇✐❧❧ ❜♦✉♥❞ t❤❡ r✐s❦ ♦✈❡rΩn.▲❡t ✉s s❡tBm={t∈Sm,ktk˜π≤1}✳ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t②✱ ✇❡ ❤❛✈❡ t❤❛t✱ ♦♥ ♦♥❡ ❤❛♥❞✱ 2 (νn+ρn) (ˆbm−bm) ≤ 2 ˆ bm−bm _π˜ sup t∈B_m| νn(t) +ρn(t)| ≤ 1 14 ˆ bm−bm 2 ˜ π+ 28 supt∈B_m ν2 n(t) +ρ2n(t) ❛♥❞ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ 2En ˆ bm−bm ≤ 3 28 ˆ bm−bm 2 n+ 28 n n X k=1 A2 k∆+Bk∆2 +Ik∆2 . ❆s✱ ❛❝❝♦r❞✐♥❣ t♦ ✭✶✾✮✱ ♦♥Ωn✱ ˆ_bm−bm 2 ˜ π≤2 ˆ_bm−bm 2 n ❛♥❞ ˆ bm−bm 2 n ≤2 ˆ_bm−bA 2 n+2kbA−bmk 2 n. ❇② ❝♦❧❧❡❝t✐♥❣ t❡r♠s✱ ✇❡ ♦❜t❛✐♥✿ ˆbm−bA 2 n≤3kbm−bAk 2 n+56 sup t∈Bm νn2(t)+ 56 n n X k=1 A2k∆+Bk∆2 +Ik∆2 +R2k∆ . ✶✶

▼♦r❡♦✈❡r✱ Ekbm−bAk2_n =kbm−bAk2_π˜ ≤π˜1kbm−bAk2_L2✳ ❯s✐♥❣ ▲❡♠♠❛ ✷✱ ✇❡ ❣❡t✿ E ˆbm−bA 2 n✶Ωn ≤3˜π1kbm−bAk2_L2+56E sup t∈B_m νn2(t) +ρ2n(t) +cτ 2 p +c ′_∆. ■t r❡♠❛✐♥s t♦ ❜♦✉♥❞E supt∈B_mνn2(t) ❛♥❞E supt∈B_mνn2(t) ✳ ▲❡t(ϕλ, λ∈Λm) ❜❡ ❛♥ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❢♦r t❤❡ L2 ˜ π✲♥♦r♠ ♦❢Sm✳ ❆♥② ❢✉♥❝t✐♦♥t∈Sm❝❛♥ ❜❡ ✇r✐tt❡♥ t=P λ∈Λmaλϕλ❛♥❞ ktkπ˜ ≤1⇔ X λ∈Λm a2λ≤1. ❍❡♥❝❡✱ E sup t∈B_m νn2(t) =E " sup P λa 2 λ≤1 νn2 X λ aλϕλ !# ≤ X λ∈Λm E νn2(ϕλ). ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛rt③ ✐♥❡q✉❛❧✐t②✿ E ν2 n(ϕλ)≤ 1 pn2_∆2 p X j=1 E   n X k=1 ϕλ Y¯(k−1)∆ Z (k+1)∆+jδ k∆+jδ σ(Xs)dWs !2 . ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ✜❧tr❛t✐♦♥s F_{t=σ(η, Ws, s≤t)} ❛♥❞ G_{t=σ(η, Ws, s≤t, εj, jδ≤t).} ✭✷✵✮ ❋♦r ❛♥②k✱Y¯(k−1)∆ ✐s G(k−1)∆✲♠❡❛s✉r❛❜❧❡ ❛♥❞ _Rt 0σ(Xs)dWs, t≥0 ✐s ❛(G_t)✲ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ q✉❛❞r❛t✐❝ ✈❛r✐❛t✐♦♥ Rt 0σ2(Xs)ds✳ ❙♦ E νn2(ϕλ) ≤ 1 pn2_∆2 p X j=1 n X k=1 E  ϕ 2 λ Y¯(k−1)∆E   Z (k+1)∆+jδ k∆+jδ σ(Xs)dWs !2 G_kpδ     + 2 pn2_∆2 p X j=1 n X 1≤l<k≤n Eϕλ Y¯(k−1)∆ ϕλ Y¯(l−1)∆ Z (l+1)pδ+jδ lpδ+jδ σ(Xs)dWsE Z (k+1)∆+jδ k∆+jδ σ(Xs)dWs G_kpδ !# ≤ _pn21_∆2 p X j=1 n X k=1 E " ϕ2λ Y¯(k−1)∆ Z (k+1)∆+jδ k∆+jδ σ2(Xs)ds # . ❚❤❛♥❦s t♦ ❆ss✉♠♣t✐♦♥ ✸✱ ✇❡ ♦❜t❛✐♥ Eνn2(ϕλ)≤ σ 2 0 n∆E h kϕλk2_n i . ❆sEhkϕλk2n i =kϕλk2π˜= 1✱ ✇❡ ❤❛✈❡ t❤❛t E sup t∈B_m νn2(t) ≤σ02 Dm n∆. ✭✷✶✮ ✶✷

