On mean discounted numbers of passage times in small balls of Ito processes observed at discrete times

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Frédéric Bernardin, Mireille Bossy, Miguel Martinez, Denis Talay

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Frédéric Bernardin, Mireille Bossy, Miguel Martinez, Denis Talay. On mean discounted numbers of passage times in small balls of Ito processes observed at discrete times. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2009, 14, pp.19. �inria-00347610v3�

a p p o r t

d e r e c h e r c h e

ISRNINRIA/RR--6813--FR+ENG

Thème NUM

On mean discounted numbers of passage times in small balls of Itô processes observed at discrete times

Frédéric Bernardin — Mireille Bossy — Miguel Martinez — Denis Talay

N° 6813

December 2008

in small balls of Itˆo processes observed at discrete times

Fr´ed´eric Bernardin^∗ , Mireille Bossy^† , Miguel Martinez^‡ , Denis Talay^§

Th`eme NUM — Syst`emes num´eriques Projets Tosca

Rapport de recherche n°6813 — December 2008 — 21 pages

Abstract: The aim of this note is to prove estimates on mean values of the number of times that Itˆo processes observed at discrete times visit small balls in R^d. Our technique, in the infinite horizon case, is inspired by Krylov’s ar- guments in [2, Chap.2]. In the finite horizon case, motivated by an application in stochastic numerics, we discount the number of visits by a locally exploding coefficient, and our proof involves accurate properties of last passage times at 0 of one dimensional semimartingales.

Key-words: last passage times,Itˆo processes observed at discrete times

∗ CETE de Lyon, Laboratoire R´egional des Ponts et Chauss´ees, Clermont-Ferrand, France; frederic.bernardin@developpement-durable.fr

† INRIA Sophia Antipolis, EPI TOSCA; Mireille.Bossy@sophia.inria.fr

‡ Universit´e Paris-Est - Marne-la-Vall´ee, France; Miguel.Martinez@univ-mlv.fr

§ INRIA Sophia Antipolis, EPI TOSCA; Denis.Talay@sophia.inria.fr

processus d’Itˆo observ´es en temps discret

R´esum´e : L’objectif de cette note est de d´emontrer des estimations sur le nombre moyen de visites dans des petites boules de R^d d’un processus d’Itˆo, observ´e `a des instants discrets. En horizon de temps infini, notre technique est inspir´e des arguments de Krylov dans [2, Chap.2]. En horizon de temps fini, motiv´e par une application en analyse num´erique stochastique, nous pond´erons le nombre de visites par un coefficient localement explosif; notre preuve utilise des propri´et´es fines sur le dernier temps de passage en z´ero d’une semimartin- gale unidimensionnelle.

Mots-cl´es : dernier temps de passage, processus d’Itˆo observ´e `a temps discret

1 Introduction

The present work is motivated by two convergence rate analyses concerning dis- cretization schemes for diffusion processes whose generators have non smooth coefficients: Bernardin et al. [1] study discretization schemes for stochastic dif- ferential equations with multivalued drift coefficients; Martinez and Talay [3]

study discretization schemes for diffusion processes whose infinitesimal gen- erators are of divergence form with discontinuous coefficients. In both cases, to have reasonable accuracies, a scheme needs to mimic the local behaviour of the exact solution in small neighborhoods of the discontinuities of the co- efficients, in the sense that the expectations of the total times spent by the scheme (considered as a piecewise constant process) and the exact solution in these neighborhoods need to be close to each other. The objective of this paper is to provide two estimates for the expectations of such total times spent by fairly general Itˆo processes observed at discrete times; these estimates concern in particular the discretization schemes studied in the above references (See section 7 for more detailed explanations).

Let (Ω,F,(F_t)t≥0,P) be a filtered probability space satisfying the usual conditions. Let (Wt)t≥0 be a m-dimensional standard Brownian motion on this space. Consider two progressively measurable processes (b_t)t≥0 and (σ_t)t≥0

taking values respectively in R^d and in the space M_d,m(R) of real d × m dimensional matrices. For X0 inR^d, consider the Itˆo process

Xt=X0 + Z t

0

bsds+ Z t

0

σsdWs. (1)

We assume the following hypotheses:

Hypotheses 1.1. There exists a positive number K ≥1 such that, P-a.s.,

∀t≥0, kb_tk ≤K, (2) and

∀0≤s ≤t, 1 K²

Z t s

ψ(u)du≤ Z t

s

ψ(u)kσ_uσ^∗_ukdu≤K² Z t

s

ψ(u)du (3) for all positive locally integrable map ψ :R⁺→R⁺.

