On Local Linear Approximations to Diffusion Processes

Abstract

Diffusion models have been used extensively in many applications. These models, such as those used in the financial engineering, usually contain unknown parameters which we wish to determine. One way is to use the maximum likelihood method with discrete samplings to devise statistics for unknown parameters. In general, the maximum likelihood functions for diffusion models are not available, hence it is difficult to derive the exact maximum likelihood estimator (MLE). There are many different approaches proposed by various authors over the past years, see, for example, the excellent books and Kutoyants (2004), Liptser and Shiryayev (1977), Kushner and Dupuis (2002), and Prakasa Rao (1999), and also the recent works by Aït-Sahalia (1999), (2004), (2002), and so forth. Shoji and Ozaki (1998; see also Shoji and Ozaki (1995) and Shoji and Ozaki (1997)) proposed a simple local linear approximation. In this paper, among other things, we show that Shoji's local linear Gaussian approximation indeed yields a good MLE.

1. Introduction

Diffusion processes are used as theoretical models in analyzing random phenomena evolved in continuous time. These models may be described in terms of Itô's type stochastic differential equations d𝑋_𝑡=𝐴𝑋_𝑡,𝜃d𝑡+𝜎𝑋_𝑡,𝜃d𝑊_𝑡, (1.1) where (𝑊_𝑡)_𝑡≥0 is a Brownian motion, with some unknown parameters 𝜃 to be determined in rational ways.

It is, however, difficulty to derive the maximum likelihood estimator for 𝜃 if the diffusion coefficient (i.e., the volatility) 𝜎 is unknown. On the other hand, in practice, the volatility is determined first by using the fact that𝜎²𝑇=lim^𝑛_𝑗=1𝑋_{(𝑗−1)𝑇/𝑛}−𝑋_{𝑗𝑇/𝑛}²inprob. (1.2) when 𝜎 is a constant. Therefore we will limit ourselves on diffusion models with constant volatility: d𝑋_𝑡=𝐴𝑋_𝑡,𝜃d𝑡+d𝑊_𝑡. (1.3) Since there is no much difference at technical level, we will consider one-dimensional models only. That is, we will assume throughout the paper that 𝑊 is a one-dimensional Brownian motion, and 𝑋 is real valued.

The distribution 𝜇^𝑋_𝑇 of (𝑋_𝑡)_𝑡≥0 over a finite time interval [0,𝑇] has a density with respect to the Wiener measure 𝜇^𝑊_𝑇 (the law of the Brownian motion 𝑊), given by the Cameron-Martin formula: d𝜇^𝑋_𝑇d𝜇^𝑊_𝑇=exp^𝑇₀𝐴𝑋_𝑡,𝜃d𝑋_𝑡−12^𝑇₀𝐴𝑋_𝑡,𝜃²d𝑡, (1.4) which is in turn the likelihood function with continuous observation. In practice, only discrete values 𝑋_𝑡0,…,𝑋_𝑡𝑛 may be observed over the duration [0,𝑇], where 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑇 and 𝑡_𝑖−𝑡_𝑖−1=𝛿. The corresponding likelihood function 𝐿(𝜃) is the conditional expectation under Wiener measure:𝔼𝐿(𝜃)∣𝑋_𝑡0,…,𝑋_𝑡𝑛=∏^𝑛_𝑗=1𝑝_𝜃𝛿,𝑋_𝑡𝑗−1,𝑋_𝑡𝑗∏^𝑛_𝑗=1𝐺𝛿,𝑋_𝑡𝑗−1,𝑋_𝑡𝑗, (1.5) where 𝑝_𝜃(𝑡,𝑥,𝑦) is the conditional probability density function of 𝑋_𝑡 given 𝑋₀=𝑥, and 𝐺(𝑡,𝑥,𝑦) is the Gaussian density 1/√2𝜋𝑡exp{−|𝑥−𝑦|²/2𝑡} (see [1]). Since the denominator of (1.5) does not depend on 𝜃, we may simply consider the numerator 𝐿𝑋_𝑡0,…,𝑋_𝑡𝑛≡^𝑛_𝑗=1𝑝_𝜃𝛿,𝑋_𝑡𝑗−1,𝑋_𝑡𝑗, (1.6) as a likelihood function. Therefore, the MLE for 𝜃 under a discrete observation may be found by solving either explicitly if possible or numerically the likelihood equation ∇𝐿𝑋_𝑡0,…,𝑋_𝑡𝑛=0. (1.7)

The difficulty with this approach is that, unless for a very special drift vector field 𝐴, an explicit formula for 𝑝_𝜃(𝑡,𝑥,𝑦) is not known. To overcome this difficulty, many approximation methods have been proposed in the literature by various authors. The idea is to replace the diffusion model (1.3) by an approximation model for which an explicit formula for the likelihood function is available. One possible candidate is of course the Euler-Maruyama approximation 𝑋_𝑡𝑗−𝑋_𝑡𝑗−1=𝐴𝑋_𝑡𝑗−1,𝜃𝛿+𝜉_𝑗√𝛿, (1.8) where {𝜉_𝑗} is an i.i.d. sequence with standard normal distribution 𝑁(0,1) and 𝑋₀=𝑋₀. However, the likelihood function 𝐿₁(𝑋₀,…,𝑋_𝑛) for this model is not, in general, close enough to that of the diffusion model if measured in terms of the ratio of their corresponding likelihood functions 𝐿𝑋_𝑡0,…,𝑋_𝑡𝑛𝐿₁𝑋_𝑡0,…,𝑋_𝑡𝑛. (1.9)

The second approach is to discretize the likelihood function d𝜇^𝑋_𝑇/d𝜇^𝑊_𝑇 for continuous observations. In order to utilize this likelihood function, we need to handle the Itô integral ∫^𝑇₀𝐴(𝑋_𝑡,𝜃)d𝑋_𝑡 which is defined only in probability sense. If 𝐴=∇𝑓 (where 𝑓 is a 𝐶¹-function) is a gradient field, then, according to Itô's formula, d𝜇^𝑋_𝑇d𝜇^𝑊_𝑇=exp𝑓𝑋_𝑇,𝜃−^𝑇₀12Δ𝑓𝑋_𝑇+1,𝜃2𝐴𝑋_𝑇,𝜃²d𝑡, (1.10) here the right-hand side involves only the sample 𝑋. This idea to get rid of Itô's integral and replace it by an ordinary one has far-reaching consequences, see the interesting paper [2] for some applications.

