We consider a stochastic differential equation of the form
dXt=θa(t,Xt)dt+σ1(t,Xt)σ2(t,Yt)dWt
with multiplicative stochastic volatility, where Y is some adapted stochastic process. We prove existence–uniqueness results for weak and strong solutions of this equation under various conditions on the process Y and the coefficients a, σ1, and σ2. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter θ. We suppose that Y is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process Y are provided supplying the strong consistency.

The goal of the paper is to study the stochastic differential equation (SDE), the diffusion coefficient of which includes an additional stochastic process:
dXt=θa(t,Xt)dt+σ(t,Xt,Yt)dWt,
where σ(t,x,y)=σ1(t,x)σ2(t,y), and to estimate the drift parameter θ by the observations of stochastic processes X and Y. Such equations often arise as models of a financial market in mathematical finance. For example, one of the first models of such a type with σ(t,x,y)=xy was proposed in [8], where Y was the square root of the geometric Brownian motion process. A similar model was considered by Heston [6], where the volatility was governed by the Ornstein–Uhlenbeck process. Fouque et al. used the model with stochastic volatility driven by the Cox–Ingersoll–Ross process; see [4, 5]. The case where σ(t,x,y)=xσ2(y) and Y is the Ornstein–Uhlenbeck process was studied in [12, 13].

In the present paper, we investigate the existence and uniqueness of weak and strong solutions to the equation (1). We adapt the approaches of Skorokhod [20], Stroock and Varadhan [21, 22], and Krylov [10, 11] to establish the weak existence and weak uniqueness. Concerning the strong existence and uniqueness, we use the well-known approaches of Yamada and Watanabe [23] (see also [2]) for inhomogeneous coefficients and Lipschitz conditions. In the present paper, we consider only the case of multiplicative stochastic volatility, where, as it was mentioned, the diffusion coefficient is factorized as σ(t,x,y)=σ1(t,x)σ2(t,y). Then we construct the maximum likelihood estimator for the unknown drift parameter and prove its strong consistency. As an example, we consider a linear model with stochastic volatility driven by a solution to some Itô’s SDE. In particular, we study in details an SDE with constant coefficients, the Ornstein–Uhlenbeck process, and the geometric Brownian motion, as the model for volatility (note that process Y can be interpreted not only as a volatility, but also as an additional source of randomness). Note that the maximum likelihood estimation in the Ornstein–Uhlenbeck model with stochastic volatility was studied in [1]. Similar statistical methods for the case of deterministic volatility can be found in [7, 9, 16, 18].

The paper is organized as follows. In Section 2, we prove the existence of weak and strong solutions under different conditions. In Section 3, we establish the strong consistency of the maximum likelihood estimator of the unknown drift parameter θ. Section 4 contains the illustrations of our results with some simulations. Auxiliary statements are gathered in Section 5.

Existence and uniqueness results for weak and strong solutions

Let (Ω,F,F‾,P) be a complete probability space with filtration F‾=Ft,t≥0 satisfying the standard assumptions. We assume that all processes under consideration are adapted to the filtration F‾.

Existence of weak solution in terms of Skorokhod conditions

Consider the following stochastic differential equation:
dXt=a(t,Xt)dt+σ1(t,Xt)σ2(t,Yt)dWt,
where X|t=0=X0∈R, W is a Wiener process, and Y is some adapted stochastic process to be specified later.

Let Y be a measurable and continuous process, a,σ1, andσ2be continuous w.r.t.x∈R,y∈R, andt∈[0,T],σ2be bounded, and|σ1(t,x)|2+|a(t,x)|2≤K(1+x2)for some constantK>00$]]>. Then Eq. (2) has a weak solution.

Consider a sequence of partitions of [0,T]: 0=t0n<t1n<⋯<tnn=T such that limn→∞maxk(tk+1n−tkn)=0. Define ξkn by ξ0n=X(0) and
ξk+1n=ξkn+a(tkn,ξkn)Δtkn+σ1(tkn,ξkn)σ2(tkn,Y(tkn))ΔWkn.
It follows from Lemma 1, Lemma 2, and Proposition 1 in Section 5 that it is possible to choose a subsequence n′ and construct processes ξ˜n′, W˜n′, and Y˜n′ such that the finite-dimensional distributions of ξ˜n′, W˜n′, and Y˜n′ coincide with those of ξn′, W, and Y and ξ˜n′→ξ˜, W˜n′→W˜ and Y˜n′→Y˜ in probability, where ξ˜, W˜, and Y˜ are some stochastic processes (evidently, W˜ is a Wiener process). It suffices to prove that ξ˜ is a solution of Eq. (2) when W and Y are replaced by W˜ and Y˜.