Notice that (3) is satisfied when (σ_t) is a bounded continuous process; our slightly more general framework is motivated by the reduction of our study, without loss of generality, to one dimensional Itˆo processes: see section 2.

The aim of this note is to prove the following two results.

Our first result concerns discounted expectations of the total number of times that an Itˆo process observed at discrete times visits small neighborhoods of an arbitrary point in R^d.

Theorem 1.2 (Infinite horizon). Let (X_t) be as in (1). Suppose that the hypotheses 1.1 are satisfied. Let h > 0. There exists a constant C, depending only on K, such that, for all ξ ∈ R^d and 0 < ε < 1/2, there exists h₀ > 0 (depending on ε) satisfying

∀h≤h₀, h

∞

X

p=0

e^−phP(kX_ph−ξk ≤δ_h)≤Cδ_h, (4) where δ_h :=h^1/2−ε.

Our second result concerns expectations of the number of times that an Itˆo process, observed at discrete times between 0 and a finite horizonT, visits small neighborhoods of an arbitrary point ξ in R^d. In this setting, the ex- pectations are discounted with a multiplying function f(t) that may tend to infinity when t tends to T, which induces technical difficulties which might seem artificial to the reader. However, this framework is necessary to analyze the convergence rate of simulation methods for Markov processes whose gen- erators are of divergence form with discontinuous coefficients: see Appendix 7 and Martinez and Talay [3]. More precisely, we consider functionsf satisfying the following hypotheses:

Hypotheses 1.3.

1. f is positive and increasing, 2. f belongs to C¹([0, T);R⁺),

3. f^α is integrable on [0, T) for all 1≤α <2,

4. there exists1< β <1 +η, where η := _4K¹4, such that Z T

0

f^2β−1(v)f⁰(v)(T −v)^1+η

v^η dv < +∞. (5) Remember that we have supposed K ≥1. An example of a suitable func- tion f is the function t→ ^√_T¹_−t (for which one can choose β = 1 + _8K¹4).

Theorem 1.4 (Finite horizon). Let (Xt) be as in (1). Suppose that the hy- potheses 1.1 are satisfied. Let f be a function on [0, T) satisfying the hypothe- ses 1.3. There exists a constant C, depending only on β, K and T, such that, for all ξ∈R^d and 0< ε <1/2, there exists h0 >0(depending on ε) satisfying

∀h≤h₀, h

N_h

X

p=0

f(ph)P(kX_ph−ξk ≤δ_h)≤Cδ_h, (6) where Nh :=bT /hc −1 and where δh :=h^1/2−ε.

Remark 1.5. Since all the norms in R^d are equivalent, without loss of gen- erality we may choose kYk = sup_1≤i≤d|Yⁱ|, Yⁱ denoting the i^th component of Y.

Remark 1.6. ChangingX₀ into X₀−ξ allows us to limit ourselves to the case ξ= 0 without loss of generality, which we actually do in the sequel.

2 Reduction to one dimensional (X_t)’s

In order to prove Theorem 1.2, it suffices to prove that, for all 1≤ i≤ d and h≤h0,

h

∞

X

p=0

e^−phP X_phⁱ

≤δ_h

≤Cδ_h,

or, similarly, for Theorem 1.4 h

Nh

X

p=0

f(ph)P |X_phⁱ | ≤δ_h

≤Cδ_h.

Set

M_tⁱ :=

m

X

j=1

Z t 0

σ_s^ijdW_s^j

.

The martingale representation theorem implies that there exist a process (f_tⁱ)t≥0

and a standard F_t-Brownian motion (B_t)_t≥0 on an extension of the original probability space, such that

M_tⁱ = Z t

0

f_sⁱdB_s,

and

∀t >0, Z t

0

(f_θⁱ)²dθ =

m

X

j=1

Z t 0

σ^ij_θ² dθ.