One can also use approximations to the probability density function 𝑝_𝜃(𝑡,𝑥,𝑦) and construct functions which are close to the maximum likelihood function. There are a great number of articles devoted to this approach, such as [3–5], for example. The difficulty, however, is that even 𝑓(𝑡,𝑥,𝑦) is a uniform approximation of 𝑝_𝜃(𝑡,𝑥,𝑦), there is no guarantee that the approximate likelihood function ∏_𝑗𝑓(𝑡,𝑥_𝑗−1,𝑥_𝑗) would tend to ∏_𝑗𝑝_𝜃(𝑡,𝑥_𝑗−1,𝑥_𝑗) when 𝑛→∞.

In this paper we consider the linear diffusion approximation proposed by Shoji and Ozaki [6] to the diffusion model (1.3), which leads to the following approximation of the likelihood function 𝐿(𝑋_𝑡0,…,𝑋_𝑡𝑛): 𝐿₂𝑋_𝑡0,…,𝑋_𝑡𝑛=^𝑛_𝑗=1ℎ_𝑗𝛿,𝑋_𝑡𝑗−1,𝑋_𝑡j, (1.11) where 𝑡_𝑗=𝑗𝑇/𝑛 so that 𝑋_𝑡𝑗 is a sample with fixed duration 𝛿=𝑡_𝑗−𝑡_𝑗−1 over [0,𝑇], and ℎ_𝑗(𝑡,𝑥,𝑦) is the probability transition density of the following linear diffusion model 𝑑𝑋_𝑡=𝐴𝑋_𝑡𝑗−1,𝜃+𝐴^′𝑋_𝑡𝑗−1,𝜃𝑋_𝑡−𝑋_𝑡𝑗−1d𝑡+d𝑊_𝑡, (1.12) when 𝑡_𝑗−1≤𝑡<𝑡_𝑗 and 𝑋_𝑡𝑗−1=𝑋_𝑡𝑗−1.

The approximation (1.12) is called the local linearization of the diffusion model (1.3), which has been studied in Shoji and Ozaki [6]. Shoji has showed numerically that the local linearizations do yield better estimates. Shoji's approximation was revisited in Prakasa Rao [7], without a definite conclusion.

The main goal of the paper is to prove Theorem 3.1 which implies that the local linear approximations (1.12) is efficient for the propose of deriving MLE with discrete samples.

The paper is organized as follows. In Section 2, we present the MLE for linear models such as (1.12). In Section 3, we state our main result for Shoji's local linear approximation, and give some comments about the conditions on the sampling data. Our main theorem provides a deterministic convergence rate for the likelihood functions. In Section 4, we prove that the likelihood function for the local linear approximation converges to the Cameron-Martin density but only in probability sense. Sections 5, 6, and 7 are devoted to the proof of our main result. In Section 5, we state the main tool, a representation formula for diffusions, established by Qian and Zheng [8]. In Section 6, we develop the main technical estimates in order to prove Theorem 3.1, whose proof is completed in Section 7. Section 8 contains a discussion about the Euler-Maruyama approximation which concludes the paper.

2. Linear Diffusions

Let us begin with the MLE of parameters 𝑎, 𝑏, and 𝜎>0 for the linear diffusion model (Mishra and Bishwal [9] discussed a similar model): 𝑑𝑋_𝑡=𝑏−𝑎𝑋_𝑡d𝑡+𝜎d𝑊_𝑡, (2.1) whose finite-dimensional distributions are Gaussian, determined through the probability transition function ℎ(𝑡,𝑥,𝑦). Fortunately we have an explicit formula for ℎ. Indeed the linear equation (2.1) may be solved explicitly and its solution is given by the formula 𝑋_𝑡=𝑒^−𝑎𝑡𝑋₀+𝑏𝑎1−𝑒^−𝑎𝑡+𝜎^𝑡₀𝑒^{−𝑎(𝑡−𝑠)}d𝑊_𝑠, (2.2) (formula (6.8) of Karatzas and Shreve [10], page 354), and therefore ℎ=1𝑡,𝑥,𝑦√𝜋𝜎𝑎1−𝑒^−2𝑎𝑡−𝑎exp1−𝑒^−2𝑎𝑡𝑦−𝑒^−𝑎𝑡𝑥−(𝑏/𝑎)1−𝑒^−𝑎𝑡2𝜎². (2.3)

Suppose we have a discrete sample observed over the equal time scale during the period [0,𝑇], 𝑋_{𝑖𝑇/𝑛}, 𝑖=0,…,𝑛. According to the Markov property, their joint distribution, or the maximum likelihood function 𝐿𝑎,𝑏,𝜎;𝑥₀,…,𝑥_𝑛=𝜇𝑥₀^𝑛_𝑗=1ℎ𝛿,𝑥_𝑗−1,𝑥_𝑗, (2.4) where 𝛿=𝑇/𝑛, and 𝜇(𝑥) is the probability density function of the initial distribution. Therefore the logarithmic of the maximum likelihood function 𝑙𝑎,𝑏,𝜎;𝑥_𝑖=log𝜇𝑥₀+^𝑛_𝑗=11logℎ𝑛,𝑥_𝑗−1,𝑥_𝑗=log𝜇𝑥₀−𝑛2𝑛log𝜋−𝑛log𝜎+2𝑛log𝑎−2log1−𝑒^{−2𝑎𝑇/𝑛}−𝑛𝑎1−𝑒^{−2𝑎𝑇/𝑛}1𝜎²𝑇^𝑛_𝑗=1𝑥_𝑗−𝑏𝑎−𝑥_𝑗−1−𝑏𝑎𝑒^{−𝑎𝑇/𝑛2}. (2.5) The maximum likelihood estimates for 𝑎, 𝑏, and 𝜎 are the stationary points of 𝑙, that is solutions to the equation ∇𝑙=0. Set 𝜌=𝑒^{−𝑎𝑇/𝑛}. Then 𝑎=−(𝑛/𝑇)log𝜌 and 𝛽=𝑏/𝑎.

Proposition 2.1. The maximum likelihood estimates for the linear diffusion model (2.1) with discrete observations are given by 𝑛𝑎=𝑇∑log^𝑛𝑗=1𝑥²_𝑗−1−(1/𝑛)∑^𝑛_𝑗=1𝑥_𝑗−1²∑^𝑛_𝑗=1𝑥_𝑗−1𝑥_𝑗−(1/𝑛)∑^𝑛_𝑗=1𝑥_𝑗−1∑^𝑛_𝑗=1𝑥_𝑗,1𝛽=𝑛^𝑛_𝑗=1𝑥_𝑗+1𝑛𝜌𝜌𝑥1−^𝑛−𝑥₀,𝜎²=1𝑇2𝑎1−𝜌^2𝑛_𝑗=1𝑥_𝑗−𝜌𝑥_𝑗−1−1−𝜌𝛽². (2.6)

As an interesting consequence we have the following.

