We study the weak limits of solutions to SDEs
dXn(t)=an(Xn(t))dt+dW(t),
where the sequence {an} converges in some sense to (c−1x<0+c+1x>0)/x+γδ00})/x+\gamma \delta _{0}$]]>. Here δ0 is the Dirac delta function concentrated at zero. A limit of {Xn} may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.
Consider the stochastic differential equation
dX(t)=a(X(t))dt+dW(t),t⩾0,
where a is a locally integrable function.
The aim of this paper is to study convergence in distribution of the sequence of processes {X(nt)/n,t⩾0} as n→∞.
Observe that
dXn(t)=na(nXn(t))dt+dWn(t),t⩾0,
where Wn(t)=W(nt)/n, t⩾0 is a Wiener process, and Xn(t)=X(nt)/n, t⩾0.
Hence, to study the sequence {X(nt)/n}, it suffices to investigate the SDEs
dXn(t)=an(Xn(t))dt+dW(t),t⩾0,
where an(x)=na(nx).
If a∈L1(R), then an converges in generalized sense to αδ0, where δ0 is the Dirac delta function at zero, where α=∫Ra(x)dx. It is well known that in this case the sequence {Xn} converges weakly to a skew Brownian motion with parameter γ=(α)=eα−e−αeα+e−α; see, for example, [14, 10]. Recall that [5, 10] the skew Brownian motion Wγ(t) with parameter γ, |γ|⩽1, is a unique (strong) solution to the SDE
dWγ(t)=dW(t)+γdLWγ0(t),
where LWγ0(t)=limε→0+(2ε)−1∫0t1|Wγ(s)|⩽εds is the local time of the process Wγ at 0. The process Wγ is a continuous Markov process with transition probability density function pt(x,y)=φt(x−y)+γ(y)φt(|x|+|y|), x,y∈R, where φt(x)=12πte−x2/2t is the density of the normal distribution N(0,t). Note also that Wγ can be obtained from excursions of a Wiener process pointing them (independently of each other) up and down with probabilities p=(1+γ)/2 and q=(1−γ)/2, respectively.
Kulinich et al. [8, 7] considered limit theorems in the case where a is nonintegrable function such that
limx→±∞1x∫0x|va(v)−c±|dv=0,|xa(x)|⩽C,
where c±>−1/2-1/2$]]> are constants. In this case, an(x) converges in some sense to c−1x<0+c+1x⩾0 as n→∞.
For instance, if a(x)=c±/xfor±x>x0x_{0}$]]>, then, for c−<1/2<c+, the sequence Xn converges weakly to a Bessel process. If c−=c+>−1/2-1/2$]]>, then |Xn| also converges weakly to a Bessel process. The problem of weak convergence of Xn for (e.g.) c−=c+>−1/2-1/2$]]> or c−<c+⩽1/2 was not considered.
In this paper, we generalize the results of [14, 8] to the case
a(x)=a˜(x)+c¯(x)x,x∈R,
where a˜ is integrable on (−∞;∞), and
c¯(x)=c+·1x>1+c−·1x<−1,x∈R.1}+c_{-}\cdot \mathbb{1}_{x<-1},\hspace{1em}x\in \mathbb{R}.\]]]>
We consider all possible limit processes (depending on c+ and c−). In particular, we show that, for c+=c−<1/2, the limit process is a skew Bessel process (see Section 2).
We recall the definition and some properties of Bessel processes.
Let δ⩾0 and x0∈R. Consider the SDE
Z(x02,t)=x02+2∫0t|Z(x02,s)|dW(s)+δt,t⩾0,
where W is a Wiener process.
It is known (see [15], XI.1, (1.1)), that there exists a unique strong solution Z(x02,·) of (3). This solution is called the squared δ-dimensional Bessel process. The process Z(x02,·) is nonnegative a.s.
The process Bc(x0,t)=Z(x02,t) with x0⩾0 is called the (nonnegative) Bessel process with parameter c=(δ−1)/2.
We will call the process Bc−(x0,t)=−Bc(x0,t)=−Z(x02,t) with x0⩽0 the nonpositive Bessel process.
Recall the following properties of the Bessel process (see [15, Chap. XI]).