Therefore it suffices to prove Theorems 1.2 and 1.4 for all one dimensional Itˆo process

X_t= Z t

0

b_sds+ Z t

0

σ_sdB_s (7)

such that,∃K ≥1, P-a.s.,

∀t ≥0, |bt| ≤K, (8) and

∀0≤s ≤t, 1 K²

Z t s

ψ(u)du≤ Z t

s

ψ(u)σ²_udu≤K² Z t

s

ψ(u)du (9) for all positive locally integrable map ψ :R⁺ →R⁺.

3 An estimate for a localizing stopping time

In this section we prove a key proposition which is a variant of Theorem 4 in [2, Chap.2]: here we need to consider time indices and stopping times which take values in a mesh with step-size h. We carefully modify Krylov’s arguments.

We start by introducing two non-decreasing sequences of random times (θ^h_p)_p and (τ_p^h)p taking values in hN as











θ₀^h =τ₀^h = 0

θ_p+1^h = inf{t≥τ_p^h, t∈hN, |Xt| ≤δh}, p≥0 τ_p^h = inf{t≥θ_p^h, t∈hN,

X_t−X_θ^h

p

≥1}, p≥1.

(10)

Our goal is then to prove the following equality : E h

+∞

X

p=0

e^−ph1_[−δh,δh](X_ph)

!

=

+∞

X

p=0

E 1_θh p<+∞

Z τ_p^h θ_p^h

e^−ηs1_[−δh,δh](X_ηs)ds

! ,

(11) where

η_s :=hbs/hc for all s≥0. (12) It is easy to see that (11) holds if the sequence θ^h_p

p takes an infinite value in a finite time, or converges to infinity when p tends to infinity, which is a consequence of Lemma 3.2 (see below). For the proof of Lemma 3.2, we need the following

Proposition 3.1. Let (Xt) be as in (7). Suppose that conditions (8) and (9) hold true. Let θ : Ω→hN be a (F_t)-stopping time. Thenτ := inf{t∈hN, t≥ θ,|X_t−X_θ| ≥1} is a (F_t)-stopping time, and

E 1_{τ <+∞}e^−τ F_θ

≤ 1

ch(µ)e^−θ1_θ<+∞ a.s., (13) where

µ >0 and 1−Kµ−(1/2)K²µ² = 0.

Proof. Set π(z) := cosh(µz) for all z in Rand Luπ(z) :=buπ⁰(z) + 1

2σ_u²π⁰⁰(z).

Let A∈ F_θ. Suppose that we have proven that, for all t >0, e^−t∧θ ≥E e^−τπ(Xt∧τ −Xt∧θ)

Ft∧θ

. (14)

Then we would have E 1_A 1θ<+∞ e^−θ

= lim

t→+∞E 1_A 1θ≤t e^−θ∧t

≥ lim

t→+∞E 1_A 1θ≤t E e^−τπ(Xt∧τ −Xt∧θ) Ft∧θ

= lim

t→+∞E 1_A 1θ≤t E e^−τπ(Xt∧τ−Xt∧θ) F_θ

= lim

t→+∞E 1_A 1θ≤t e^−τπ(Xt∧τ−Xt∧θ)

= lim

t→+∞E 1_A 1θ<+∞ e^−τπ(1)

≥E 1_A cosh(µ)1τ <+∞e^−τ ,

and thus the desired result would have been obtained. We now prove the inequality (14).

Fix t≥0 arbitrarily and set

B₁ :={t ≤θ} ∩ {t ≤τ}={t≤θ}, and

C₁ :={t > θ} ∩ {t≤τ}={t > θ} ∩ {∀s ∈[θ, t]∩hN, |X_s−X_θ|<1}. The sets B1 and C1 are Ft-measurable. Consequently the set {t ≤ τ} is Ft- measurable and thusτ is a (F_t)-stopping time.

To show (14), we start with checking that, P-a.s, for all 0 ≤ s ≤ t and z ∈R,

Z t s

ψ(u) (π(z)−L_uπ(z))du≥cosh(µz)

1−Kµ− 1 2K²µ²

Z t s

ψ(u)du = 0 (15) for all positive locally bounded map ψ :R⁺ → R⁺. Indeed, for all z ∈ R and u≥0,

π(z)−L_uπ(z) = cosh(µz)−µ b_usinh(µz)− 1

2µ²σ_u²cosh(µz).

As sinh(µ|z|)≤cosh(µz), we get (15) by using (8) and (9).