We have that ξ˜n′ satisfies the equation
ξ˜n′(t)=ξ˜n′(0)+∑tk+1n′≤ta(tkn′,ξ˜n′(tkn′))Δtkn′+∑tk+1n′≤tσ1(tkn′,ξ˜n′(tkn′))σ2(tkn′,Y˜n′(tkn′))ΔW˜kn′.
Since σ2 is bounded and σ1 is of linear growth, their product is of linear growth:
|σ1(t,x)σ2(t,y)|≤K1(1+x),
where K1>00$]]> is a constant. Therefore,
P(sup0≤t≤Tσ1(t,ξ˜n′(t))σ2(t,Y˜n′(t))>C)≤P(sup0≤t≤TK1(1+ξ˜n′(t))>C)=P(sup0≤t≤Tξ˜n′(t)>CK1−1)=P(sup0≤t≤T|ξn′(t)|>CK1−1).C\Big)\\{} & \displaystyle \hspace{1em}\le \mathbb{P}\Big(\underset{0\le t\le T}{\sup }K_{1}\big(1+\left|\widetilde{\xi }_{{n^{\prime }}}(t)\right|\big)>C\Big)=\mathbb{P}\bigg(\underset{0\le t\le T}{\sup }\left|\widetilde{\xi }_{{n^{\prime }}}(t)\right|>\frac{C}{K_{1}}-1\bigg)\\{} & \displaystyle \hspace{1em}=\mathbb{P}\bigg(\underset{0\le t\le T}{\sup }\big|{\xi }^{{n^{\prime }}}(t)\big|>\frac{C}{K_{1}}-1\bigg).\end{array}\]]]>
Using Lemma 1, we get that
P(sup0≤t≤Tσ1(t,ξ˜n′(t))σ2(t,Y˜n′(t))>C)→0asC→∞.C\Big)\to 0\hspace{1em}\text{as}\hspace{2.5pt}C\to \infty .\]]]>
Moreover, we have that σ1(t,x)σ2(t,y) is continuous w.r.t. t∈[0,T], x,y∈R. Then, for any ε>00$]]>, there exists δ>00$]]> such that
|σ1(t1,x1)σ2(t1,y1)−σ1(t2,x2)σ2(t2,y2)|<ε
whenever t1−t2<δ, x1−x2<δ, y1−y2<δ. Therefore,
P(σ1(t1,ξ˜n′(t1))σ2(t1,Y˜n′(t1))−σ1(t2,ξ˜n′(t2))σ2(t2,Y˜n′(t2))>ε)≤P(ξ˜n′(t1)−ξ˜n′(t2)<δ,Y˜n′(t1)−Y˜n′(t2)<δ,σ1(t1,ξ˜n′(t1))σ2(t1,Y˜n′(t1))−σ1(t2,ξ˜n′(t2))σ2(t2,Y˜n′(t2))>ε)+P(ξ˜n′(t1)−ξ˜n′(t2)≥δ)+P(Y˜n′(t1)−Y˜n′(t2)≥δ)=P(ξn′(t1)−ξn′(t2)≥δ)+P(Y(t1)−Y(t2)≥δ),\varepsilon \big)\\{} & \displaystyle \le \mathbb{P}\big(\left|\widetilde{\xi }_{{n^{\prime }}}(t_{1})-\widetilde{\xi }_{{n^{\prime }}}(t_{2})\right|<\delta ,\left|\widetilde{Y}_{{n^{\prime }}}(t_{1})-\widetilde{Y}_{{n^{\prime }}}(t_{2})\right|<\delta ,\\{} & \displaystyle \hspace{1em}\left|\sigma _{1}\big(t_{1},\widetilde{\xi }_{{n^{\prime }}}(t_{1})\big)\sigma _{2}\big(t_{1},\widetilde{Y}_{{n^{\prime }}}(t_{1})\big)-\sigma _{1}\big(t_{2},\widetilde{\xi }_{{n^{\prime }}}(t_{2})\big)\sigma _{2}\big(t_{2},\widetilde{Y}_{{n^{\prime }}}(t_{2})\big)\right|>\varepsilon \big)\\{} & \displaystyle \hspace{1em}+\mathbb{P}\big(\left|\widetilde{\xi }_{{n^{\prime }}}(t_{1})-\widetilde{\xi }_{{n^{\prime }}}(t_{2})\right|\ge \delta \big)+\mathbb{P}\big(\left|\widetilde{Y}_{{n^{\prime }}}(t_{1})-\widetilde{Y}_{{n^{\prime }}}(t_{2})\right|\ge \delta \big)\\{} & \displaystyle =\mathbb{P}\big(\left|\xi _{{n^{\prime }}}(t_{1})-\xi _{{n^{\prime }}}(t_{2})\right|\ge \delta \big)+\mathbb{P}\big(\left|Y(t_{1})-Y(t_{2})\right|\ge \delta \big),\end{array}\]]]>
and the last relation implies the following one:
limh→0limn′→∞supt1−t2≤hP(|σ1(t1,ξ˜n′(t1))σ2(t1,Y˜n′(t1))−σ1(t2,ξ˜n′(t2))σ2(t2,Y˜n′(t2))|>ε)=0.\varepsilon \big)=0.\end{array}\]]]>
Applying Lemma 1, we get that
∑tk+1n′≤tσ1(tkn′,ξ˜n′(tkn′))σ2(tkn′,Y˜n′(tkn′))ΔWkn′→∫0Tσ1(s,ξ˜(s))σ2(s,Y˜(s))dW˜(s)
in probability as n′→∞, and we also have that
∑tk+1n′≤ta(tkn′,ξ˜n′(tkn′))Δtkn′→∫0Ta(s,ξ˜(s))ds,
whence the proof follows. □