The Bessel process ξ(t)=Bc(x0,t) satisfies the SDE
dξt=dWt+cξtdt,t<T0,
where T0 is the first hitting time of 0. If δ⩾2 (i.e., c⩾1/2), then the Bessel process with probability 1 does not hit 0.
If 0<δ<2 (i.e., −1/2<c<1/2), then with probability 1 the Bessel process hits 0 but spends zero time at 0. In particular, if δ=1 (i.e., c=0), then the Bessel process is a reflecting Brownian motion.
If δ=0 (i.e., c=−1/2), then with probability 1 the process attains 0 and remains there forever.
The scale function of the Bessel process Bc equals
ψc(x)=−x−2c+1ifc>1/2,lnxifc=1/2,x−2c+1ifc<1/2,1/2,\\{} \ln x\hspace{1em}& \text{if}\hspace{2.5pt}\hspace{2.5pt}c=1/2,\\{} {x}^{-2c+1}\hspace{1em}& \text{if}\hspace{2.5pt}\hspace{2.5pt}c<1/2,\end{array}\right.\]]]>
that is,
Px(Ta<Tb)=ψc(b)−ψc(x)ψc(b)−ψc(a)for any0<a<x<b,
where Ty=inf{t⩾0:Bc(t)=y}.
The transition density for c>−1/2-1/2$]]>, x,y>00$]]>, and t>00$]]> equals
ptc(x,y)=t−1(y/x)νyexp(−(x2+y2)/2t)Iν(xy/t),
where Iν is a Bessel function of index ν=c−1/2.
Let c∈(−1/2,1/2), and let pt0,c(x,y) be the transition density of the Bessel process Bc killed at 0.
Set
ptskew(x,y)=pt0,c(|x|,|y|)·1xy>0+1+γy2(ptc(|x|,|y|)−pt0,c(|x|,|y|)),x,y∈R.0}\\{} & \displaystyle \hspace{1em}+\frac{1+\gamma y}{2}\big({p_{t}^{c}}\big(|x|,|y|\big)-{p_{t}^{0,c}}\big(|x|,|y|\big)\big),\hspace{1em}x,y\in \mathbb{R}.\end{array}\]]]>
It is easy to verify that this function satisfies the Chapman–Kolmogorov equation, is nonnegative, and ∫Rptskew(x,y)dy=1,x∈R.
A time-homogeneous Markov process with the transition density ptskew is called the skew Bessel processBc,γskew with parameters c and γ∈[−1,1].
We do not consider the skew Bessel process for c⩾1/2 because Bc(x0,·) does not hit 0 if x0≠0.
The skew Bessel process Bskew can be obtained from a nonnegative Bessel process by pointing its excursions up with probability p=1+γ2 and down with probability q=1−γ2, similarly to the case of a skew Brownian motion; see arguments in [1], Section 2.
Thus, the scale function of the skew Bessel process equals
ψskew(x)=(q1x⩾0−p1x<0)|x|−2c+1,x∈R.
For other properties of the skew Bessel process, we refer to [2].
If x0>00$]]> and p=1 (i.e., γ=1), then Bc,γskew(x0,·) is a (nonnegative) Bessel process Bc(x0,·) with parameter c: Bc,1skew(x0,·)=dBc(x0,·).
Also, the absolute value of the skew Bessel process |Bc,γskew| is a (nonnegative) Bessel process Bc(x0,·) with parameter c: |Bc,γskew(x0,·)|=dBc(x0,·).
If c=0, then Bc,γskew is a skew Brownian motion: B0,γskew(·)=dWγ(·).
Main results
Let
a(x)=a˜(x)+c¯(x)x,x∈R,
where a˜∈L1(R) and
c¯(x)=c+·1x>1+c−·1x<−1,x∈R.1}+c_{-}\cdot \mathbb{1}_{x<-1},\hspace{1em}x\in \mathbb{R}.\]]]>
Let Xn(t),t⩾0, be the solution of the SDE
dXn(t)=na(nXn(t))dt+dW(t)=(na˜(nXn(t))+c¯(nXn(t))Xn(t))dt+dW(t),t⩾0,Xn(0)=x0.
