In this paper, we provide strong L2-rates of approximation of the integral-type functionals of Markov processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its transition probability density, we get the accuracy that coincides with that obtained in [3] for a one-dimensional diffusion process.
Markov processesintegral functionalrates of convergencestrong approximation60H0760H35Introduction
Let Xt, t≥0, be a Markov process with values in Rd. Consider the following objects:
the integral functional
IT(h)=∫0Th(Xt)dt
of this process;
the sequence of integral sums
IT,n(h)=Tn∑k=0n−1h(X(kT)/n),n≥1.
In this paper, we establish strongL2-approximation rates, that is, the bounds for
E|IT(h)−IT,n(h)|2.
The current research is mainly motivated by the recent papers [2] and [3].
In [3], strong Lp-approximation rates are considered for an important particular case where X is a one-dimensional diffusion. The approach developed in this paper contains both the Malliavin calculus tools and the Gaussian bounds for the transition probability density of the process X, and relies substantially on the structure of the process.
Another approach to that problem has been developed in [2]. This approach is, in a sense, a modification of Dynkin’s theory of continuous additive functionals (see [1], Chap. 6) and also involves the technique similar to that used in the proof of the classical Khasminskii lemma (see, e.g., [4, Lemma 2.1]). This approach allows us to obtain strong Lp-approximation rates under assumptions on the process X formulated only in terms of its transition probability density.
For a bounded function h, the strong Lp-rates of approximation of the integral functional IT(h) obtained in [2] essentially coincide with those established in [3]. However, under additional regularity assumptions on the function h (e.g., when h is Hölder continuous), the rates obtained in [3] are sharper (see [2, Thm. 2.2] and [3, Thm. 2.3]).
In this note, we improve the method developed in [2], so that under the assumption of the Hölder continuity of h, the strong L2-approximation rates coincide with those obtained in [3], preserving at the same time the advantage of the method that the assumptions on the process X are quite general and do not essentially rely on the structure of the process.
Main result
In what follows, Px denotes the law of the Markov process X conditioned by X0=x, and Ex denotes the expectation with respect to this law. Both the absolute value of a real number and the Euclidean norm in Rd are denoted by |·|.
We make the following assumption on the process X.
A. The process X possesses a transition probability density pt(x,y) that is differentiable with respect to t and satisfies the following estimates:
pt(x,y)≤CTt−d/αQ(t−1/α(x−y)),t≤T,|∂tpt(x,y)|≤CTt−1−d/αQ(t−1/α(x−y)),t≤T,|∂tt2pt(x,y)|≤CTt−2−d/αQ(t−1/α(x−y)),t≤T,
for some fixed α∈(0,2] and some distribution density Q such that ∫Rd|z|2γQ(z)dz<∞. Without loss of generality, we assume that in (1)–(3) CT≥1.
We assume that the function h satisfies the Hölder condition with exponent γ∈(0,α/2], that is,
‖h‖γ:=supx≠y|h(x)−h(y)||x−y|γ<∞.
Now we formulate the main result of the paper.
Suppose that AssumptionAholds. ThenEx|IT(h)−IT,n(h)|2≤DT,γ,α,QCγ,α‖h‖γ2n−(1+2γ/α),γ≠α/2,DT,γ,α,Q‖h‖γ2n−2lnn,γ=α/2,whereDT,γ,α,Q=8CT2T2+2γ/α∫Rd|z|2γQ(z)dz,Cγ,α=max{(1−2γ/α)−1(2γ/α)−1,maxn≥1((lnn)2n1−2γ/α)}.
We provide the proof of Theorem 1 in Section 3.
Any diffusion process satisfies conditions (1)–(3) with α=2, Q(x)=c1e−c2|x|2, and properly chosen c1,c2 (see [2]). In the case where X is a one-dimensional diffusion, Theorem 1 provides the same rates of convergence as those obtained in [3] (see Theorem 2.3 in [3]).