Existence and uniqueness of weak solution in terms of Stroock–Varadhan conditions

In this approach, we assume additionally that the process Y also is a solution of some diffusion stochastic differential equation. Let W1 and W2 be two Wiener processes, possibly correlated, so that dWt1dWt2=ρdt for some |ρ|≤1. In this case, we can represent Wt2=ρWt1+1−ρ2Wt3, where W3 is a Wiener process independent of W1.

Consider the system of stochastic differential equationswhere all coefficients a, α,σ1,σ2, and β are nonrandom measurable and bounded functions,σ1,σ2, and β are continuous in all arguments. Let|ρ|<1, and letσ1(t,x)>00$]]>,σ2(t,y)>00$]]>,β(t,y)>00$]]>for allt,x,y. Then the weak existence and uniqueness in law hold for system (3)–(4), and in particular, the weak existence and uniqueness in law hold for Eq. (3) with Y being a weak solution of Eq. (4).

Equations (3) and (4) are equivalent to the two-dimensional stochastic differential equation
dZ(t)=A(t,Zt)dt+B(t,Zt)dW(t),
where Z(t)=(X(t)Y(t)), W(t)=(W1(t)W3(t)) is a two dimensional Wiener process,
A(t,x,y)=a(t,x)α(t,y),andB(t,x,y)=σ1(t,x)σ2(t,y)0ρβ(t,y)1−ρ2β(t,y).
It follows from the measurability and boundedness of a and α and from the continuity and boundedness of σ1, σ2, and β that the coefficients of matrices A and B are nonrandom, measurable, and bounded, and additionally the coefficients of B are continuous in all arguments. Then we can apply Theorems 4.2 and 5.6 from [21] (see also Prop. 1.14 in [3]) and deduce that we have to prove the following relation: for any (t,x,y)∈R+×R2, there exists ε(t,x,y)>00$]]> such that, for all λ∈R2,
‖B(t,x,y)λ‖≥ε(t,x,y)‖λ‖.
Relation (5) is equivalent to the following one (we omit arguments):
σ12σ22λ12+β2(ρλ1+1−ρ2λ2)2≥ε2(λ12+λ22)
or
(σ12σ22+β2ρ2)λ12+β2(1−ρ2)λ22+2ρ1−ρ2β2λ1λ2≥ε2(λ12+λ22).
The quadratic form
Q(λ1,λ2)=(σ12σ22+β2ρ2)λ12+β2(1−ρ2)λ22+2ρ1−ρ2β2λ1λ2
in the left-hand side of (6) is positive definite since its discriminant
D=ρ2(1−ρ2)β4−β2(1−ρ2)(σ12σ22+β2ρ2)=−β2(1−ρ2)σ12σ22<0.
The continuity of Q(λ1,λ2) implies the existence of minλ12+λ22=1Q(λ1,λ2)>00$]]>. Then, putting ε=minλ12+λ22=1Q(λ1,λ2) and using homogeneity, we get (6). □

Existence of strong solution in terms of Yamada–Watanabe conditions

Now we consider strong existence–uniqueness conditions for Eq. (2), adapting the Yamada–Watanabe conditions for inhomogeneous coefficients from [2].

Let a,σ1, andσ2be nonrandom measurable bounded functions such that

There exists a positive increasing functionρ(u),u∈(0,∞), satisfyingρ(0)=0such that|σ1(t,x)−σ1(t,y)|≤ρ(x−y),t≥0,x,y∈R,and∫0∞ρ−2(u)du=+∞.