The existence and uniqueness of a strong solution of this SDE follows from [3, Thm. 4.53].
Ifc+andc−>−1/2-1/2$]]>, then the sequence of processes{Xn}converges weakly to a limit processX∞. In particular:
If
x0>00$]]>andc+⩾1/2, or
x0⩾0andc−<c+<1/2, or
x0=0andc−<1/2⩽c+,
thenX∞(t)=Bc++(x0,t),t⩾0.
Similarly, if
x0<0andc−⩾1/2, or
x0⩽0andc+<c−<1/2, or
x0=0andc+<1/2⩽c−,
thenX∞(t)=Bc−−(x0,t),t⩾0.
Ifx0<0,c−<1/2, andc−<c+, then the limiting process evolves asBc−−until hitting 0 and then proceeds asBc++indefinitely, that is,X∞(t)=Bc−−(x0,t)·1t⩽τ+Bc++(0,t−τ)·1t>τ,t⩾0,\tau },\hspace{1em}t\geqslant 0,\]]]>whereτ=inf{t:X∞(t)⩾0}andBc±±are independent (positive and negative) Bessel processes.
Ifc+=c−=:c<1/2, then, for anyx0,X∞(t)=Bc,γskew(x0,t),t⩾0,whereγ=(∫−∞+∞a˜(z)dz)=1−exp{−2∫−∞+∞a˜(z)dz}1+exp{−2∫−∞+∞a˜(z)dz}.
Finally, ifx0=0,c+⩾1/2, andc−⩾1/2, then the distribution of the limit processX∞equalsp·PBc+++(1−p)·PBc−−,wherep=∫0∞A(−y)(y∨1)−2c−dy∫0∞(A(−y)(y∨1)−2c−+A(y)(y∨1)−2c+)dy,A(y)=exp{−2∫0ya˜(z)dz}, andPBc±±are the distributions of positive and negative Bessel processesBc±±(0,·)starting from 0.
Some results of the theorem follow from [8]. However, we apply here the general approach applicable to all cases simultaneously. Condition (2) is somewhat weaker than a˜∈L1(R). However, we do not assume that supx|xa˜(x)|<∞, contrary to the paper [8].
Proof
It follows from [9, Section 3] or [11, Section 3.7] that if A1 is satisfied, then, for any α>00$]]>, we have the convergence
Xn(·∧τn,α)⇒Bc++(x0,·∧τ0,α),n→∞,
where τn,α=inf{t⩾0:Xn(t)⩽α} and τ0,α=inf{t⩾0:Bc++(x0,t)⩽α}. Since the process Bc++(x0,·) does not hit 0, this yields the proof. Case A2 is considered similarly.
To prove all other items of Theorem 1, we use the method proposed in [13].
Let {ξ(n),n⩾0} be a sequence of continuous homogeneous strong Markov processes. For α>00$]]>, set
τn,α:=inf{t⩾0:|ξ(n)(t)|⩽α},σn,α:=inf{t⩾0:|ξ(n)(t)|⩾α}.
We denote by ξx0(n) a process that has the distribution of ξ(n) conditioned by ξ(n)(0)=x0.
The next statement is a particular case of Theorem 2 of [13].
Assume that the sequence{ξ(n),n⩾0}satisfies the following conditions: ξ(n)(0)⇒ξ(0)(0);∀T>0∀ε>0∃δ>0∃n0∀n⩾n0P(sup|s−t|<δ,s,t∈[0,T]|ξ(n)(t)−ξ(n)(s)|⩾ε)⩽ε;0\hspace{2.5pt}\forall \varepsilon >0\hspace{2.5pt}\exists \delta >0\hspace{2.5pt}\exists n_{0}\hspace{2.5pt}\forall n\geqslant n_{0}\\{} & \displaystyle \hspace{1em}\mathrm{P}\Big(\underset{\begin{array}{c} |s-t|<\delta ,\\{}s,t\in [0,T]\end{array}}{\sup }\big|{\xi }^{(n)}(t)-{\xi }^{(n)}(s)\big|\geqslant \varepsilon \Big)\leqslant \varepsilon ;\end{array}\]]]>∀T>0limε→0+supnE∫0T1|ξ(n)(t)|⩽εdt=0;0\hspace{2em}\underset{\varepsilon \to 0+}{\lim }\underset{n}{\sup }\mathrm{E}{\int _{0}^{T}}\mathbb{1}_{|{\xi }^{(n)}(t)|\leqslant \varepsilon }\hspace{0.1667em}dt=0;\end{array}\]]]>∫0∞1ξ(0)(t)=0dt=0a.s.Assume that, for anyα>00$]]>,x0∈R, and any sequence{xn}such thatlimn→∞xn=x0, we have(ξxn(n)(·∧τn,α),τn,α)⇒(ξx0(0)(·∧τ0,α),τ0,α),n→∞;ξxn(n)(σn,α)⇒ξx0(0)(σ0,α),n→∞.Thenξ(n)⇒ξ(0)inC([0,∞))asn→∞.