■♥ t❤❡ s❛♠❡ ✇❛②✱ ✇❡ ❤❛✈❡ t❤❛t E sup t∈B_m ρ2n(t) ≤ X λ∈Λm Eρ2n(ϕλ) ≤ 2 n2_∆2 X λ∈Λm E   X k even ϕλ( ¯Y(k−1)∆) (¯εk+1−ε¯k) !2  + 2 n2_∆2 X λ∈Λm E   X k odd ϕλ( ¯Y(k−1)∆) (¯εk+1−ε¯k) !2 . ❚❤❡ s♣❧✐tt✐♥❣ ✐♥t♦ ♦❞❞ ❛♥❞ ❡✈❡♥ ✐♥❞❡①❡s ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❞✈❛♥t❛❣❡✿ ❡❛❝❤ s✉♠ ✐s ❝♦♠♣♦s❡❞ ♦❢ ✉♥❝♦rr❡❧❛t❡❞ ✈❛r✐❛❜❧❡s ✇✐t❤ ✐❞❡♥t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✳ ▼♦r❡♦✈❡r✱ s✐♥❝❡_Y¯_{(k−1)∆= ¯X(k−1)∆+ ¯εk−1✱ t❤❛♥❦s t♦ t❤❡ ❧❛❣✱ϕλ( ¯Y(k−1)∆)✐s ✐♥❞❡♣❡♥❞❡♥t} ♦❢(¯εk+1−ε¯k)✳ ❲❡ ❝❛♥ ✇r✐t❡✿ E sup t∈Bm ρ2n(t) ≤ _n∆2₂ X λ∈Λm E ϕ2λ( ¯Y(k−1)∆)E h (¯εk+1−ε¯k)2 i ≤ 2Dmτ 2 np∆2. ❙♦ E ˆbm−bA 2 n✶Ωn ≤3˜π1kbm−bAk2L2+cσ2₀ Dm n∆ +cτ 2 Dm np∆2+ cτ2 p +c ′_∆. ✭✷✷✮ ■t r❡♠❛✐♥s t♦ ❜♦✉♥❞ t❤❡ r✐s❦ ♦✈❡r Ωc n✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ✇✐❧❧ ❜❡ ♣r♦✈❡❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐①✿ ▲❡♠♠❛ ✸✳ P(Ωcn)≤ 1 n2. ▲❡t ✉s s❡te= (e∆, . . . , en∆)∗✱ ✇❤❡r❡ek∆=Tk∆−b( ¯Y(k−1)∆)✇❤❡r❡ Tk∆ ✐s ❞❡✜♥❡❞ ❜② ✭✶✸✮ ❛♥❞ΠmT = Πm(T∆, . . . , Tn∆)∗ = ˆ bm( ¯Y0), . . . ,ˆbm( ¯Y(n−1)∆) ∗ ✇❤❡r❡Πm✐s t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ♦✈❡rSm✳ ❚❤❡♥ ˆbm−bA 2 n = kΠmT−bAk 2 n ≤2kbA−ΠmbAk 2 n+ 2kΠmbA−ΠmTk 2 n ≤ 2kbAk2n+ 2kΠmek2n≤2kbAk2n+ 2kek 2 n. ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t② ❛♥❞ ▲❡♠♠❛ ✸✱ EkbAk2n✶Ωc n ≤ E b4A( ¯Y0)P[Ωcn] 1/2 ≤ _nc q E b4 A( ¯Y0). ❆❝❝♦r❞✐♥❣ t♦ ❆ss✉♠♣t✐♦♥ ✻✱ E b4 A Y¯0=Rb4A(x)˜π(x)dx≤π˜1kbAk4_∞✳ ❯s✐♥❣ ✭✶✹✮✱ Ekek2n✶Ωc n ≤ c n E I4 k∆ +E A4 k∆ +E B4 k∆ +E R4 k∆ +E Z4 k∆ . ✶✸

❇② ▲❡♠♠❛ ✷✱ ✇❡ ♦❜t❛✐♥✿ E ˆ bm−bA 2 n✶Ω c n ≤ 1 n C+C∆ + C p + τ2 p∆2 + cσ2 0 ∆ . ❍❡♥❝❡✱ ❛s∆→0❛♥❞p→ ∞✿ E ˆbm−bA 2 n✶Ω c n ≤K 1 n∆+ 1 np∆2 . ✭✷✸✮ ❈♦❧❧❡❝t✐♥❣ t❡r♠s✱ ✇❡ ♦❜t❛✐♥ ✭s❡❡ ✭✷✷✮ ❛♥❞ ✭✷✸✮✮✿ E ˆbm−bA 2 n ≤ 3˜π1kbm−bAk2L2+cD_m σ2 0 N δ + τ2 N p2_δ2 + K 1 N δ + 1 N p2_δ2 +C pδ+1 p .