Corollary 2.2. The maximum likelihood estimators (𝑎,𝑏̂,𝜎) to the linear diffusion model (2.1) are not sufficient statistics while (𝑎,𝑏̂,𝜎,𝑋₀,𝑋_𝑛) are sufficient.

3. Diffusion Models

We consider the diffusion model (1.3). Our approach and our conclusions are applicable to multidimensional cases as long as the diffusion coefficients are constant. For simplicity, we only consider one-dimensional case. The question is to estimate 𝜃 under a discrete observation {𝑥₀,…,𝑥_𝑛} over the time scale 𝛿 in the time interval [0,𝑇]. Then, up to a constant factor, its maximum likelihood function 𝐿𝑥₀,…,𝑥_𝑛=^𝑛_𝑗=1𝑝𝛿,𝑥_𝑗−1,𝑥_𝑗, (3.1) where 𝑝(𝑡,𝑧,𝑦) is the transition probability density of (𝑋_𝑡) (we have dropped the subscript 𝜃 for simplicity). The approximation maximum likelihood function, proposed in [6], is given by 𝐿₂𝑥₀,…,𝑥_𝑛=^𝑛_𝑗=1ℎ_𝑗𝛿,𝑥_𝑗−1,𝑥_𝑗, (3.2) where ℎ_𝑗(𝑡,𝑥,𝑦) is the transition density function to the linear diffusion model d𝑋_𝑡=𝐴𝑥_𝑗−1,𝜃+𝐴^′𝑥_𝑗−1,𝜃𝑋_𝑡−𝑥_𝑗−1d𝑡+d𝑊_𝑡, (3.3) which is the first-order approximation to (1.3).

In what follows we assume that 𝐴 has bounded first and second derivatives and 𝐴^′(𝑥,𝜃),𝐴^′′(𝑥,𝜃)≤2𝐶₀, (3.4) for some constant 𝐶₀>0 independent of parameters 𝜃.

The main result of the paper is follows.

Theorem 3.1. Assume that 𝐴^′(⋅,𝜃) and 𝐴^′′(⋅,𝜃) are bounded uniformly in 𝜃. Let 𝑇>0 be a fixed time and 𝐶>0 be a constant. Suppose {𝑥^𝑛_𝑗}_𝑗≤𝑛 (𝑛=1,2,…) is a family of discrete samples such that 𝑥^𝑛_𝑗−𝑥^𝑛_𝑗−1²≤𝐶𝛿(𝑛),𝑥^𝑛_𝑗≤𝐶, (3.5) for all pair (𝑗,𝑛) such that 𝑗≤𝑛, 𝑛=1,2,…, where 𝛿(𝑛)=𝑇/𝑛. Then lim_𝑛→∞𝐿𝑥^𝑛₀,…,𝑥^𝑛_𝑛𝐿₂𝑥^𝑛₀,…,𝑥^𝑛_𝑛=1, (3.6) where 𝐿 and 𝐿₂ are defined in (3.1) and (3.2) with 𝛿=𝛿(𝑛)=𝑇/𝑛.

The convergence in (3.6) happens in a deterministic sense, and therefore conditions such as |𝑥^𝑛_𝑗−𝑥^𝑛_𝑗−1|²≤𝐶𝛿(𝑛) and |𝑥^𝑛_𝑗|≤𝐶 are reasonable. The first condition, that is |𝑥^𝑛_𝑗−𝑥^𝑛_𝑗−1|²≤𝐶𝛿(𝑛), just says the “variance” of the sample cannot be too big. Since _𝑗𝑋_{𝑗𝑇/𝑛}−𝑋_{(𝑗−1)𝑇/𝑛}²⟶𝑇inprobability, (3.7) so that on average we should have |𝑥^𝑛_𝑗−𝑥^𝑛_𝑗−1|²≤𝐶𝛿(𝑛). Since (𝑋_𝑡) has continuous sample paths, so that {𝑋(𝜔)_𝑡∶𝑡∈[0,𝑇]} for a fixed sample point 𝜔 is bounded. Since 𝑥^𝑛_𝑗 are sampled from the fixed duration [0,𝑇], thus we can assume that {𝑥^𝑛_𝑗} is bounded, though here we have a countable many samples. It is possible to relax this constraint, for example, we may impose that |𝑥^𝑛_𝑗|≤𝐶𝑛^𝛼 with 𝛼<1/2, but for simplicity we only consider the bounded case. This condition is placed as a kind of “integrability” condition on the samples.

From the asymptotic of the transition density function 𝑝(𝑡,𝑥,𝑦), it is easy to see that lim_𝛿→0𝑝𝛿,𝑥_𝑗−1,𝑥_𝑗ℎ_𝑗𝛿,𝑥_𝑗−1,𝑥_𝑗=1, (3.8) for each 𝑗, while, as our observation {𝑥₀,…,𝑥_𝑛} happens over a fixed time interval [0,𝑇], the ratio (3.6) as 𝑛→∞ is really an infinite product, its behavior thus depends on the global behavior of 𝑝(𝑡,𝑥,𝑦). Although there are many results about bounds of 𝑝(𝑡,𝑥,𝑦) in the literature (see [2, 11] e.g.), the best we could find are those which yield (3.8) uniformly in 𝑥_𝑗, none of them yields the precise limit (3.6). In fact, the proof of (3.6) depends on careful estimates on 𝑝(𝑡,𝑥,𝑦) through a representation formula established in [8].