We apply this lemma for ξ(n)=Xn,n⩾1, and ξ(0)=X∞ in cases A1–A5 of the theorem. Case A6 will be considered separately.
Condition (12) is the only condition that is not true in case A6. It fails if x0=0. Indeed, for any x>00$]]>, the process Bc++(x,·) does not hit 0. So, we may select a sequence {xn}⊂(0,∞) that converges to 0 sufficiently slowly and such that, given Xn(0)=xn, we have Xn(·)⇒B+(0,·) and limn→∞P(∃t≥0:Xn(t)=0)=0. The concrete selection of {xk} can be done using formulas (15) and (16). Since B+(0,σ0,α)=α a.s., we get Xn(σn,α)⇒α. However, if all xn were negative, then the limit might be −α.
Conditions (7) and (10) are obvious.
The convergence
∀α>0ξxn(n)(·∧τn,α)⇒ξx0(0)(·∧τ0,α),n→∞,0\hspace{2.5pt}\hspace{2.5pt}\hspace{2.5pt}{\xi _{x_{n}}^{(n)}}\big(\cdot \wedge {\tau }^{n,\alpha }\big)\hspace{1em}\Rightarrow \hspace{1em}{\xi _{x_{0}}^{(0)}}\big(\cdot \wedge {\tau }^{0,\alpha }\big),\hspace{1em}n\to \infty ,\]]]>
follows from [9, Section 3] or [11, Section 3.7]. Since
P(∀ε>0∃t∈(τ0,α,τ0,α+ε):|ξx0(0)(t)|<α|τ0,α<∞)=1,0\hspace{2.5pt}\exists t\in \big({\tau }^{0,\alpha },{\tau }^{0,\alpha }+\varepsilon \big):\hspace{2.5pt}\big|{\xi _{x_{0}}^{(0)}}(t)\big|<\alpha \hspace{2.5pt}|\hspace{2.5pt}{\tau }^{0,\alpha }<\infty \big)=1,\]]]>
convergence (13) yields the convergence of pairs (11).
Let us check condition (8). Set
A(y)=exp{−2∫0ya˜(z)dz},An(y)=exp{−2∫0yna˜(nz)dz}=A(ny),y∈R,Φn(x)=∫0xAn(y)dy,x∈R.
Observe that Φn:R→R is a bijection, Φn(0)=0, and
∃L>0∀n∀x,y∈RL−1|x−y|⩽|Φn(x)−Φn(y)|⩽L|x−y|.0\hspace{2.5pt}\hspace{2.5pt}\forall n\hspace{2.5pt}\hspace{2.5pt}\forall x,y\in \mathbb{R}\hspace{2em}{L}^{-1}|x-y|\leqslant \big|\varPhi _{n}(x)-\varPhi _{n}(y)\big|\leqslant L|x-y|.\]]]>
Itô’s formula yields
dΦn(Xn(t))=A(nXn(t))(c¯(nXn(t))Xn(t)dt+dW(t)).
So
|Xn(t)−Xn(s)|⩽L|Φn(Xn(t))−Φn(Xn(s))|⩽L|∫stA(nXn(z))c¯(nXn(z))Xn(z)dz|+L|∫stA(nXn(z))dW(z)|.