✻✳✷ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷

❆spen1(m)−pen1( ˆm) + 28p1(m,mˆ)≤2pen1(m)✱ ✐t r❡♠❛✐♥ t♦ ❜♦✉♥❞ t❤❡ t❡r♠ E " sup t∈B_m,mˆ νn2(t)−p1(m,mˆ) ! ✶Ωn # . ■♥ t❤❡ s❛♠❡ ✇❛②✱ ✇❡ ❤❛✈❡ t❤❛t F := E " 28 sup t∈B_m,mˆ ρ2n(t) + (pen2(m)−pen2( ˆm)) ! ✶Ωn # ✭✷✺✮ ≤ E " sup t∈B_m,mˆ ρ2n(t)−p2(m,mˆ) ! ✶Ωn # + 2pen2(m). ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ✇✐❧❧ ❜❡ ❧❛t❡r ♣r♦✈❡❞✿ ▲❡♠♠❛ ✹✳ ❯♥❞❡r ❆ss✉♠♣t✐♦♥s ✶✲✼✱ ✇❡ ❤❛✈❡✿ Pνn(t)≥η,ktk2n ≤ζ2 ≤2 exp −n∆ η 2 8σ2 0ζ2 . ❚❤❡ ❢✉♥❝t✐♦♥ νn(t)✐s ❞❡✜♥❡❞ ❜② ✭✶✽✮✳ ▲❡♠♠❛ ✺✳ ■❢ ❆ss✉♠♣t✐♦♥s ✶✲✽ ❛r❡ s❛t✐s✜❡❞✱ t❤❡♥ Pρn(t)≥η,ktk2n≤ζ2 ≤2 exp −np∆2 η2 8υ2_ζ2 ✇❤❡r❡ρn(t) ✐s ❞❡✜♥❡❞ ❜② ✭✶✽✮✳ ▲❡♠♠❛ ✻✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ❢✉♥❝t✐♦♥ fn :Sm,m′ →R✱ f_n(t) =P n k=1akt Y¯(k−1)∆ ✱ ✇✐t❤ (ak, k= 1, . . . , n)r❡❛❧ ♥✉♠❜❡rs✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t P|fn(t)| ≥η,ktk2n≤ζ 2 ≤Kexp −nη 2 aζ2 . ❚❤❡♥✱ ❞❡♥♦t✐♥❣Dm,m′ =D m+Dm′✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t❛♥t κ >0 s✉❝❤ t❤❛t E " sup t∈B_m,m′ fn2(t)− κaDm,m′ n # + ≤5.2Kκae −Dm,m′ n . ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② ❢✉♥❝t✐♦♥fn :Sm+m′ →R " sup t∈B_m,mˆ fn2(t)−κa Dm+Dmˆ n # + ≤ X m′_∈M n " sup t∈B_m,m′ fn2(t)−κa Dm+Dm′ n # + . ❆♣♣❧②✐♥❣ t❤❡ ▲❡♠♠❛s✱ ❛sP me−Dm <1✱ ✇❡ ❣❡t t❤❛t t❤❡r❡ ❡①✐st t✇♦ ❝♦♥st❛♥ts κ1❛♥❞ κ2 s✉❝❤ t❤❛t E " sup t∈B_m,mˆ νn2(t)−p1(m,mˆ) # + ≤10.4κ1σ 2 0 n∆ ✭✷✻✮ ✶✺