There exists a positive increasing concave functionk(u),u∈(0,∞), satisfyingk(0)=0such that|a(t,x)−a(t,y)|≤k(x−y),t≥0,x,y∈R,and∫0∞k−1(u)du=+∞. Also, let Y be an adapted continuous stochastic process. Then the pathwise uniqueness of solution holds for Eq. (2), and hence it has a unique strong solution.

Let 1>a1>a2>⋯>an>⋯>0a_{1}>a_{2}>\cdots >a_{n}>\cdots >0$]]> be defined by
∫a11ρ−2(u)du=1,∫a2a1ρ−2(u)du=2,…,∫anan−1ρ−2(u)du=n,….
We have that an→0 as n→∞. Let ϕn(u), n=1,2,…, be a continuous function with support contained in (an,an−1) such that 0≤ϕn(u)≤2ρ−2(u)n and ∫anan−1ϕn(u)du=1. Such a function obviously exists. Set
φn(x)=∫0x∫0yϕn(u)dudy,x∈R.
Clearly, φn∈C2(R), φn′(x)≤1, and φn(x)↗x as n→∞.

Let X1 and X2 be two solutions of Eq. (2) on the same probability space with the same Wiener process and such that X1(0)=X2(0). Then we can present their difference as
X1(t)−X2(t)=∫0tσ2(s,Y(s))(σ1(s,X1(s))−σ1(s,X2(s)))dW(s)+∫0t(a(s,X1(s))−a(s,X2(s)))ds.
By the Itô formula,
φn(X1(t)−X2(t))=∫0tφn′(X1(s)−X2(s))σ2(s,Y(s))(σ1(s,X1(s))−σ1(s,X2(s)))dW(s)+∫0tφn′(X1(s)−X2(s))(a(s,X1(s))−a(s,X2(s)))ds+12∫0tφn″(X1(s)−X2(s))σ2(s,Y(s))2(σ1(s,X1(s))−σ1(s,X2(s)))2ds=J1+J2+J3.
We have that E(J1)=0,
E(J2)≤∫0tEa(s,X1(s))−a(s,X2(s))ds≤∫0tE(k(X1(s)−X2(s))ds≤∫0tk(E(X1(s)−X2(s))ds
by Jensen’s inequality, and
E(J3)≤C22∫0tE(2nρ−2(|X1(s)−X2(s)|)ρ2(|X1(s)−X2(s)|))ds≤tn→0asn→∞.
So by letting n→∞ we get
E(|X1(s)−X2(s)|)≤∫0tk(E(|X1(s)−X2(s)|)ds.
We have that ∫0∞k−1(u)du=+∞. Then we get E(X1(s)−X2(s))=0, and hence X1(s)=X2(s) a.s. □

Existence and uniqueness for strong solution in terms of Lipschitz conditions

Let a,σ1, andσ2be nonrandom measurable functions, and let Y be an adapted continuous stochastic process. Consider the following assumptions:

There existsK>00$]]>such that, for allt≥0andx∈R,|σ1(t,x)|2+|a(t,x)|2≤K2(1+x2);

For anyn∈N, there existsKN>00$]]>such that, for allt≥0and for all(x,y)satisfyingx≤Nandy≤N,|a(t,x)−a(t,y)|+|σ1(t,x)−σ1(t,y)|≤KNx−y;

sups≥0supx≤Nσ2(s,x)≤CN.

Then Eq. (2) has a unique strong solution.

This result can be proved by using the successive approximation method; see, for example, [19, Thm. 1.2].

Drift parameter estimationGeneral results

Let (Ω,F,F‾,P) be a complete probability space with filtration F‾=Ft,t≥0 satisfying the standard assumptions. We assume that all processes under consideration are adapted to the filtration F‾. Consider a parameterized version of Eq. (2)
dXt=θa(t,Xt)dt+σ1(t,Xt)σ2(t,Yt)dWt,
where W is a Wiener process. Assume that Eq. (7) has a unique strong solution X={Xt,t∈[0,T]}. Our main problem is to estimate the unknown parameter θ by continuous observations of X and Y.