Similarly to [2], we formulate the assumption on the process X only in terms of its transition probability density. Condition A, compared with condition X (cf. [2]), contains the additional assumption (3).
Proof of Theorem <xref rid="j_vmsta29_stat_001">1</xref>
For t∈[kT/n,(k+1)T/n), denote
ηn(t)=kTn,ζn(t)=(k+1)Tn,
and put Δn(s):=h(Xs)−h(Xηn(s)), s∈[0,T].
By the Markov property of X, for any r<s, we have
Ex|Xs−Xr|2γ=Ex∫Rdps−r(Xr,z)|Xr−z|2γdz≤CTEx∫Rd(s−r)−d/αQ((s−r)−1/α(Xr−z))|Xr−z|2γdz=CT(s−r)2γ/α∫Rd|z|2γQ(z)dz.
Therefore, using the inequality s−ηn(s)≤T/n, s∈[0,T] and the Hölder continuity of the function h, we obtain:
Ex|Δn(s)|2≤CTT2γ/α(∫Rd|z|2γQ(z)dz)‖h‖γ2n−2γ/α.
Split
Ex|IT(h)−IT,n(h)|2=2Ex∫0T∫sTΔn(s)Δn(t)dtds=J1+J2+J3,
where
J1=2Ex∫0T∫sζn(s)+T/nΔn(s)Δn(t)dtds,J2=2Ex∫0T/n∫ζn(s)+T/nTΔn(s)Δn(t)dtds,J3=2Ex∫T/nT∫ζn(s)+T/nTΔn(s)Δn(t)dtds.
For |J1| and |J2|, the estimates can be obtained in the same way. Indeed, using the Cauchy inequality and (4), we get
|J1|≤2∫0T∫sζn(s)+T/n(Ex|Δn(s)|2)1/2(Ex|Δn(t)|2)1/2dtds≤2CTT2γ/α‖h‖γ2(∫Rd|z|2γQ(z)dz)n−2γ/α∫0T(T/n+ζn(s)−s)ds≤4CTT2+2γ/α‖h‖γ2(∫Rd|z|2γQ(z)dz)n−(1+2γ/α).
In the last inequality, we have used the inequality ζn(s)−s≤T/n, s∈[0,T]. Similarly,
|J2|≤2CTT2+2γ/α‖h‖γ2(∫Rd|z|2γQ(z)dz)n−(1+2γ/α).
Now we proceed to the estimation of |J3|, which is the main part of the proof. Observe that the following identities hold: ∫Rd∂uv2pu(x,y)pv−u(y,z)dz=∂uv2pu(x,y)∫Rdpv−u(y,z)dz=∂uv2pu(x,y)=0,y∈Rd,∫Rd∂uv2pu(x,y)pv−u(y,z)dy=∂uv2∫Rdpu(x,y)pv−u(y,z)dy=∂uv2pv(x,z)=0,z∈Rd, where in (6) we used that ∫Rdpr(y,z)dz=1, r>00$]]>, y∈Rd, and in (7) we used the Chapman–Kolmogorov equation.
We have:
J3=2∫T/nT∫ζn(s)+T/nT∫Rd∫Rdh(y)h(z)[ps(x,y)pt−s(y,z)−pηn(s)(x,y)pt−ηn(s)(y,z)−ps(x,y)pηn(t)−s(y,z)+pηn(s)(x,y)pηn(t)−ηn(s)(y,z)]dzdydtds=2∫T/nT∫ζn(s)+T/nT∫Rd∫Rd∫ηn(s)s∫ηn(t)th(y)h(z)∂uv2(pu(x,y)×pv−u(y,z))dvdudzdydtds=−∫T/nT∫ζn(s)+T/nT∫Rd∫Rd∫ηn(s)s∫ηn(t)t(h(y)−h(z))2∂uv2(pu(x,y)×pv−u(y,z))dvdudzdydtds,
where in the last identity we have used (6) and (7).