❛♥❞ E " sup t∈B_m,mˆ ρ2n(t)−p2(m,mˆ) # + ≤10.4κ2υ 2 n∆ . ❙♦✱ ❛❝❝♦r❞✐♥❣ t♦ ✭✷✹✮ ❛♥❞ ✭✷✺✮✱ ✇❡ ❤❛✈❡ t❤❛t E≤300κ1 σ2 0 n∆ + 2pen1(m) ❛♥❞ F ≤300κ2 υ2 np∆2 + 2pen2(m) ❋♦r t❤❡ st✉❞② ♦♥Ωc n✱ t❤❡ ❡♥❞ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶ ❝❛♥ ❜❡ ✉s❡❞✳ ❈♦❧❧❡❝t✐♥❣ t❡r♠s✱ ✇❡ ♦❜t❛✐♥✿ Ekˆbmˆ −bAk2n ≤ inf m∈M_n 3˜π1kbm−bAk 2 L2+ 4pen(m) + C n∆+ C′ np∆2+C 2_∆+C3 p . ❙❡tt✐♥❣ p=δ−1/2_{✱ ✇❡ ♦❜t❛✐♥ t❤❡ r❡s✉❧t✳} ✻✳✷✳✶ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✹ ▲❡t ✉s ❞❡♥♦t❡ Hs(1)= X k even, k≤n t Y¯(k−1)∆ ✶]k∆,(k+2)∆](s)   1 p p X j=1 ✶]k∆+jδ,(k+1)∆+jδ](s)σ(Xs)   ❛♥❞ Hs(2)= X k odd, k≤n t Y¯(k−1)∆ ✶]k∆,(k+2)∆](s)   1 p p X j=1 ✶]k∆+jδ,(k+1)∆+jδ](s)σ(Xs)   ❚❤❡ ♣r♦❝❡ss❡s Mt(1)= Z t 0 Hs(1)dWs, Mt(2) = Z t 0 Hs(2)dWs ❛r❡ t✇♦(G_t)✲♠❛rt✐♥❣❛❧❡s✳ ❲❡ ❤❛✈❡ t❤❛t M_(n+2)∆(1) = X k even, k≤n t Y¯(k−1)∆   1 p p X j=1 Z (k+1)∆+jδ k∆+jδ σ(Xs)dWs   ❛♥❞ _D M(1)E (n+2)∆≤ X k even, k≤n σ20∆t2 Y¯(k−1)∆ . ❲❡ ♦❜t❛✐♥ s✐♠✐❧❛r r❡s✉❧ts ❢♦r M_(n+2)∆(2) ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② t ≥ 0✱ i ∈ {1,2}✱ ✇❡ ❤❛✈❡ t❤❛t M(i) t ≤ σ 2 0n∆ktk 2 n✱ s♦ exp λMt(i)−λ2 M(i) t/2 ❛r❡ G_t✲ ♠❛rt✐♥❣❛❧❡s ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✜❧tr❛t✐♦♥G_t✳ ❚❤❡r❡❢♦r❡✱ ❢♦r ❛♥②_{λ >0}✱_{η, ζ >0}✱ i∈ {1,2}✿ PMs(i)≥η, D M(i)E s≤ζ ≤ P exp λMs(i)− λ2 2 D M(i)E s ≥exp λη−λ 2 2 ζ ≤ exp −λη+λ 2 2 ζ . ✶✻

▼✐♥✐♠✐s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦λ✱ ✇❡ ♦❜t❛✐♥ t❤❛t PMs(i)≥η, D M(i)E s≤ζ ≤exp −η 2 2ζ . ❆s Pνn(t)≥η,ktk2n≤ζ ≤ X i=1,2 P M_(n+2)∆(i) ≥n∆₂η,DM(i)E (n+2)∆≤σ 2 0n∆ζ , ✇❡ ❞❡❞✉❝❡ t❤❛t Pνn(t)≥η,ktk2n ≤ζ 2 ≤2 exp −n∆ η 2 8σ2 0ζ2 . ✻✳✷✳✷ Pr♦♦❢ ♦❢ t❤❡♦r❡♠ ✺ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ t✇♦ ♠❛rt✐♥❣❛❧❡s N_k(1)= k X l=1 t Y¯(l−1)∆ ¯ εl N_k(2)= k X l=1 t Y¯(l−1)∆ ¯ εl+1. ❲❡ ❝❛♥ ♥♦t✐❝❡ t❤❛t N_k(1) ✐s ❛ G_(k+1)∆✲♠❛rt✐♥❣❛❧❡ ✇❤❡r❡❛s _N(2) k ✐s ❛ G(k+2)∆✲ ♠❛rt✐♥❣❛❧❡✳ ▼♦r❡♦✈❡r E eλ¯ε1₌ Eeλε0/p p ≤exp _λ2_υ2 2p , s♦✱ ❢♦r i∈ {1,2}✱ ✇❡ ❤❛✈❡ Eexp(λN_k+1(i) ) Gk∆