Now, we observe that sup_θ≤te^γXθ is integrable for all γ > 0 since by (9), (b_t) and (hXi_t) are bounded. This justifies the following equality deduced from the Itˆo’s formula : for all y∈R,

e^−t∧θπ(X_t∧θ+y) = E e^−t∧τπ(X_t∧τ +y) F_t∧θ +E

Z t∧τ t∧θ

e^−s(π(X_s+y)−L_sπ(X_s+y))ds

Ft∧θ

a.s..

Therefore, in view of (15), there exists a set N_y with P(N_y) = 0 such that e^−t∧θπ(X_t∧θ+y)≥E e^−τπ(X_t∧τ +y)

F_t∧θ

on Ω− N_y. (16) We now use the continuity ofπ in order to replacey by−Xt∧θ. For all positive integer q, let k_q(y) be the unique integer ` such that `/2^q ≤ y < (`+ 1)/2^q. Then

π(Xt∧τ −Xt∧θ) = lim

q→+∞π

Xt∧τ − k_q(Xt∧θ) 2^q

.

Therefore, by Fatou’s lemma and (16), E e^−τπ(X_t∧τ −X_t∧θ)

F_t∧θ

≤lim inf

q→+∞ E

e^−τπ

X_t∧τ +k_q(−Xt∧θ) 2^q

F_t∧θ

≤lim inf

q→+∞ e^−t∧θπ

X_t∧θ+ k_q(−Xt∧θ) 2^q

=e^−t∧θ.

We end this section by the following Lemma 3.2. For all p≥0,

E

1_θ^h

p<+∞e^−θhp

≤

1 cosh(µ)

p

, (17)

where µ is defined as in Proposition 3.1.

Proof. Apply Proposition 3.1 to the stopping timesθ_p^h and τ_p^h. It comes E

1_τh

p<+∞e^−τph F_θh

p

≤ 1

cosh(µ)e^−θph1_θh

p<+∞ P−a.s., from which

E

1_θ^h

p+1<+∞e^−θhp+1

≤E

1_τ^h

p<+∞e^−τph

=E

E

1_τ^h

p<+∞e^−τph F_θ^h

p

≤ 1 cosh(µ)E

1_θ^h_p<+∞e^−θhp

≤ · · · ≤

1 cosh(µ)

p+1

.

In the next section we carefully estimate the right-hand side of (11).

4 Estimate for the localized problem in the infinite horizon setting

In this section we aim to prove the following lemma.

Lemma 4.1.Consider a one dimensional Itˆo process (7) satisfying (8) and (9).

Let θ : Ω→ hN be a Ft-stopping time such that Xθ ∈[−δh, δh] almost surely.

Let τ be the stopping time τ := inf{t ≥ θ, t ∈ hN, |X_t−X_θ| ≥ 1}. There exists a constantC, independent of θ andτ, such that, for allh small enough,

E

1θ<+∞

Z τ θ

e^−ηs1[−δ_h,δh](X_ηs)ds

≤Cδ_hp

E(1θ<+∞e^−θ). (18) Proof. To start with, we construct a function I whose second derivative is 1_{[−2δh,2δh]} in order to be able to apply Itˆo’s formula. LetI :R→R be defined as

I(z) =











0 if z≤ −2δ_h, 1

2z² + 2δ_hz+ 2δ_h² if −2δ_h ≤z ≤2δ_h, 4δ_hz if 2δ_h ≤z.

(19)

The mapIis of classC¹(R) and of classC²(R\({−2δ_h}∪{2δ_h})). In addition, for all z ∈ R, I⁰⁰(z) = 1[−2δ_h,2δh](z) (we set I⁰⁰(−2δ_h) = I⁰⁰(2δ_h) = 1). For h small enough, there exists a constant C independent ofh such that

sup

−1−δh≤z≤1+δh

|I(z)| ≤Cδ_h, sup

z∈R

|I⁰(z)| ≤Cδ_h. (20) Set

q_h :=1θ<+∞

Z τ θ

e^−ηs1[−δ_h,δh](X_ηs)ds, (21) and observe that, for all s≥0,

{X_ηs ∈[−δ_h, δ_h]}

= ({X_ηs ∈[−δ_h, δ_h]} ∩ {X_s∈[−2δ_h,2δ_h]})∪({X_ηs ∈[−δ_h, δ_h]} ∩ {X_s∈/ [−2δ_h,2δ_h]}), so that