Denote
f(t,x,y)=a(t,x)σ12(t,x)σ22(t,y),g(t,x,y)=a(t,x)σ1(t,x)σ2(t,y).
Assume that, for all t>00$]]>, σ1(t,Xt)σ2(t,Yt)≠0a.s.,∫0tg2(s,Xs,Ys)ds<∞a.s.,∫0∞g2(s,Xs,Ys)ds=∞a.s. Then a likelihood function for Eq. (7) has the form
dPθ(T)dP0(T)=expθ∫0Tf(t,Xt,Yt)dXt−θ22∫0Tg2(t,Xt,Yt)dt;
see [15, Ch. 7]. Hence, the maximum likelihood estimator of parameter θ constructed by observations of X and Y on the interval [0,T] has the form
θˆT=∫0Tf(t,Xt,Yt)dXt∫0Tg2(t,Xt,Yt)dt=θ+∫0Tg(t,Xt,Yt)dWt∫0Tg2(t,Xt,Yt)dt.

Under assumptions (8)–(10), the estimatorθˆTis strongly consistent asT→∞.

Note that, under condition (9) the process Mt=∫0tg(s,Xs,Ys)dWs is a square-integrable local martingale with quadratic variation ⟨M⟩t=∫0tg2(s,Xs,Ys)ds. According to the strong law of large numbers for martingales [14, Ch. 2, § 6, Thm. 10, Cor. 1], under the condition ⟨M⟩T→∞ a.s. as T→∞, we have that MT⟨M⟩T→0 a.s. as T→∞. Therefore, it follows from representation (11) that θˆT is strongly consistent. □

Linear equation with stochastic volatility

As an example, let us consider the model
dXt=θXtdt+Xtσ2(Yt)dWt,X0=x0∈R,
where Wt is a Wiener process, and Yt is a continuous stochastic process with values from an open interval J=(l,r) (further, in examples, we will consider J=R or J=(0,+∞)). By Theorem 4, under the assumption

σ2(y) is locally bounded on J,

there exists a unique strong solution of (12).

Let Y be a J-valued solution of the equation
dYt=α(Yt)dt+β(Yt)dWt1,Y0=y0∈J,
where W1 is a Wiener process, possibly correlated with W.

By Lloc1(J) we denote the set of Borel functions J→[−∞,∞] that are locally integrable on J, that is, integrable on compact subsets of J. By Lloc1(l+) we denote the set of Borel functions f:J→[−∞,∞] such that ∫lzf(y)dy<∞ for some z∈J. The notation Lloc1(r−) is introduced similarly.

Assume that coefficients α and β satisfy the Engelbert–Schmidt conditions

β(y)≠0 for all y∈J, and

β−2,αβ−2∈Lloc1(J).

Let us introduce the following notation:
ρ(y)=exp−2∫cyα(u)β2(u)du,y∈J,s(y)=∫cyρ(u)du,y∈J¯=[l,r],
for some c∈J. Assume additionally that

s(r)=∞ or s(r)−sρβ2∉Lloc1(r−),

s(l)=−∞ or s−s(l)ρβ2∉Lloc1(l+).

Under (A2)–(A3), the SDE (13) has a weak solution, unique in law, which possibly exits J at some time ζ. Moreover, ζ=∞ a.s. if and only if conditions (A4)–(A5) are satisfied, see, for example, [17, Prop. 2.6].

The maximum likelihood estimator (11) for model (12) equals
θˆT=∫0TXt−1σ2−2(Yt)dXt∫0Tσ2−2(Yt)dt.

Under assumptions (A1)–(A8), the estimatorθˆTis strongly consistent asT→∞.

We need to verify conditions (8)–(10) of Theorem 5. For model (12), they read as follows: Xtσ2(Yt)≠0,t≥0,a.s.,∫0tσ2−2(Ys)ds<∞,t>0,a.s.,0,\hspace{1em}\text{a.s.},\end{array}\]]]>∫0∞σ2−2(Ys)ds=∞a.s. Note that (15) is assumption (A8). By [17, Thm. 2.7] the local integrability condition (A6), together with (A2)–(A5), implies (16). Further, if assumption (A7)(i) holds, then (17) is satisfied by [17, Thm. 2.11]. In the remaining case s(r)<∞ or s(l)>−∞-\infty $]]>, we have that
Ω=limt↑∞Yt=r∪limt↑∞Yt=l;
see [17]. Moreover, if s(r)=∞, then P(limt↑∞Yt=r)=0 by [17, Prop. 2.4]. If s(r)<∞ and s(r)−sρβ2σ22∉Lloc1(r−), then ∫0∞σ2−2(Ys)ds=∞ a.s. on {limt↑∞Yt=r} by [17, Thm. 2.12]. The similar statements hold for {limt↑∞Yt=l}. This implies that (17) is satisfied under each of conditions (ii)–(iv) of assumption (A7). □

Now we consider several examples of the process Y, namely the Bachelier model, the Ornstein–Uhlenbeck model, the geometric Brownian motion, and the Cox–Ingersoll–Ross model. We concentrate on verification of assumption (A7) for these models, assuming that other conditions of Theorem 6 are satisfied.