4. Linear Diffusion Approximations

Without losing generality, we may assume that 𝑇=1. Let 𝑋_𝑗/𝑛 be a discrete observation of the diffusion model (1.3) at 𝑡_𝑗=𝑗/𝑛 (𝑗=0,…,𝑛). For simplicity, write 𝑋_𝑗/𝑛 as 𝑋_𝑗 if no confusion may arise. Consider the family of linear diffusions d𝑋^𝑗_𝑡=𝐴𝑋_𝑗−1,𝜃+𝐴^′𝑋_𝑗−1,𝜃𝑋^𝑗_𝑡−𝑋_𝑗−1d𝑡+d𝑊_𝑡, (4.1) with 𝑋^𝑗_{(𝑗−1)/𝑛}=𝑋_𝑗−1. Let 𝑏_𝑗=𝐴𝑋_𝑗−1,𝜃−𝐴^′𝑋_𝑗−1,𝜃𝑋_𝑗−1,𝑎_𝑗=𝐴^′𝑋_𝑗−1,𝜃. (4.2) Then 𝑋^𝑗_𝑡=e^{𝑎𝑗(𝑡−𝑡𝑗−1)}𝑋_𝑗−1+e^{𝑎𝑗(𝑡−𝑡𝑗−1)}−1𝑎_𝑗𝑏_𝑗+^𝑡_𝑡𝑗−1e^{𝑎𝑗(𝑡−𝑠)}d𝑊_𝑠, (4.3) so that ℎ_𝑗𝛿,𝑋_𝑗−1,𝑋_𝑗=ℙ𝑋^𝑗_𝑡𝑗=𝑋_𝑗∣𝑋^𝑗_𝑡𝑗−1=𝑋_𝑗−1=2𝑎_𝑗𝑒^{2𝑎𝑗𝛿}1−1√e2𝜋^{−(𝑎𝑗/(e𝑗𝛿2𝑎−1))(𝑋𝑗−𝑋𝑗−1−((e𝑎𝑗𝛿−1)/𝑎𝑗)(𝑏𝑗+𝑎𝑗𝑋𝑗−1))2}, (4.4) where 𝛿=1/𝑛. The approximating likelihood function is 𝐿₂(𝛿)=^𝑛_𝑗=1ℎ_𝑗𝛿,𝑋_𝑗−1,𝑋_𝑗. (4.5) We need to compare this function to the likelihood function with continuous observation—the Cameron-Martin density, which, however, should be discounted with respect to the Wiener measure. Thus we have to renormalize 𝐿₂(𝛿) against the discrete version of Brownian motion, which is given by 𝐿₂(𝛿)=^𝑛_𝑗=1ℎ_𝑗𝛿,𝑋_𝑗−1,𝑋_𝑗𝐺𝛿,𝑋_𝑗−1,𝑋_𝑗, (4.6) where 𝐺=1𝛿,𝑥,𝑦√−2𝜋𝛿exp𝑥−𝑦²2𝛿. (4.7) Hence its logarithmic 𝑙₂̂(𝛿)=12_𝑗log2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}+1−12𝛿_𝑗𝑋_𝑗−𝑋_𝑗−1²−12_𝑗2𝑎_𝑗e^{2𝑎𝑗𝛿}𝑋−1^𝑗−𝑋_𝑗−1−e^𝑎𝑗𝛿−1𝑎_𝑗𝑏_𝑗+𝑎_𝑗𝑋_𝑗−1². (4.8)

Proposition 4.1. One has lim_𝛿↓0𝑙₂̂(𝛿)=𝑙 (4.9) uniformly in 𝜃, in probability with respect to the Wiener measure, where 𝑙 is the log of the Cameron-Martin density (1.4).

Proof. Let 𝐷_𝑗=𝑋_𝑗−𝑋_𝑗−1. Then 𝑙₂̂(𝛿)=12_𝑗log2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}+1−12𝛿_𝑗1−2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}𝐷−1^2𝑗−_𝑗1e^𝑎𝑗𝛿𝑏+1^𝑗+𝑎_𝑗𝑋_𝑗−1²e^𝑎𝑗𝛿−1𝑎_𝑗+2_𝑗1e^𝑎𝑗𝛿𝑏+1^𝑗+𝑎_𝑗𝑋_𝑗−1𝐷_𝑗. (4.10) Since 𝑏_𝑗=𝐴(𝑋_𝑗−1,𝜃)−𝑎_𝑗𝑋_𝑗−1 and 𝑎_𝑗=𝐴^′(𝑋_𝑗−1,𝜃), so that 𝑙₂̂(𝛿)=12_𝑗log2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}+1−12𝛿_𝑗1−2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}𝐷−1^2𝑗−_𝑗1e^𝑎𝑗𝛿𝐴𝑋+1^𝑗−1,𝜃²e^𝑎𝑗𝛿−1𝑎_𝑗+2_𝑗1e^𝑎𝑗𝛿𝐴𝑋+1^𝑗−1,𝜃𝐷_𝑗. (4.11) However, lim_𝛿↓02_𝑗1e^𝑎𝑗𝛿𝐴𝑋+1^𝑗−1,𝜃𝐷_𝑗=¹₀𝐴𝑋_𝑡,𝜃d𝑋_𝑡,lim_𝛿↓0𝑗1e^𝑎𝑗𝛿𝐴𝑋+1^𝑗−1,𝜃²e^𝑎𝑗𝛿−1𝑎_𝑗=12¹₀|𝐴|²𝑋_𝑡,𝜃d𝑡,lim_𝛿↓012_𝑗log2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}1−1=−2¹₀𝐴^′𝑋_𝑡,𝜃d𝑡,lim_𝛿↓012𝛿_𝑗1−2𝛿𝑎_𝑗e^{2𝑎𝑗𝛿}𝐷−1^2𝑗=12¹₀𝐴^′𝑋_𝑡,𝜃d𝑡 (4.12) in probability. The claim thus follows immediately.

5. A Representation Formula

From this section, we develop necessary estimates in order to prove Theorem 3.1. In this section, we recall the main tool in our proof, a representation formula proved by Qian and Zheng [8]. Based on this formula, we prove the main estimate (6.65), which has independent interest, in the next section. We conclude the proof of Theorem 3.1 in Section 7.

Let 𝑥∈ℝ. Consider the linear diffusion 𝑑𝑋_𝑡=𝐴(𝑥,𝜃)+𝐴^′(𝑥,𝜃)𝑋_𝑡−𝑥d𝑡+d𝑊_𝑡, (5.1) whose probability transition function is also denoted by ℎ(𝑡,𝑧,𝑦). Recall that 𝑝(𝑡,𝑧,𝑦) is the probability transition function of the diffusion defined by (1.3). The strong solution of (5.1) is given by 𝑋_𝑡=𝑋₀+𝜎(𝑎,𝑡)²𝑎𝑋₀+𝑏+^𝑡₀e^{𝑎(𝑡−𝑠)}d𝑊_𝑠, (5.2) so that ℎ=1𝑡,𝑧,𝑦√()−12𝜋𝜎2𝑎,𝑡exp2𝜎(2𝑎,𝑡)²𝑦−𝑧−𝜎(𝑎,𝑡)²(𝑏+𝑎𝑧)², (5.3) where 𝑏=𝐴(𝑥,𝜃)−𝑥𝐴^′(𝑥,𝜃), 𝑎=𝐴^′(𝑥,𝜃) and 𝜎(𝑎,𝑡)=(e^𝑎𝑡−1)/𝑎.