Let |s−t|<δ, and let Δ>00$]]> be fixed. Denote fn(t):=∫0tA(nXn(z))dW(z).
a) Assume that |Xn(z)|>Δ,z∈[s,t]\varDelta ,z\in [s,t]$]]>. Then
|∫stA(nXn(z))c¯(nXn(z))Xn(z)dz|⩽Cδ/Δ,
where C=‖A‖∞max(|c−|,|c+|)<∞. Hence, we have the estimate
|Xn(t)−Xn(s)|⩽LCδ/Δ+Lωfn(δ),
where ωf(δ)=sup|s−t|<δ,s,t∈[0,T]|f(t)−f(s)| is the modulus of continuity.
b) Assume that |Xn(z0)|⩽Δ for some z0∈[s,t].
Denote τ:=inf{z⩾s:|Xn(z)|⩽Δ} and σ:=sup{z⩽t:|Xn(z)|⩽Δ}. Then
|Xn(t)−Xn(s)|⩽|Xn(s)−Xn(τ)|+|Xn(σ)−Xn(t)|+2Δ⩽2LCδ/Δ+2Lωfn(δ)+2Δ.
Thus, in any case, we have the following estimate of the modulus of continuity:
ωXn(δ)⩽2LCδ/Δ+2Lωfn(δ)+2Δ.
Let Δ⩽ε/6. Then, for δ⩽εΔ6LC, we have
supnP(ωXn(δ)⩾ε)⩽supnP(ε/3+2Lωfn(δ)+ε/3⩾ε)=supnP(ωfn(δ)⩾ε/6L)→0,δ→0+.
The last convergence follows from the fact that the sequence of distributions of {fn(·)=∫0.A(nXn(z))dW(z)}n⩾1 in the space of continuous functions is weakly relatively compact because the function A is bounded.
Let us prove (12) in cases A1–A5.
The proof below yields that condition (12) is true if xn=0 for all n⩾0.
Let |x|<α. It is easy to see that Px(σn,α<∞)=1,n∈N∪{∞}. Since the process Xn is continuous, we have |Xn(σn,α)|=α a.s.
By pxn=Px(Xn(σn,α)=α),n∈N∪{∞}, we denote the probability to reach α before reaching −α when starting from x.
Using formulas (4) and (5) for the scale of a Bessel process and a skew Bessel process, it is easy to check that
px∞=1x⩾0−(1−ψc−(−x)ψc−(α))1x<0in cases A1, A3,ψc+(x)ψc+(α)·1x>0in cases A2, A4,ψc(|x|)ψc(α)(q1x⩾0−p1x<0)+pin case A5,0}\hspace{1em}& \text{in cases A2, A4},\\{} \frac{\psi _{c}(|x|)}{\psi _{c}(\alpha )}(q\mathbb{1}_{x\geqslant 0}-p\mathbb{1}_{x<0})+p\hspace{1em}& \text{in case A5},\end{array}\right.\]]]>
where ψc is given in (4).
For n∈N, we have (see [4, Section 15] and [15])
pxn=φn(x)−φn(−α)φn(α)−φn(−α),
where
φn(x)=∫0xexp{−2∫0yan(z)dz}dy=∫0xexp{−2∫0yna(nz)dz}dy=1n∫0nxexp{−2∫0ya(z)dz}dy=1nφ(nx),φ(x):=∫0xexp{−2∫0ya(z)dz}dy.
The function φ is increasing. It follows from the definition of a that φ is bounded from above (below) iff c+>1/21/2$]]> (c−>1/21/2$]]>). The function φ has the following asymptotic behavior:
φ(x)∼±A(±∞)|x|1−2c±1−2c±ifc±<1/2,±A(±∞)ln|x|ifc±=1/2,x→±∞,
where
A(y)=exp{−2∫0ya˜(z)dz},y∈R,
and
limx→±∞φ(x)=φ(±∞)∈Rifc±>1/2.1/2.\]]]>
Condition (12) follows from (14), (15), (16), (17), (18) in cases A1–A5 (and in case A6 if xn=0,n⩾0).
Set τn=inf{t⩾0:|Xn(t)|⩾1}.