(Bachelier model).

Let Y be a solution of the SDEdYt=αdt+βdWt1,Y0=y0∈R,whereα∈Randβ≠0are some constants. Assume thatσ2−2(y)∈Lloc1(R)and one of the following assumptions holds:

α=0,

α>00$]]>andσ2−2(y)∉Lloc1(+∞),

α<0andσ2−2(y)∉Lloc1(−∞).

Then estimator (14) is strongly consistent.

Indeed, in this case, J=R,
ρ(y)=exp−2αβ2y,ands(y)=∫0yexp−2αβ2udu.
If α=0, then s(y)=y, s(+∞)=∞, s(−∞)=−∞, and assumption (A7)(i) is satisfied. Otherwise, we have
s(y)=β22α(1−exp−2αβ2y).
If α>00$]]>, then s(+∞)=β22α, s(−∞)=−∞, and
s(+∞)−s(y)ρ(y)β2σ22(y)=12ασ22(y)∉Lloc1(+∞),
and hence (A7)(ii) holds. The case α<0 is considered similarly.

(Ornstein–Uhlenbeck or Vasicek model).

Let Y be a solution of the SDEdYt=a(b−Yt)dt+γdWt1,Y0=y0∈R,wherea,b∈R, andγ>00$]]>are some constants. Assume thatσ2−2∈Lloc1(R)and one of the following assumptions holds:

a≥0,

a<0,y−1σ2−2(y)∉Lloc1(+∞)∪Lloc1(−∞).

Then estimator (14) is strongly consistent.

In this case, we also take J=R. Then
ρ(y)=exp−2∫bya(b−u)γ2du=expaγ2(y−b)2,s(y)=∫byexpaγ2(u−b)2du.

If a≥0, then exp{aγ2(u−b)2}≥1, and we get that s(+∞)=∞, s(−∞)=−∞.

If a<0, then
s(y)=γ−a∫0−aγ(y−b)e−z2dz.
Therefore, s(+∞)=−s(−∞)=γπ2−a<∞, and we need to verify (A7)(iv). Since ∫x∞e−z2dz∼12xe−x2 as x→∞, we see that
s(+∞)−s(y)ρ(y)γ2σ22(y)=γ−a∫−aγ(y−b)∞e−z2dzexpaγ2(y−b)2γ2σ22(y)∼1−2a(y−b)σ22(y)
as y→∞. Then s(+∞)−s(y)ρ(y)γ2σ22(y)∉Lloc1(+∞) if y−1σ2−2(y)∉Lloc1(+∞). The condition s−s(−∞)ρβ2σ22∉Lloc1(−∞) is considered similarly.

(Geometric Brownian motion).

Let Y be a solution of the SDEdYt=αYtdt+βYtdWt1,Y0=y0>0,0,\]]]>whereα∈Randβ≠0are some constants. Assume thaty−2σ2−2(y)∈Lloc1((0,+∞))and one of the following assumptions holds:

β2=2α2,

β2<2α2andy−1σ2−2(y)∉Lloc1(+∞),

β2>2α22{\alpha }^{2}$]]>andy−1σ2−2(y)∉Lloc1(0+).

Then estimator (14) is strongly consistent.

In this case, the process Y is positive, and hence J=(0,∞). We have
ρ(y)=exp−2∫1yα2β2udu=y−2α2β2,s(y)=∫1yu−2α2β2du=y1−2α2β2−11−2α2β2,β2≠2α2,lny,β2=2α2.
If β2=2α2, then s(0)=−∞ and s(+∞)=∞. If β2<2α2, then s(0)=−∞, s(+∞)<∞, and
s(+∞)−s(y)ρ(y)β2y2σ22(y)=1(2α2−β2)yσ22(y)∉Lloc1(+∞).
If β2>2α22{\alpha }^{2}$]]>, then s(0)>−∞-\infty $]]>, s(+∞)=∞, and
s(y)−s(0)ρ(y)β2y2σ22(y)=1(β2−2α2)yσ22(y)∉Lloc1(0+).

(Cox–Ingersoll–Ross model).

Let Y be a solution of the SDEdYt=a(b−Yt)dt+γYtdWt1,Y0=y0∈R,where a, b, γ are positive constants, and2ab≥γ2. Assume thaty−1σ2−2(y)∈Lloc1((0,+∞)).Then estimator (14) is strongly consistent.