Observe that for any 𝑎∈ℝ, 𝑡→𝜎(𝑎,𝑡) is increasing, and lim_𝑡↓0𝜎(𝑎,𝑡)√𝑡=1forany𝑎. (5.4) We will also use the fact that 𝜎(2𝑎,𝑡)=e^2𝑎𝑡−1=2𝑎e^𝑎𝑡+12𝜎(𝑎,𝑡). (5.5)

Lemma 5.1. For 𝑥∈ℝ, and 𝐶(𝑧)=𝐴(𝑧,𝜃)−𝐴(𝑥,𝜃)−𝐴^′(𝑥,𝜃)(𝑧−𝑥). Then |𝐶(𝑧)|≤𝐶₀2||,1min𝑧−x2|𝑧−𝑥|². (5.6)

Our main tool is a representation formula (5.7) discovered in [8]. Let (𝑋_𝑡,ℙ^𝑥) be the solution to the linear stochastic differential equation (5.1).

Proposition 5.2. For 𝑥,𝑦∈ℝ and 𝑇>0 one has 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦=1+^𝑇₀ℙ^𝑥𝑈_𝑡𝐶𝑋_𝑡∇ℎ𝑇−𝑡,𝑋_𝑡,𝑦ℎ𝑇,𝑥,𝑦d𝑡, (5.7) where 𝑈_𝑡=exp^𝑡₀𝐶𝑋_𝑠d𝑊_𝑠−12^𝑡₀|𝐶|²𝑋_𝑠d𝑠, (5.8) which is a martingale under the probability ℙ^𝑥.

To prove (3.6), we need to estimate the double integral appearing on the right-hand side of (5.7), which requires a precise estimate for 𝐼_𝑡=ℙ^𝑥𝑈_𝑡𝐶𝑋_𝑡∇ℎ𝑇−𝑡,𝑋_𝑡,𝑦ℎ𝑇,𝑥,𝑦, (5.9) which can be achieved since we know the precise form ℎ(𝑇,𝑥,𝑦). Of course, if we knew the joint distribution of (𝑈_𝑡,𝑋_𝑡), our task would be easy, but unfortunately it is rarely the case. Our arguments are based on the fact that (𝑈_𝑡) is a martingale under ℙ^𝑥, together with some delicate estimates for the functional integral ℙ^𝑥∇ℎ𝑇−𝑡,𝑋_𝑡,𝑦ℎ𝑇,𝑥,𝑦^𝑝, (5.10) which will be done in the next section.

6. Main Estimates

We use the notations established in the previous section. Let 𝑇>0, 𝑥, 𝑦∈ℝ and 𝑑=𝑦−𝑥. Then ∇_𝑧ℎ𝑇−𝑡,𝑧,𝑦ℎ=𝑇−𝑡,𝑧,𝑦2𝑎e^{𝑎(𝑇−𝑡)}𝑒^{2𝑎(𝑇−𝑡)}−1𝑦−e^{𝑎(𝑇−𝑡)}𝑧−𝜎(𝑎,𝑇−𝑡)²𝑏, (6.1) and therefore ∇_𝑧ℎ𝑇−𝑡,𝑋_𝑡,𝑦ℎ=𝑇,𝑥,𝑦2𝑎e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}ℎ−1𝑇−𝑡,𝑋_𝑡,𝑦ℎ×𝑇,𝑥,𝑦𝑦−e^{𝑎(𝑇−𝑡)}𝑋_𝑡−𝜎(𝑎,𝑇−𝑡)²𝑏=𝜎(2𝑎,𝑇)e^{𝑎(𝑇−𝑡)}𝜎(2𝑎,𝑇−𝑡)³e^{(1/2𝜎(2𝑎,𝑇)2)𝑆(𝑇)2}×𝑦−e^{𝑎(𝑇−𝑡)}𝑋_𝑡−𝜎(𝑎,𝑇−𝑡)²𝑏×e^{−(1/2𝜎(2𝑎,𝑇−t)2)|𝑦−e𝑎(𝑇−𝑡)𝑋𝑡−𝜎(𝑎,𝑇−𝑡)2𝑏|2,} (6.2) where 𝑆(𝑇)=𝑦−𝑥−𝜎(𝑎,𝑇)²(𝑎𝑥+𝑏). (6.3)

For 𝑡∈(0,𝑇) and 𝑝>1 we set 𝛽_𝑝(𝑡)=𝑎𝑝e^2𝑎𝑇−1−𝑝−1e^{2𝑎(𝑇−𝑡)}𝜎()−1𝑎,𝑇²(),𝛼𝑏+𝑎𝑥−𝑑_𝑝(𝑡)=e^{2𝑎(𝑇−𝑡)}−1𝑝e^2𝑎𝑇−1−𝑝−1e^{2𝑎(𝑇−𝑡)},−1 (6.4) for simplicity.

Lemma 6.1. For any 𝑝>1 one has 1𝑝𝜎(𝑎,𝑇)²(𝑏+𝑎𝑥)−𝑑≤e^2𝑎𝑇−1𝑎𝛽_𝑝(𝑡)≤𝜎(𝑎,𝑇)²(𝑏+𝑎𝑥)−𝑑,1𝑝e^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇−1≤𝛼_𝑝(𝑡)≤e^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇,−1 (6.5) for all 𝑡∈[0,𝑇].

Proof. The two inequalities follow from the fact that e^2𝑎𝑇−1𝑝e^2𝑎𝑇−1−𝑝−1e^{2𝑎(𝑇−𝑡)}−1 (6.6) assumes its maximum 1 and minimum 1/𝑝.

Since 𝛽_𝑝(𝑡)𝛼_𝑝(𝑡)=𝑎e^{2𝑎(𝑇−𝑡)}𝜎()−1𝑎,𝑇²(),e𝑏+𝑎𝑥−𝑑_{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇𝛽−1^2𝑝(𝑡)²𝛼_2𝑝(𝑡)²=𝑎e^2𝑎𝑇𝜎()−1𝑎,𝑇²(𝑏+𝑎𝑥)−𝑑², (6.7) so that ∇_𝑧ℎ𝑇−𝑡,𝑋_𝑡,𝑦ℎ=𝜎()𝑇,𝑥,𝑦2𝑎,𝑇𝜎(2𝑎,𝑇−𝑡)³e^{((e2𝑎(𝑇−𝑡)−1)/(e2𝑎𝑇−1))(𝛽2𝑝(𝑡)2/𝛼2𝑝(𝑡)2)+𝑎(𝑇−𝑡)}×𝑦−e^{𝑎(𝑇−𝑡)}𝑋_𝑡−𝜎(𝑎,𝑇−𝑡)²𝑏×e^{−(1/2𝜎(2𝑎,𝑇−𝑡)2)|𝑦−e𝑎(𝑇−𝑡)𝑋𝑡−𝜎(𝑎,𝑇−𝑡)2𝑏|2}, (6.8) which yield, together with (5.7), the following.