Assume thatlimε→0+sup|x|⩽1supnEx∫0τn1|Xn(t)|⩽εdt=0.Then (9) is satisfied, that is,∀T>0limε→0+supnE∫0T1|Xn(t)|⩽εdt=0.0\hspace{1em}\underset{\varepsilon \to 0+}{\lim }\underset{n}{\sup }\mathrm{E}{\int _{0}^{T}}\mathbb{1}_{|X_{n}(t)|\leqslant \varepsilon }\hspace{0.1667em}dt=0.\]]]>
Introduce the notations
Sn,±0:=0,Tn,±k:=inf{t≥Sn,±k−1:Xn(t)=±1},Sn,±k:=inf{t≥Tn,±k:Xn(t)=±ε},T˜n,±k:=inf{t≥Sn,±k:|Xn(t)|=1},βn,±k:=∫Sn,±kT˜n,±k1|Xn(t)|⩽εdt,αn,±k:=Sn,±k−Tn,±k,k⩾1.
Then
∫0T1|Xn(t)|⩽εdt⩽∫0τn1|Xn(t)|⩽εdt+∑k(βn,+1+⋯+βn,+k)1αn,+1<T,…,αn,+k<T,αn,+k+1⩾T+∑k(βn,−1+⋯+βn,−k)1αn,−1<T,…,αn,−k<T,αn,−k+1⩾T.
It follows from the strong Markov property that
∑kE(βn,+1+⋯+βn,+k)1αn,+1<T,…,αn,+k<T,αn,+k+1⩾T=∑kkEε∫0τn1|Xn(t)|⩽εdt(1−pn,+)kpn,+=(pn,+)−1Eε∫0τn1|Xn(t)|⩽εdt,
where pn,+=P1(Sn,+1⩾T).
Considering the last term similarly, we get the inequality
E∫0T1|Xn(t)|⩽εdt⩽(1+(pn,+)−1+(pn,−)−1)sup|x|⩽1supnEx∫0τn1|Xn(t)|⩽εdt.
It is not difficult to see that supn(pn,±)−1<∞. The lemma is proved. □
Let us verify (19). It is known [6, Chap. 4.3] that
un,ε(x):=Ex∫0τn1|Xn(t)|⩽εdt
is of the form
un,ε(x)=∫−11Gn(x,y)1|y|⩽εmn(dy),
where Green’s function Gn equals
Gn(x,y)=(φn(x)−φn(−1))(φn(1)−φn(y))φn(1)−φn(−1),x⩽y,Gn(y,x),x⩾y,
with φn given by formula (16), and
mn(dx)=exp{2∫0xan(z)dz}dx.
The function un,ε(x) is a generalized solution (because an may be discontinuous) of the equation
1/2un,ε″(x)+an(x)un,ε′(x)=−1|x|⩽ε(x),|x|⩽1,
with boundary conditions un,ε(±1)=0.
A direct verification of the condition limε→0+sup|x|⩽1supnun,ε(x)=0 is possible but cumbersome. We prove the corresponding convergence using the comparison theorem. We consider only the case where an satisfies the Lipschitz condition. The general case follows by approximation.
It follows from the Itô–Tanaka formula that
d|Xn(t)|=(Xn(t))an(Xn(t))dt+(Xn(t))dW(t)+dln(t)=(Xn(t))an(Xn(t))dt+dWn(t)+dln(t),
where Wn is a new Wiener process, and ln is the local time of Xn at zero.
Let −1/2<c<min(c−,c+,0). It follows from the arguments of [12] on comparison of reflecting SDEs that |Xn(t)|⩾Yn(t),t⩾0, where Yn satisfies the following SDE with reflection at zero:
dYn(t)=a¯n(Yn(t))dt+dWn(t)+dl˜n(t).
Here Wn(t)=∫0t(Xn(s))dW(s) is a Wiener process, l˜n is the local time of Yn at 0, a¯n(x)=na¯(nx),a¯(x)=−(|a(x)|+|a(−x)|)−cx1|x|>1+r(x)1}+r(x)$]]>, and r is any nonpositive function such that a¯ satisfies Lipschitz condition. We will also assume that ∫R|r(x)|dx⩽∫R|b(x)|dx. The Lipschitz property is used only for application of comparison theorem.