1_[−δh,δh](X_ηs)≤1_{[−2δh,2δh]}(X_s) +1_[−δh,δh](X_ηs)1_{R\[−2δh,2δh]}(X_s). (22) From (21) and (22),

qh ≤A^h +B^h, (23)

where

A^h =1θ<+∞

Z τ θ

e^−ηs1_{[−2δh,2δh]}(X_s)ds (24) and

B^h =1_θ<+∞

Z τ θ

e^−ηs1_[−δh,δh](X_ηs)1_{R\[−2δh,2δh]}(X_s)ds. (25) An upper bound for A^h. In view of (9), one has

A^h ≤1_θ<+∞K²e^−h Z τ

θ

e^−s1_{[−2δh,2δh]}(X_s)σ²_sds=1_θ<+∞K²e^−h Z τ

θ

e^−sI⁰⁰(X_s)σ²_sds.

Notice that (Rt

0 e^−sI⁰(X_s)σ_sdW_s) is a uniformly square integrable martingale.

Thus the limit R∞

0 e^−sI⁰(X_s)σ_sdW_s exists a.s. As I⁰⁰ is continuous except

at points −2δ_h and 2δ_h, an extended Itˆo’s formula (see, e.g., Protter [4, Thm.71,Chap.IV]) implies

A^h ≤1_θ<+∞K²e^−h

e^−τI(X_τ)−e^−θI(X_θ) + Z τ

θ

e^−sI(X_s)ds

− Z τ

θ

e^−sI⁰(X_s)b_sds− Z τ

θ

e^−sI⁰(X_s)σ_sdW_s

. (26) In view of (20) we have |I(X_s)| ≤Cδ_h even on the random set {τ = +∞}. In addition,

E

1τ <∞

Z τ 0

e^−sI(Xs)ds

≤E

1τ <∞

Z τ 0

e^−s|I(Xηs)|ds

+E

1τ <∞

Z +∞

0

e^−s|I(Xs)−I(Xηs)|ds

.

The first expectation is bounded from above by Cδ_h since, for all integer k such thatθ ≤kh≤τ−h, one has |X_kh| ≤1 +δ_h. The second one is obviously bounded from above by C√

h. Thus

E A^h

≤Cδ_hE 1θ<+∞e^−θ

. (27)

An upper bound for B^h. Setting M_s :=Rs

0 σ_udW_u, one has B^h ≤1_θ<+∞

Z τ θ

e^−ηs1_[δh,+∞)(|X_s−X_ηs|)ds

=1θ<+∞e^−θ Z τ−θ

0

e^−ηs1_[δh,+∞)(|Xs+θ−Xηs+θ|)ds

≤1θ<+∞e^−θ Z +∞

0

e^−ηs1_[δh,+∞)

Z s+θ ηs+θ

b_udu+M_s+θ−M_ηs+θ

ds

≤1θ<+∞e^−θ Z +∞

0

e^−ηs1_[δh−Kh,+∞)(|M_s+θ−M_ηs+θ|)ds

≤1θ<+∞e^−θ Z +∞

0

e^−ηs1_[δh/2,+∞)(|M_s+θ−M_ηs+θ|)ds,

where we have used (8) and h is assumed small enough. We now successively use the Cauchy-Schwarz, Jensen and Bernstein inequalities (see, e.g., Revuz

and Yor [5, Ex.3.16,Chap.IV] for the Bernstein inequality for martingales):

EB^h ≤ q

E(1θ<+∞e^−θ)² s

E

Z +∞

0

e^−ηs1_[δh/2,+∞)(|M_s+θ−M_ηs+θ|)ds 2

≤p

E(1θ<+∞e^−θ) s

E

Z +∞

0

e^−ηs1_[δh/2,+∞)(|M_s+θ−M_ηs+θ|)ds

≤p

E(1θ<+∞e^−θ) s

Z +∞

0

e^−ηsP

sup

ηs+θ≤u≤s+θ

|Mu−Mηs+θ| ≥ ^δ₂^h

ds

≤Cp

E(1θ<+∞e^−θ) s

Z +∞

0

e^−ηsexp

− 1 8K²

δ_h² h

ds

≤Cp

E(1θ<+∞e^−θ) exp

− 1 16K²h^−2ε

.

(28) One then deduces (18) from (21), (23), (27) and (28).