≤expλN_k(i)exp

λ2_υ2_t2_{( ¯Y} k∆) 2p . ✭✷✼✮ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❚❝❤❡❜✐t❝❤❡✈ ✐♥❡q✉❛❧✐t②✱ ❢♦r ❛♥②λ >0✱ PNn(i)≥cn,ktk 2 n≤c ′ ≤ P " exp λNn(i)− λ2_υ2_nktk2 n 2p ! ≥exp λcn−λ 2_υ2_c′_n 2p # ≤ exp −λcn+λ 2_υ2_c′_n 2p E " exp λNn(i)− λ2υ2nktk2n 2p !# . ❆❝❝♦r❞✐♥❣ t♦ ✭✷✼✮✱ ✇❡ ❤❛✈❡ t❤❛t E " exp λNn(i)− λ2υ2nktk2n 2p !# ≤ E exp −λ 2_υ2_n ktk2n 2p ! EhexpλNn(i) Gn∆ i ! ≤ E exp λN_n(i)₋₁−λ 2_υ2 2p n−1 X k=1 t2 _Y¯ (k−1)∆ !! . ❲❡ ❞❡❞✉❝❡ ❜② ✐♥❞✉❝t✐♦♥ t❤❛t EexpλNn(i)−λ2υ2nktk2_n/(2p) ≤ 1✱ t❤❡r❡✲ ❢♦r❡✱ PN(i) n ≥cn,ktk 2 n≤c′ ≤exp −λcn+λ 2_υ2_c′_n 2p . ✶✼

▼✐♥✐♠✐s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦λ✱ ✇❡ ♦❜t❛✐♥ t❤❛t PNn(i)≥cn,ktk 2 n ≤c ′ ≤exp −npc 2 2υ2_c′ . ❆s Pρn(t)≥η,ktk2n≤ζ2 ≤ X i=1,2 P N(i) n ≥ ∆nη 2 ,ktk 2 n≤ζ2 , ✇❡ ♦❜t❛✐♥ t❤❛t P ρn(t)≥η,ktk2n≤ζ2 ≤2 exp −np∆2 η 2 8υ2_ζ2 . ✻✳✷✳✸ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✻ ❲❡ ❤❛✈❡ t❤❛t E " sup t∈B_m,m′ fn2(t)− κaD n # + = Z ∞ 0 P " sup t∈B_m,m′ fn2(t)− κaD n ≥x # dx = κa Z ∞ 0 P " sup t∈B_m,m′ f2 n(t)≥ κa(y+D) n # dy. ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ♣r♦♦❢ ✐s ❞♦♥❡ ✐♥ ❇❛r❛✉❞ ❡t ❛❧✳ ✭✷✵✵✶✮ ❜② ❛ L2_{✲❝❤❛✐♥✐♥❣} t❡❝❤♥✐q✉❡ ✭s❡❡ t❤❡✐r ❙❡❝t✐♦♥ ✼✱ ♣✳ ✹✹✲✹✼✱ ▲❡♠♠❛ ✼✳✶✱ ✇✐t❤s2_{=a❛♥❞ν} n =fn✮✳

✻✳✸ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✸✳

✇❤✐❝❤ ♣r♦✈❡s ▲❡♠♠❛ ✶✳ ◆♦✇ ❧❡t ✉s ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ p(m, m′) =κσ 2 0(Dm+Dm′) n∆ ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ♣❡♥❛❧t② pen(m) =24κσ 2 0Dm n∆ . ▲❡t ✉s s❡t E′ :=E " 24 sup t∈Bm,mˆ νn2(t) + (pen(m)−pen( ˆm)) ! ✶Ωn # . ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢✱ ✇❡ ♦❜t❛✐♥ t❤❛t t❤❡r❡ ❡①✐stsκs✉❝❤ t❤❛tE′ _≤ 250κσ 2 0 n∆+ 2pen(m)✳ ▼♦r❡♦✈❡r✱ ✇❡ ❦♥♦✇ t❤❛t ♦♥Ωcn✱ Ekˆbmˆ −bAk2n✶Ωc n ≤Cσ 2 0 n∆. ❚❤❡♥✱ ❝♦❧❧❡❝t✐♥❣ t❡r♠s✱ ✇❡ ❤❛✈❡✿ Ekˆbmˆ −bAk2n ≤ inf m∈M_n 3˜π1kbm−bAk 2 L2+ 4pen(m) +Cσ 2 0 n∆ +C ′_{∆ +} C2 p∆2. ❯s✐♥❣ t❤❛t N=pn ❛♥❞∆ =pδ✱ ✇❡ ♦❜t❛✐♥ t❤❛t Ekˆbmˆ −bAk2n ≤ inf m∈M_n 3˜π1kbm−bAk 2 L2+ 4pen(m) +Cσ 2 0 N δ+C ′_pδ+ C2 p3_δ2. ❚❤✐s ✐♥❡q✉❛❧✐t② ✐s ♠✐♥✐♠✉♠ ✐❢pδ∼1/(p3_δ2_{)✱ t❤❛t ✐s ✐❢p∼δ}−3/4_✳