Under the condition 2ab≥γ2, the process Y is positive, and hence J=(0,∞). Further,
ρ(y)=exp−2∫1ya(b−u)γ2udu=y−2abγ2e2aγ2(y−1),s(y)=e−2aγ2∫1yu−2abγ2e2aγ2udu.
Since u−2abγ2e2aγ2u→∞ as u→∞, we see that s(+∞)=∞. Moreover, using the inequality e2aγ2u≥1, we get
s(0)=−e−2aγ2∫01u−2abγ2e2aγ2udu≤−e−2aγ2∫01u−2abγ2du=−∞
since 2abγ2>11$]]>. Thus, assumption (A7)(i) is satisfied.

Simulations

We illustrate the quality of the estimator θˆT in model (12)–(13) by simulation experiments. We simulate the trajectories of the Wiener processes W and W1 at the points t=0,h,2h,3h,… and compute the approximate values of the process Y and X as solutions to SDEs using Euler’s approximations. For each set of parameters, we simulate 100 sample paths with step h=0.0001. The initial values of the processes are x0=y0=1, and the true value of the parameter is θ=2. The results are reported in Table 1.

The means and standard deviations of θˆT

T

α(y)

β(y)

σ2(y)

10

50

100

200

1

1

y1/4

Mean

1.9455

1.9431

1.9711

1.9762

Std.dev.

0.4260

0.2576

0.2367

0.2022

y

2y

y

Mean

2.0104

2.0000

2.0000

2.0000

Std.dev.

0.1225

5.7·10−5

4.7·10−8

1.6·10−14

y

y

(1+y)−1

Mean

2.0008

2.0001

2.0000

2.0000

Std.dev.

0.0769

0.0010

2.2·10−12

1.4·10−14

y

1

2+siny

Mean

1.9358

1.9819

1.9927

1.9939

Std.dev.

0.5436

0.2437

0.1679

0.1077

−y

1

2+siny

Mean

1.9061

1.9684

1.9700

1.9786

Std.dev.

0.5994

0.2472

0.1781

0.1254

2−y

y

y

Mean

1.9923

2.0039

1.9796

1.9872

Std.dev.

0.3540

0.1604

0.1173

0.0782

2−y

y

y

Mean

2.0830

1.9835

1.9803

1.9886

Std.dev.

0.4347

0.1974

0.1205

0.0840

Appendix

The next two propositions are taken from [20].

Assume that we have r sequences of stochastic processesξn(1),…,ξn(r)such that, for alli=1,…,r,

for everyδ>00$]]>,limh→0lim‾n→∞supt1−t2≤hP(|ξn(i)(t1)−ξn(i)(t2)|>δ)=0\delta )=0$]]>,

limC→∞limn→∞sup0≤t≤TP(|ξn(i)(t)|>C)=0C)=0$]]>.

Then, for some sequencenkwe can construct processesXnk(1),…,Xnk(r)on the probability space(Ω′,F′,P′), whereΩ′=[0,1],F′=B([0,1]), andP′is the Lebesgue measure, such that the finite-dimensional distributions ofXnk(1),…,Xnk(r)coincide with those ofξnk(1),…,ξnk(r)and each of the sequencesXnk(1),…,Xnk(r)converges in probability to some limit.

Letηn(t)be a sequence of martingales such thatηn(t)→W(t)in probability for all t andEηn(t)2→tasn→∞. Letfn(t)be a sequence such that∫0Tfn(t)dηn(t)exists for all n,fn(t)→f(t)in probability for all t, and∫0Tf(t)dW(t)exists. Suppose that, in addition, the following conditions hold:

for allε>00$]]>, there existsC>00$]]>such that, for all n,P(sup0≤t≤T|fn(t)|>C)≤ε,C\Big)\le \varepsilon ,\]]]>

for allε>00$]]>,limh→0limn→∞supt1−t2≤hP(|fn(t2)−fn(t1)|>ε)=0.\varepsilon \big)=0.\]]]>

Then∫0Tfn(t)dηn(t)→∫0Tf(t)dW(t)in probability.

In the next two lemmas, we modify the corresponding auxiliary results from [20] to Eq. (1) with multiplicative diffusion. Consider a sequence of partitions 0=t0n<t1n<⋯<tnn=T of [0,T] such that limn→∞maxk(tk+1n−tkn)=0. Define ξkn by ξ0n=X(0) and
ξk+1n=ξkn+a(tkn,ξkn)Δtkn+σ1(tkn,ξkn)σ2(tkn,Y(tkn))ΔWkn.

The random variablessupk|ξkn|are bounded in probability uniformly w.r.t. n.