Lemma 6.2. One has 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦=1+^𝑇₀2𝑎e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇−1e^{2𝑎(𝑇−𝑡)}e−1^{((e2𝑎(𝑇−𝑡)−1)/(e𝑎𝑇−1))(𝛽2𝑝(𝑡)2/𝛼2𝑝(𝑡)2)}×ℙ^𝑥𝑈_𝑡𝐶𝑋_𝑡𝐾(𝑡)d𝑡, (6.9) where 𝐾(𝑡)=𝑦−e^{𝑎(𝑇−𝑡)}𝑋_𝑡−e^{𝑎(𝑇−𝑡)}−1𝑎𝑏×e^{−(𝑎/(e2𝑎(𝑇−𝑡)−1))|𝑦−e𝑎(𝑇−𝑡)𝑋𝑡−((e𝑎(𝑇−𝑡)−1)/𝑎)𝑏|2}. (6.10)

Let 𝐽𝑝_𝑡=^𝑝ℙ^𝑥|𝐾(𝑡)|^𝑝𝐷𝑝_𝑡=^2𝑝ℙ^𝑥e^{(2𝑝/(2𝑝−1))∫𝑡0|𝐶|2(𝑋𝑠)d𝑠}𝐶𝑋_𝑡^2𝑝. (6.11)

Lemma 6.3. Choose 𝜁>0 such that ℙexp𝜁sup_[]0,1𝑊_𝑡²=𝑐<+∞. (6.12) Then for any 𝑇>0 and 𝜆>0, such that 4𝜆𝐶₀e^𝑎𝑇e^2𝑎𝑇−1𝑎≤𝜁, (6.13) one has ℙ^𝑥e^{𝜆∫𝑡0|𝐶|2(𝑋𝑠)d𝑠}≤𝑐∀𝑡≤𝑇. (6.14)

Proof. Let 𝑌_𝑡=𝑋_𝑡−𝑥−𝜎(𝑎,𝑡)²(𝑎𝑥+𝑏)=^𝑡₀e^{𝑎(𝑡−𝑠)}d𝑊_𝑠. (6.15) Then |𝐶|²𝑋_𝑠≤4𝐶₀𝑋_𝑠−𝑥²≤4𝐶₀𝑌_𝑠+𝜎(𝑎,𝑠)²(𝑎𝑥+𝑏)²≤4𝐶₀𝜎(𝑎,𝑠)⁴|(𝑎𝑥+𝑏)|²+4𝐶₀𝐶₂𝑌_𝑠², (6.16) and therefore ℙ^𝑥e^{𝜆∫𝑡0|𝐶|2(𝑋𝑠)d𝑠}≤e^{4𝜆𝐶0𝜎(𝑎,𝑡)4|(𝑎𝑥+𝑏)|2𝑡}ℙ^𝑥e^{4𝜆𝐶0∫𝑡0|𝑌𝑠|2d𝑠}. (6.17) On the other hand 𝑌_𝑠=𝑒^𝑎𝑠𝑠₀e^−𝑎𝑟d𝑊_𝑟, (6.18) so that 𝑀_𝑠≡𝑒^−𝑎𝑠𝑌_𝑠 is a martingale with ⟨𝑀⟩_𝑠=^𝑠₀e^−2𝑎𝑟ed𝑟=^2𝑎𝑠−12𝑎. (6.19) Thus 𝑀_𝑠 is a time change of a standard Brownian motion, and 𝑀_𝑠=𝐵_{(e2𝑎𝑠−1/2𝑎)}, (6.20) for some standard Brownian motion (𝐵_𝑡)_𝑡≥0. Since 2𝜆𝐶₀1e^𝑎𝑡e^2𝑎𝑡−12𝑎≤𝜁, (6.21) so we have ℙ^𝑥e^{4𝜆𝐶0∫𝑡0e𝑎𝑠|𝑀𝑠|2d𝑠}=ℙexp4𝜆𝐶₀^𝑡₀𝑒^𝑎𝑠𝐵_{(e2𝑎𝑠−1)/2𝑎}²d𝑠=ℙexp4𝜆𝐶₀^(e2𝑎𝑡₀^−1)/2𝑎𝑠√𝐵2𝑎𝑠+1^𝑠2d𝑠≤ℙexp4𝜆𝐶₀1e^𝑎𝑡e^2𝑎𝑡−12𝑎²sup_{0,e2𝑎𝑡−1/2𝑎}𝐵_𝑠²=ℙexp4𝜆𝐶₀1e^𝑎𝑡e^2𝑎𝑡−12𝑎sup_[]0,1𝐵_𝑠²≤𝑐. (6.22)

Corollary 6.4. For 𝑝>1 and 𝑇>0 to be such that 16𝑝𝐶₀12𝑝−1e^𝑎𝑇e^2𝑎𝑇−12𝑎≤𝜁, (6.23) one has ℙ^𝑥e^{(4𝑝/(2𝑝−1))∫𝑡0|𝐶|2(𝑋𝑠)d𝑠}≤𝑐∀𝑡≤𝑇. (6.24)

In what follows, we always assume that 𝑇>0 is chosen such that the condition (6.23) is satisfied. Next we estimate 𝐷(𝑝)_𝑡, which is provided in the following.

Lemma 6.5. Let 𝑝>1. Then 𝐷𝑝_𝑡≤𝐶₁𝜎(2𝑎,𝑡)²+𝜎(𝑎,𝑡)²(𝑎𝑥+𝑏)², (6.25) where the positive constant 𝐶₁ depends only on 𝑝, 𝜁, and 𝐶₀.