❆ ❆❞❞✐t✐♦♥❛❧ ♣r♦♦❢s

❆✳✶ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✷

❛♥❞E(I4 k∆)≤b4Lc(4)∆2/3.❯s✐♥❣ t❤❛tb✐s ▲✐♣s❝❤✐t③ ❛♥❞ t❤❡ ❘♦s❡♥t❤❛❧ ✐♥❡q✉❛❧✲ ✐t②✱ ✇❡ ♦❜t❛✐♥✿ EB2k∆ = Eb X¯(k−1)∆ −b X¯(k−1)∆+ ¯εk−1 2 ≤ b2LE ¯ ε2k−1 ≤b 2 Lτ2 p . ❆s✱ ♠♦r❡♦✈❡r✱ E(B4k∆) =E h b X¯(k−1)∆−b X¯(k−1)∆+ ¯εk−1 4i ≤b4LE ¯ ε4k−1 , ✇❡ ❤❛✈❡ t❤❛tE(B4 k∆)≤cp−3E ε41 +cτ4_p−2_{.❲❡ ❞❡r✐✈❡ ❢r♦♠ ✭✾✮ t❤❛t} E R2k∆ ≤4E " 1 ∆ε¯k 2# = 4τ 2 p∆2 ❛♥❞✱ ❢r♦♠ t❤❡ ❘♦s❡♥t❤❛❧ ✐♥❡q✉❛❧✐t②✿ E R4 k∆ ≤ c ∆4 E ε¯4 1 ≤ cE ε 4 1 p3_∆4 +c τ4 p2_∆4. ❲❡ ❤❛✈❡ t❤❛t E Zk∆2 = 2 3+ 1 p2 E σ02(X0) ∆ . ❚❤❡ ❇✉r❦❤ö❧❞❡r✲❉❛✈✐s✲●✉♥❞② ✐♥❡q✉❛❧✐t② ❛❧❧♦✇s ✉s t♦ ✇r✐t❡✿ E(Zk∆4 ) ≤ c p∆4 p X j=1 E   Z (k+1)∆+jδ k∆+jδ σ2(Xs)ds !2  ≤ _∆c₂E σ4(X0).

❆✳✷ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✸

✸✳ P ∃k, Y¯∗ k∆6= ¯Yk∆, k= 1, . . . , n≤pnβX(qn∆)✳ ❚♦ ♣r♦✈❡ t❤✐s ❧❡♠♠❛✱ ✇❡ ✉s❡ ❛ ❇❡r❜❡❡ ❝♦✉♣❧✐♥❣ ♠❡t❤♦❞ ❧✐❦❡ ✐♥ Pr♦♣♦s✐t✐♦♥ ✺✳✶ ♦❢ ❱✐❡♥♥❡t ✭✶✾✾✼✮✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡r❡ ❡①✐sts ✈❛r✐❛❜❧❡sX∗ kδ s✉❝❤ t❤❛t ❼ ❋♦rl= 1, . . . , pn✱ t❤❡ r❛♥❞♦♠ ✈❡❝t♦rs ~ Ul,1= X[2(l−1)pqn+1]δ, . . . , X(2l−1)pqnδ ❛♥❞ _U~∗ l,1= X[2(l∗ −1)pqn+1]δ, . . . , X ∗ (2l−1)pqnδ ❤❛✈❡ s❛♠❡ ❧❛✇✱ ❛s t❤❡ ✈❡❝t♦rs ~ Ul,2= X[(2l−1)pqn+1]δ, . . . , X2lpqnδ ❛♥❞ _U~∗ l,2= X[(2l∗ −1)pqn+1]δ, . . . , X ∗ 2lpqnδ . ❼ ❋♦rl= 1, . . . , pn✱ a∈ {1,2}✱P ~ Ul,a6=U~l,a∗ ≤βX(qnpδ) =βX(qn∆)✳ ❼ ❋♦r ❡❛❝❤a∈ {1,2}✱ r❛♥❞♦♠ ✈❡❝t♦rsU~∗ 1,a, ..~Up∗n,a ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❆s t❤❡εk ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✈❛r✐❛❜❧❡sY¯k∆∗ = 1p Pp j=1Xk∆+jδ∗ +εkp+j s❛t✐s❢② ✭✶✮ ❛♥❞ ✭✷✮✳ ▲❡t ✉s s❡t Ω∗_={X kδ =Xkδ∗ , k= 1, . . . , N} ⊂ _¯ Y∗ k∆= ¯Yk∆ . ❲❡ ❤❛✈❡ t❤❛t P(Ω∗c_{) ≤ P(∃k, X} kδ6=Xkδ∗ ) ≤ pnβX(qn∆)✱ ✇❤✐❝❤ ♣r♦✈❡s ✭✸✮✳ ❚❤❡ ❡♥❞ ♦❢ t❤❡ ♣r♦♦❢ ✐s ❞♦♥❡ ❢♦r ❡①❛♠♣❧❡ ✐♥ ❈♦♠t❡ ❡t ❛❧✳ ✭✷✵✵✼✮✱ s❡❝t✐♦♥ ✼ ❛s ❧♦♥❣ ❛sln(n)/(n∆)→0❛♥❞D_n≤c0 n∆ ln2 (n)✳