Further, we have
∂uv2pu(x,y)pv−u(y,z)=pu(x,y)∂rr2pr(y,z)|r=v−u+∂upu(x,y)∂rpr(y,z)|r=v−u.
Then, using condition A and the Hölder continuity of the function h, we obtain
∫Rd∫Rd(h(y)−h(z))2|∂uv2(pu(x,y)pv−u(y,z))|dzdy≤CT2‖h‖γ2(∫Rd|z|2γQ(z)dz)((v−u)2γ/α−2+(v−u)2γ/α−1u−1).
Therefore, according to (8) and (9),
|J3|≤CT2‖h‖γ2(∫Rd|z|2γQ(z)dz)×∫T/nT∫ζn(s)+T/nT∫ηn(s)s∫ηn(t)t((v−u)2γ/α−2+(v−u)2γ/α−1u−1)dvdudtds.
Denote aα,γ(u,v):=(v−u)2γ/α−2+(v−u)2γ/α−1u−1. Then
∫T/nT∫ζn(s)+T/nT∫ηn(s)s∫ηn(t)taα,γ(u,v)dvdudtds=∑i=1n−1∑j=i+2n−1∫iT/n(i+1)T/n∫jT/n(j+1)T/n∫iT/ns∫jT/ntaα,γ(u,v)dvdudtds=∑i=1n−1∑j=i+2n−1∫iT/n(i+1)T/n∫jT/n(j+1)T/n∫u(i+1)T/n∫v(j+1)T/naα,γ(u,v)dtdsdvdu≤T2n−2∑i=1n−1∑j=i+2n−1∫iT/n(i+1)T/n∫jT/n(j+1)T/naα,γ(u,v)dvdu=T2n−2∑i=1n−1∫iT/n(i+1)T/n∫(i+2)T/nTaα,γ(u,v)dvdu,
where in the fourth line we used that, for u∈[iT/n,(i+1)T/n) and v∈[jT/n,(j+1)T/n), we always have (i+1)T/n−u≤T/n and (j+1)T/n−v≤T/n.
Thus, from (10) we obtain
|J3|≤CT2T2‖h‖γ2(∫Rd|z|2γQ(z)dz)n−2(S1+S2),
where
S1=∑i=1n−1∫iT/n(i+1)T/n∫(i+1)T/nT(v−u)2γ/α−2dvdu,S2=∑i=1n−1∫iT/n(i+1)T/n∫(i+2)T/nT(v−u)2γ/α−1u−1dvdu.
We estimate each term separately. In what follows, we consider the case γ<α/2; the case of γ=α/2 is similar and therefore omitted. We have
S1≤(1−2γ/α)−1∑i=1n−1∫iT/n(i+1)T/n((i+1)T/n−u)2γ/α−1du=(1−2γ/α)−1(2γ/α)−1∑i=1n−1((i+1)T/n−iT/n)2γ/α≤(1−2γ/α)−1(2γ/α)−1T2γ/αn1−2γ/α≤Cγ,αT2γ/αn1−2γ/α.
Finally, since v−u≤T for 0≤u<v≤T, we have
S2≤T2γ/α∑i=1n−1∫iT/n(i+1)T/n∫(i+2)T/nT(v−u)−1u−1dvdu≤T2γ/α∑i=1n−1(∫iT/n(i+1)T/nu−1du)(∫(i+2)T/nT(v−(i+1)T/n)−1dv)≤T2γ/αlnn∑i=1n−1(∫iT/n(i+1)T/nu−1du)=T2γ/α(lnn)2≤Cγ,αT2γ/αn1−2γ/α.
Combining inequality (11) with (12) and (13), we derive
|J3|≤2Cγ,αCT2T2+2γ/α‖h‖γ2(∫Rd|z|2γQ(z)dz)n−(1+2γ/α).
□
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