Proof. Let 𝐹(𝑡)=ℙ^𝑥e^{(2𝑝/(2𝑝−1))∫𝑡0|𝐶|2(𝑋𝑠)d𝑠}𝐶𝑋_𝑡^2𝑝. (6.26) Then, by the Hölder inequality 𝐹(𝑡)=ℙ^𝑥e^{(4𝑝/(2𝑝−1))∫𝑡0|𝐶|2(𝑋𝑠)d𝑠}ℙ^𝑥𝐶𝑋_𝑡^4𝑝≤𝐶₀ℙ^𝑥𝐶𝑋_𝑡^4𝑝. (6.27) Next we estimate the expectation ℙ^𝑥|𝐶(𝑋_𝑡)|^4𝑝. Since 𝐶𝑋_𝑡≤𝐶₀2𝑋_𝑡−𝑥²≤𝐶₀𝑌_𝑡²+𝐶₀𝜎(𝑎,𝑇)²(𝑎𝑥+𝑏)², (6.28) so that 𝐶𝑋_𝑡^4𝑝≤2^4𝑝−1C₀^4𝑝𝑌_𝑡^8𝑝+2^4𝑝−1𝐶₀^4𝑝𝜎(𝑎,𝑇)^8𝑝(𝑎𝑥+𝑏)^8𝑝. (6.29) On the other hand ℙ^𝑥𝑌_𝑡^8𝑝=1√()|𝑧|2𝜋𝜎2𝑎,𝑡^8𝑝−|𝑧|exp²𝜎(2𝑎,𝑡)²()d𝑧=𝜎2𝑎,𝑡^8𝑝, (6.30) so that ^4𝑝ℙ^𝑥𝐶𝑋_𝑡^4𝑝≤2^{(4𝑝−1)/4𝑝}𝐶₀𝜎(2𝑎,𝑡)²+2^{(4𝑝−1)/4𝑝}𝐶₀𝜎(𝑎,𝑡)²(𝑎𝑥+𝑏)². (6.31)

Lemma 6.6. Let 𝑇>0 satisfy condition (6.23), 𝑥,𝑦∈ℝ, and 𝑝>1 and 𝑞>1 such that (1/𝑝)+(1/𝑞)=1. Then 𝑝𝑇,𝑥,𝑦ℎe𝑇,𝑥,𝑦≤1+exp^{2𝑎(𝑇−𝑡)}−1e^𝑎𝑇𝛽−1^2𝑝(𝑡)²𝛼_2𝑝(𝑡)²×^𝑇₀2𝑎e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇−1e^{2𝑎(𝑇−𝑡)}𝐷𝑝−1^𝑡𝐽2𝑝_𝑡d𝑡. (6.32)

Proof. Since 𝑈_𝑡e^{−((𝑞−1)/2)∫𝑡0|𝐶|2(𝑋𝑠)d𝑠𝑞}=exp^𝑡₀𝑞𝐶𝑋_𝑠d𝑊_𝑠−12𝑞^2𝑡₀|𝐶|²𝑋_𝑠d𝑠 (6.33) is a martingale under ℙ^𝑥, so that ℙ^𝑥𝑈_𝑡e^{−((𝑞−1)/2)∫𝑡0|𝐶|2(𝑋𝑠)d𝑠𝑞}=1. (6.34) By the Hölder inequality we deduce that ℙ^𝑥𝑈_𝑡𝐶𝑋_𝑡𝐾(𝑡)≤𝐷𝑝_𝑡𝐽2𝑝_𝑡. (6.35)
Equation (6.32), follows from the representation (6.9).

Lemma 6.7. Let 𝑝>1. Then 𝐽𝑝_𝑡≤^𝑝2^𝑝−1𝛼_𝑝(𝑡)^1+(1/𝑝)e^−𝛽2𝑝×^𝑝𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}+2𝑎e^{2𝑎(𝑇−𝑡)}−1𝑎𝛽_𝑝(𝑡). (6.36)