❆✳✸ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✶

s♦ σ2 9be b∆/3 ≤σ¯2∆≤ σ2 b e b∆_. ❚❤❡♥✱ t❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② ♦❢ t❤❡ ✈❛r✐❛❜❧❡s_X¯_k∆✱ f(x) =√ 1 2πσ¯∆ exp − x 2 2¯σ2 ∆ , ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❛♥❞ ❛❜♦✈❡✳ ❆ss✉♠♣t✐♦♥ ✻ ✐s s❛t✐s✜❡❞✳ ❲❡ ✇✐❧❧ ♥♦✇ ♣r♦✈❡ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥✳ ■♥ ❛ ✜rst st❡♣✱ ✐t ✐s ❛ss✉♠❡❞ t❤❛t X0=x✳ ❚❤❡ ♣r♦❝❡ssyu = √ ∆−1(Xu∆−1)✱ ❢♦r0 ≤u≤1✱ ✐s s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❉❊ dyt= ˜b(yt)dt+ ˜σ(yt)dW˜t, y0= 0, ✇❤❡r❡˜_{b(t) =}√_∆b t√∆ +x✱σ˜(t) =σ(t√∆ +x)❛♥❞W˜u=Wu∆/ √ ∆✳ ❋✉♥❝✲ t✐♦♥s˜_{b(x)❛♥❞σ˜(x)❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❜♦✉♥❞❡❞ ❛♥❞˜σ(x)✐s ❜♦✉♥❞❡❞ ❢r♦♠} ❜❡❧♦✇ ❜②σ0✳ Pr♦❝❡ss❡s(Xt)❛♥❞(yt)❛r❡ ❧✐♥❦❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥s✿ ¯ X∆, X∆= x+√∆U, x+√∆V ✇❤❡r❡ U = 1 p p X k=1 yk/p ❛♥❞ V =y1. ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ✐s ❡①❛❝t❧② ❚❤❡♦r❡♠ ✹ ♣✳✶✵✾ ✐♥ ●❧♦t❡r ❛♥❞ ●♦❜❡t ✭✷✵✵✽✮ ▲❡♠♠❛ ✽✳ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss (xt)s♦❧✉t✐♦♥ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ dxt= ¯b(xt) + ¯σ(xt)dWt, ❛♥❞ x0=x ✇✐t❤ σ¯(x) ❛♥❞¯b(x) ❛r❡ C3_{✱ σ¯(x)✱ ¯b(x) ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s ❛r❡ ❜♦✉♥❞❡❞ ❛♥❞} t❤❡r❡ ❡①✐stσ¯0>0 s✉❝❤ t❤❛t σ¯(x)≥σ¯0✳ ▲❡tµ ❜❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥[0,1] s✉❝❤ t❤❛t µ(]0,1[)>0✳ ❚❤❡♥✱ ✐❢ ✇❡ s❡t ( ¯U ,V¯) = Z 1 0 xsdµ(s), x1 , t❤❡r❡ ❡①✐st ❝♦♥st❛♥ts c1 ❛♥❞ c2 ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ t❤❡ ❜♦✉♥❞s ♦❢¯b✱σ¯ ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s s✉❝❤ t❤❛t t❤❡ ❞❡♥s✐t② ♦❢ ( ¯U ,V¯)✱p¯x(¯u,v¯)s❛t✐s✜❡s✿ c−11e−c1(¯u 2 +¯v2 ) ≤p¯x(¯u,v¯)≤c−21e−c 2(u¯ 2 +¯v2 )_. ❆s µ(du) = p−1Pp j=1δj/p(du) ✐s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ t❤❡r❡ ❡①✐sts ❝♦♥✲ st❛♥tsc1 ❛♥❞c2s✉❝❤ t❤❛t t❤❡ ❞❡♥s✐t② px(u, v)♦❢(U, V)s❛t✐s✜❡s✿ c−11e−c 1(u 2 +v2 ) ≤px(u, v)≤c−21e−c 2(u 2 +v2 )_. ■♥t❡❣r❛t✐♥❣ t❤✐s r❡s✉❧t ✇✐t❤ r❡s♣❡❝t t♦v✱ ✇❡ ❡❛s✐❧② ♦❜t❛✐♥ ❜♦✉♥❞s ❢♦r t❤❡ ❞❡♥s✐t② px(u)♦❢U✿ t❤❡r❡ ❡①✐sts ❝♦♥st❛♥tsK1 ❛♥❞K2 ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ t❤❡ ❜♦✉♥❞s ♦❢ ¯ b✱σ¯ ❛♥❞ t❤❡✐r ❞❡r✐✈❛t✐✈❡s s✉❝❤ t❤❛t K₁−1exp −c1u2≤px(u)≤K2−1exp −c2u2. ✷✷