Let η0n=ξ0n1ξ0n≤N and
ηk+1n=ηkn+aN(tkn,ηkn)Δtkn+σ1N(tkn,ηkn)σ2(tkn,Y(tkn))ΔWkn,
where aN(t,x)=a(t,x)1x≤N, σ1N(t,x)=σ1(t,x)1x≤N. If |ηkn|>NN$]]>, then ηk+1n=ηkn, and if |ηkn|≤N, then
ηk+1n≤N+|aN(tkn,ηkn)|Δtkn+|σ1N(tkn,ηkn)σ2(tkn,Y(tkn))|ΔWkn.
Then, for any 1≤k≤n,
ηkn≤N+∑r=0k−1|aN(trn,ηrn)|Δtrn+∑r=0k−1|σ1N(trn,ηrn)σ2(trn,Y(trn))|ΔWrn
and is square-integrable. Furthermore,
Eηk+1n2=Eηkn2+2E(aN(tkn,ηkn)ηknΔtkn)+EaN(tkn,ηkn)2(Δtkn)2+E((σ1N(tkn,ηkn))2(σ2(tkn,Y(tkn)))2(ΔWkn)2)≤Eηkn2+2E(aN(tkn,ηkn)ηknΔtkn)+EaN(tkn,ηkn)2(Δtkn)2+C2E((σ1N(tkn,ηkn))2Δtkn).
Then there exists a constant H=H(T,K) such that
Eηk+1n2≤Eηkn2(1+HΔtkn)+HΔtkn≤Eηkn2eHΔtkn+HΔtkn,Eηk+1n2+1≤(Eηkn2+1)eHΔtkn≤(Eη0n2+1)eHT,Eηk+1n2≤(Eη0n2+1)eHT−1.
We have that
supkηkn≤η0n+∑j=0n−1|aN(tjn,ηjn)|Δtjn+supk∑j=0k−1σ1N(tjn,ηjn)σ2(tjn,Y(tjn))ΔWjn.
So, since
Esup0≤r≤n∑k=0rσ1N(tkn,ηkn)σ2(tkn,Y(tkn))ΔWkn2≤4∑k=0nE|σ1N(tkn,ηkn)σ2(tkn,Y(tkn))|2Δtkn≤4K∑k=0n(Eηkn2+1)Δtkn
and
E(∑k=0n−1|aN(tkn,ηkn)|Δtkn)2≤KT∑k=0n−1(Eηkn2+1)Δtkn,
we get
Esupkηkn2≤A+Bη0n2.
We have that η0n is bounded uniformly w.r.t. n and N. Then supk|ηkn|2 is bounded in L2 uniformly w.r.t. n, N.

For supk|ηkn|<N, we have supk|ηkn|=supk|ξkn|. Hence, supk|ξkn| is bounded in probability uniformly w.r.t. n. □

Using Lemma 1, we have that, for all ε>00$]]>, there exists N>00$]]> such that, for every n≥N,P(supk|ξkn−ηkn|>0)<ε0)<\varepsilon $]]>.

Letξn(t)=ξknfort∈[tkn,tk+1n). Then, for allδ>00$]]>,limh→0lim‾n→∞supt1−t2≤hP(|ξn(t1)−ξn(t2)|>δ)=0.\delta \big)=0.\]]]>

Let ηn(t)=ηkn, t∈[tkn,tk+1n). Then
supt1−t2≤hP(|ξn(t1)−ξn(t2)|>δ)≤supt1−t2≤hP(|ηn(t1)−ηn(t2)|>δ)+P(supkξkn−ηkn>0).\delta \big)& \displaystyle \le \underset{\left|t_{1}-t_{2}\right|\le h}{\sup }\mathbb{P}\big(\big|\eta _{n}(t_{1})-\eta _{n}(t_{2})\big|>\delta \big)\\{} & \displaystyle \hspace{1em}+\mathbb{P}\Big(\underset{k}{\sup }\left|{\xi _{k}^{n}}-{\eta _{k}^{n}}\right|>0\Big).\end{array}\]]]>
From (18) and the boundedness of aN, σ1N, and σ2 we have that
limh→0lim‾n→∞supt1−t2≤hP(|ηn(t1)−ηn(t2)|>δ)=0.\delta \big)=0.\]]]>
Therefore,
limh→0lim‾n→∞supt1−t2≤hP(|ξn(t1)−ξn(t2)|>δ)≤lim‾n→∞P(supkξkn−ηkn>0).\delta \big)\le \varlimsup_{n\to \infty }\mathbb{P}\Big(\underset{k}{\sup }\left|{\xi _{k}^{n}}-{\eta _{k}^{n}}\right|>0\Big).\]]]>
The proof follows now from Remark 1. □

Acknowledgment

The authors are grateful to the anonymous referee for his useful remarks and suggestions, which contributed to a substantial improvement of the text.

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