5 End of the proof of Theorem 1.2

We apply Lemmas 4.1 and 3.2 and the equality (11) and deduce hE

+∞

X

p=0

e^−ph1[−δ_h,δh](X_ph)

!

=

+∞

X

p=0

E 1_θ^h

p<+∞

Z τ_p^h θ^h_p

e^−ηs1[−δ_h,δh](X_ηs)ds

!

≤Cδ_h

+∞

X

p=0

r E

1_θ^h

p<+∞e^−θph

≤Cδ_h

+∞

X

p=0

1 cosh(µ)

p 2

,

which ends the proof of Theorem 1.2.

6 Proof of Theorem 1.4

In the finite horizon setting, we do not need to localize by means of an appro- priate sequence of stopping times as in the preceding section. We may start from

E h

Nh

X

p=0

f(ph)1_[−δh,δh](X_ph)

!

=

Nh

X

p=0

E

Z (p+1)h ph

f(η_s)1_[−δh,δh](X_ηs)ds

! .

(29) In view of (22) we have

E h

Nh

X

p=0

f(ph)1_[−δh,δh](X_ph)

!

≤E Z T

0

f(η_s)1_{[−2δh,2δh]}(X_s)ds

+

Nh

X

p=0

E(D_h^p)

=:A^h(T) +

Nh

X

p=0

E D_p^h ,

(30) where for all p∈ {0, . . . , Nh}

D^h_p :=

Z (p+1)h ph

f(η_s)1[−δ_h,δh](X_ηs)1_R\[−2δ_h,2δh](X_s)ds. (31) An upper bound for A^h(T).

In view of (9), as the function f is increasing, one has A^h(T)≤K²E

(

Z T 0

f(s)1[−2δ_h,2δh](X_s)σ_s²ds

.

Since we may have lim

s↑T f(s) = +∞, the proof of Lemma 4.1 does not apply.

Set τ_y := inf{s > 0 : X_s− y = 0}. The generalized occupation time formula (see, e.g., Revuz and Yor [5, Ex.1.15,Chap.VI]) implies that

E Z T

0

f(s)1[−2δ_h,2δh](X_s)σ_s²ds

= Z

R

dy1[−2δ_h,2δh](y)E Z T

0

f(s)dL^y_s(X)

= Z

dy1_{[−2δ ,2δ ]}(y)E 1_{τ ≤T} Z T

f(s)dL^y_s(X)

! ,

where {L^y_s(X) : s ∈ [0, T], y ∈ R} denotes a bi-continuous c`adl`ag version of the local time of the process (X_t)t≥0.

Our aim now is to bound from above E

1_τy≤T

RT

τyf(s)dL^y_s(X)

uniformly in y in [−2δh,2δh]. It clearly suffices to show that there exists a constant CT

depending only β,K and T, such that E

Z T 0

f(s)dL⁰_s(Y)

≤C_T, (33)

where the semimartingale Y is defined asY :=X−y.

Because f is not square integrable over [0, T), we arbitrarily choose γ >0.

Now f is square integrable on [0, T −γ] and the function t 7→Rt

0 f(s)dL⁰_s(Y) is the local time of the local semimartingale

{f(g_t)Y_t, t∈[0, T −γ]},

whereg_t:= sup{s < t : Y_s = 0}(see, e.g., Revuz and Yor [5, Sec.4,Chap.VI]).

The Stieltjes measure induced by the function s 7→ f(g_s) has support Z :=

{s≥0 : Ys = 0}, and therefore Z T−γ

0

|Y_θ|df(g_θ) = 0, so that Itˆo-Tanaka formula implies

E

Z T−γ 0

f(s)dL⁰_s(Y)

=E(|f(gT−γ)YT−γ|)−f(g0)|Y0|−E

Z T−γ 0

f(gs)sgn((Ys))dYs

. (34)

Apply Theorem 4.2 of Chap. VI in [5]. It comes:

E(|f(g_T−γ)Y_T−γ|) =E

|f(0)Y₀ + Z T−γ

0

f(g_s)dY_s |

,

from which

E(|f(g_T−γ)Y_T−γ|)≤|f(0)Y₀ |+E

| Z T−γ

0

f(g_s)σ_sdW_s|

+E

| Z T−γ

0

f(s)b_sds|

≤C_T +K² Z T

0

E f²(g_s) ds

1 2

.