▲❡t ✉s ❞❡♥♦t❡ ❜② π¯x(y) t❤❡ ❞❡♥s✐t② ♦❢ X¯∆ ❣✐✈❡♥ X0 = x✳ ❆s X¯∆ = x+ √ ∆U✱ ¯πx(y) = ∆−1/2px (y−x) ∆−1/2❛♥❞✱ s✉❜st✐t✉t✐♥❣u❜②(y−x) ∆−1/2✱ ✇❡ ♦❜t❛✐♥✿ K_√1−1 ∆ exp −c1 (y−x)2 ∆ ! ≤π¯x(y)≤K −1 2 √ ∆ exp −c2 (y−x)2 ∆ ! . ❚❤❡ ✐♥✈❛r✐❛♥t ❞❡♥s✐t② ♦❢_X¯_{∆✱π¯✱ s❛t✐s✜❡sπ¯(y) =}R ¯ πx(y)π(x)dx.■♥t❡❣r❛t✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✐♥❡q✉❛❧✐t②✱ ✇❡ ❤❛✈❡✿ K1−1 Z R exp −c1z2π y+z√∆dz≤π¯(y)≤K2−1 Z R exp −c2z2π y+z√∆dz. ❚❤❡♥✱ ✇❡ ❞❡r✐✈❡ ❢r♦♠ ✭✻✮ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C2 ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ ❢✉♥❝t✐♦♥s b❛♥❞σs✉❝❤ t❤❛t✱ ❢♦r ❛♥②y∈R✱ ¯ π(y)≤C2π1. ■♥ t❤❡ s❛♠❡ ✇❛②✱ ✇❡ ❞❡❞✉❝❡ ❢r♦♠ ✭✼✮ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥tC1 s✉❝❤ t❤❛t✱ ❢♦r ❛♥②y∈[a0−1, a1+ 1]✱ ¯ π(y)≥C1π0.

❆✳✹ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✷

❆✳✺ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸

P[z,z+1](x) =Prj=0aj(x−z)j+ (br−ar) (x−z)r✱ ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t c s✉❝❤ t❤❛t P[z,z+1](x) =P[z−1,z](x) +c(x−z)r. ✭✷✽✮ ◆♦✇✱ ❧❡tR❛♥❞Q❜❡ t✇♦ ❢✉♥❝t✐♦♥s ♦❢P_r❡q✉❛❧ ♦♥_{[z−1, z]}✳ ❆❝❝♦r❞✐♥❣ t♦ ✭✷✽✮✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥tc′ _{s✉❝❤ t❤❛t} R[z,z+1](x)−Q[z,z+1](x) =c′(x−z) r . ✭✷✾✮ ▼♦r❡♦✈❡r✱ t❤❡ s♣❧✐♥❡ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡r✱ ❞❡♥♦t❡❞ ❜②gr✱ ❜❡❧♦♥❣s t♦Pr❛♥❞✱ ♦♥ [0,1]✱ ✐s ❡q✉❛❧ t♦xr_{/r!✳ ❲❡ ♣r♦✈❡ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ kt❤❛t✱ ♦✈❡r [−r−1, r+k]✱} t❤❡r❡ ❡①✐st s♦♠❡ r❡❛❧ ♥✉♠❜❡rs (cj, 0≤j ≤k−1)s✉❝❤ t❤❛t P(x) = k−1 X j=0 cjgr(x+r−j). ▲❡t ✉s ❞❡♥♦t❡fk(x) =Pj=0k−1cjgr(x+r−j)✳ ❆sP[−r−1,−r](x) = 0✱ t❤❡ ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s ✐s tr✉❡ ❢♦rk= 0✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t ✐t ✐s s❛t✐s✜❡❞ ❢♦rk✳ ❋✉♥❝t✐♦♥s fk ❛♥❞P ❜❡❧♦♥❣s t♦Pr ❛♥❞ ❛r❡ ❡q✉❛❧ ♦♥[−r+k−1,−r+k]✱ s♦✱ ❛❝❝♦r