To prove (19), it suffices to verify that
limε→0supx∈[0,1]supnu¯n,ε(x):=limε→0supx∈[0,1]supnEx∫0τ¯n1Yn(s)∈[0,ε]ds=0,
where τ¯n is the entry time of Yn into [1,∞).
It is known [6] that
u¯n,ε(x)=2∫x1exp{−2∫1ya¯n(z)dz}∫0y1[0,ε](z)exp{2∫1ya¯n(z)dz}dy
is a (generalized) solution of the equation
1/2u¯n,ε″(x)+a¯n(x)u¯n,ε′(x)=−1[0,ε],x∈[0,1],
with boundary conditions un,ε′(0)=0,un,ε(1)=0. So
supx∈[0,1]supnu¯n,ε(x)=u¯n,ε(0)=2∫01exp{−2∫1ya¯n(z)dz}∫0y1[0,ε](z)exp{2∫1ya¯n(z)dz}dy⩽K∫01exp{∫0yy¯−2cdz}∫0y1[0,ε](z)y2cdy,
where K is a constant that depends only on ∫R|b(x)|dx and c (and is independent of n). By our choice, c∈(−1/2,0), so the right-hand side of (21) tends to 0 as ε→0+ by the Lebesgue dominated convergence theorem.
The theorem is proved in cases A1–A5.
Consider case A6. Note that conditions (7)–(11) are satisfied for ξ(n)=Xn, n⩾1, and ξ(0)=X∞, where X∞ is given in the theorem. In particular, this implies that the sequence of distributions of stochastic processes {Xn} in the space of continuous functions is weakly relatively compact. Choosing an arbitrary convergent subsequence, without loss of generality, we may assume that {Xn} itself converges weakly to a continuous process X. Let δ>00$]]>, and let σn,δ=inf{t⩾0:Xn(t)=δ},σδ=inf{t⩾0:X(t)=δ}. It follows from formulas for the scale function of the processes {Xn} that limn→∞P(Xn(σn,δ)=δ)=p,limn→∞P(Xn(σn,δ)=−δ)=1−p, where p is given by (6). Formulas (9) and (11) imply that the limit process exits from the interval [−δ,δ] with probability 1.
Observe that, for almost all δ>00$]]>, with respect to the Lebesgue measure, the distribution of Xn(σn,δ+·) converges weakly as n→∞ to the distribution of X(δ+·). Indeed, by the Skorokhod theorem on a single probability space (see [16]), without loss of generality, we may assume that the sequence {Xn} converges to X uniformly on compact sets with probability 1. For simplicity, we will assume that the convergence holds for all ω and that also σn,δ,σδ<∞ for all ω,n,δ>00$]]>. So we show convergence
Xn(σn,δ+·)→X(σδ+·)
if we prove that
σn,δ→σδ,n→∞.
Convergence (23) may fail only if σδ is a point of a local maximum of X. It follows from the definition that σδ is a point of a strict local maximum of X from the left. The set of points of local maximums that are strict maximums from the left is at most countable. This yields that, for almost all ω and almost all δ>00$]]> with respect to the Lebesgue measure, we have convergence (23) and hence (22).
On the other hand, the distribution of Xn(σn,δ+·) converges weakly as n→∞ to the distribution of the process 1Ω−Bc−−(−δ,·)+1Ω+Bc++(δ,·), where P(Ω−)=1−p,P(Ω+)=p, and the σ-algebra {∅,Ω−,Ω+,Ω} is independent of σ(Bc±±(±δ,t),t⩾0).
Recall that assumptions of the theorem yield
P(∃t⩾0:Bc±±(±δ,t)=0)=0.
It follows from (9) that
P(∫0∞1X(s)=0ds=0)=1.
Thus, we have the almost sure convergence in C([0,∞))X(σδ+·)→X(·),δ→0.
The processes 1Ω−Bc−−(−δ,·)+1Ω+Bc++(δ,·) converge in distribution to
1Ω−Bc−−(0,·)+1Ω+Bc++(0,·),
where the σ-algebras {∅,Ω−,Ω+,Ω} and σ(Bc±±(0,t),t⩾0) are independent.
This completes the proof of Theorem 1.
Acknowledgments
The authors thank the anonymous referee for valuable comments that helped improving the presentation.
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