Proof. We have ℙ^𝑥|𝐾(𝑡)|^𝑝≤ℙ^𝑥𝑦−𝑒^{𝑎(𝑇−𝑡)}𝑋_𝑡−𝑒^{𝑎(𝑇−𝑡)}−1𝑎𝑏^𝑝×e^{−(𝑝𝑎/(𝑒2𝑎(𝑇−𝑡)−1))|𝑦−𝑒𝑎(𝑇−𝑡)𝑋𝑡−((𝑒𝑎(𝑇−𝑡)−1)/𝑎)𝑏|2}. (6.37) Under the probability ℙ^𝑥, 𝑍_𝑡≡𝑒^{𝑎(𝑇−𝑡)}𝑋_𝑡−𝑒^𝑎𝑡𝑒𝑥−^𝑎𝑡−1𝑎𝑏=^𝑡₀𝑒^{𝑎(𝑇−𝑠)}d𝑊_𝑠 (6.38) is a central normal distribution with variance ^𝑡₀𝑒^{2𝑎(𝑇−𝑠)}ed𝑠=^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}2𝑎. (6.39) In terms of 𝑍_𝑡 and 𝑑=𝑦−𝑥ℙ^𝑥|𝐾(𝑡)|^𝑝≤ℙ^𝑥𝑍_𝑡+𝑒^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑^𝑝×e^{−(𝑝𝑎/(𝑒2𝑎(𝑇−𝑡)−1))|𝑍𝑡+((𝑒𝑎𝑇−1)/𝑎)(𝑏+𝑎𝑥)−𝑑|2}. (6.40) Making change of variable 𝑁=2𝑎e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑍_𝑡. (6.41) Then, under ℙ^𝑥, 𝑁 has the standard normal distribution 𝑁(0,1), so that ℙ^𝑥|𝐾(𝑡)|^𝑝≤ℙe^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑒2𝑎𝑁+^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑^𝑝×e^{−(𝑝𝑎/(𝑒2𝑎(𝑇−𝑡)−1))|((e2𝑎𝑇−e2𝑎(𝑇−𝑡))/2𝑎)𝑁+((𝑒𝑎𝑇−1)/𝑎)(𝑏+𝑎𝑥)−𝑑|2}=1√2𝜋^ℝe^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑒2𝑎𝑧+^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑^𝑝×e^{−(𝑝𝑎/(𝑒2𝑎(𝑇−𝑡)−1))|((e2𝑎𝑇−e2𝑎(𝑇−𝑡))/2𝑎)𝑧+((𝑒𝑎𝑇−1)/𝑎)(𝑏+𝑎𝑥)−𝑑|2−(𝑧2/2)}𝑑𝑧. (6.42) Let us simplify the last integral. Indeed, set 𝜂=𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1𝑒^{2𝑎(𝑇−𝑡)}𝐸−1𝑧,_𝑡=2𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}2𝑒^{2𝑎(𝑇−𝑡)}𝑒−1^𝑎𝑇−1𝑎()×𝑏+𝑎𝑥−𝑑𝑎𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}.−1 (6.43) Then we rewrite the term appearing in the exponential in the last line of (6.42) −𝑝𝑎𝑒^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑒2𝑎𝑧+^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑²1=−2𝜂²+2𝐸_𝑡𝜂−𝑝𝑎𝑒^{2𝑎(𝑇−𝑡)}𝑒−1^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑²1=−2𝜂+𝐸_𝑡²+12𝐸²_𝑡−𝑝𝑎𝑒^{2𝑎(𝑇−𝑡)}𝑒−1^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑², (6.44) together with 12𝐸²_𝑡−𝑝𝑎𝑒^{2𝑎(𝑇−𝑡)}𝑒−1^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑²=−𝑝𝑎𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}𝑒−1^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑² (6.45) the inequality (6.42) may be rewritten as follows: ℙ^𝑥|𝐾(𝑡)|^𝑝≤1√e2𝜋^{−(𝑝𝑎/(𝑝e2𝑎𝑇−(𝑝−1)e2𝑎(𝑇−𝑡)−1))(((𝑒𝑎𝑇−1)/𝑎)(𝑏+𝑎𝑥)−𝑑)2}×_ℝe^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}e2𝑎𝑧+^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑^𝑝×e^{−(1/2)(𝜂+𝐸𝑡)2}d𝑧. (6.46) Making change of variable in the last integral 𝑣=𝜂+𝐸_𝑡=𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1𝑒^{2𝑎(𝑇−𝑡)}−1𝑧+𝐸_𝑡, (6.47) so that d𝑧=𝑒^{2𝑎(𝑇−𝑡)}−1𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1d𝑣,e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑒2𝑎𝑧+^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑=𝑒^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}2𝑎𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}𝑣+e−1_{2𝑎(𝑇−𝑡)}−1𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}𝑒−1^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑑. (6.48) Thus (6.46) yields that ℙ^𝑥|𝐾(𝑡)|^𝑝≤e^{2𝑎(𝑇−𝑡)}−12𝑎^𝑝+12𝑎𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1×e^{−(𝑝𝑎/(𝑝e2𝑎𝑇−(𝑝−1)e2𝑎(𝑇−𝑡)−1))(((𝑒𝑎𝑇−1)/𝑎)(𝑏+𝑎𝑥)−𝑑)2}𝑄_𝑡, (6.49) where 𝑄_𝑡=_ℝe^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}𝑧+−12𝑎e^{2𝑎(𝑇−𝑡)}−1𝑒^𝑎𝑇−1/𝑎(𝑏+𝑎𝑥)−𝑑𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1^𝑝e^{−(1/2)𝑧2}√2𝜋d𝑧≤2^𝑝−1𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1^𝑝+2^𝑝−12𝑎e^{2𝑎(𝑇−𝑡)}−1𝑒^𝑎𝑇−1/𝑎(𝑏+𝑎𝑥)−𝑑𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−t)}−1^𝑝𝜀_𝑝=1√2𝜋^ℝ|𝑧|^𝑝e^{−(1/2)𝑧2}d𝑧. (6.50) Therefore 𝐽𝑝_𝑡≤e^{2𝑎(𝑇−𝑡)}−12𝑎^1+(1/𝑝)2𝑎𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1^1/(2𝑝)×e^{−(𝑎/(𝑝e2𝑎𝑇−(𝑝−1)e2𝑎(𝑇−𝑡)−1))(((𝑒𝑎𝑇−1)/𝑎)(𝑏+𝑎𝑥)−𝑑)2}×^𝑝2^𝑝−1𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−t)}+−1_𝑝2^𝑝−12e^{2𝑎(𝑇−𝑡)}−1𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)}−1𝑎𝑒^𝑎𝑇−1/𝑎(𝑏+𝑎𝑥)−𝑑²𝑝e^2𝑎𝑇−𝑝−1e^{2𝑎(𝑇−𝑡)},−1 (6.51) which is equivalent to the required inequality.

Lemma 6.8. Let 𝐻𝑝_𝑡e≡exp^{2𝑎(𝑇−𝑡)}−1e^𝑎𝑇𝛽−1^2𝑝(𝑡)²𝛼_2𝑝(𝑡)²𝐽2𝑝_𝑡. (6.52) Then 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≤1+^𝑇₀2𝑎𝑒^{𝑎(𝑇−𝑡)}𝑒^{2𝑎(𝑇−𝑡)}−1𝑒^2𝑎𝑇−1𝑒^{2𝑎(𝑇−𝑡)}𝐷𝑝−1^𝑡𝐻𝑝_𝑡𝐻𝑝d𝑡,_𝑡≤^2𝑝2^2𝑝−1𝛼_2𝑝(𝑡)^1+(1/2𝑝)e^{−(2𝑝−1)((e2𝑎𝑇−e2𝑎(𝑇−𝑡))2/(e2𝑎(𝑇−𝑡)−1)(e2𝑎𝑇−1))𝛽2𝑝(𝑡)2}×^𝑝𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}+2𝑎e^{2𝑎(𝑇−𝑡)}−1𝑎𝛽_2𝑝(𝑡). (6.53)(6.54)

Proof. Indeed, by Lemma 6.7 we have 𝐻𝑝_𝑡≤^2𝑝2^2𝑝−1𝛼_2𝑝(𝑡)^1+(1/2𝑝)e^{−𝛽22𝑝+((e2𝑎(𝑇−𝑡)−1)/(e2𝑎𝑇−1))(𝛽2𝑝(𝑡)2/𝛼2𝑝(𝑡)2)}×^𝑝𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}+2𝑎e^{2𝑎(𝑇−𝑡)}−1𝑎𝛽_𝑝(𝑡). (6.55) On the other hand −𝛽²_2𝑝+e^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇𝛽−1^2𝑝(𝑡)²𝛼_2𝑝(𝑡)²=𝑝−1𝛽_2𝑝(𝑡)²e^2𝛼𝑇−e^{2𝛼(𝑇−𝑡)}e^2𝑎𝑇−1, (6.56) thus 𝐻𝑝_𝑡≤^2𝑝2^2𝑝−1𝛼_2𝑝(𝑡)^1+(1/2𝑝)e^{(𝑝−1)𝛽2𝑝(𝑡)2((e2𝛼𝑇−e2𝛼(𝑇−𝑡))/(e2𝑎𝑇−1))}×^𝑝𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}+2𝑎e^{2𝑎(𝑇−𝑡)}−1𝑎𝛽_2𝑝(𝑡). (6.57)

Let 𝐵𝑝_𝑡≡e^{(𝑝−1)𝛽2𝑝(𝑡)2((e2𝛼𝑇−e2𝛼(𝑇−𝑡))/(e2𝑎𝑇−1))}×^𝑝𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}+2𝑎e^{2𝑎(𝑇−𝑡)}−1𝑎𝛽_2𝑝(𝑡). (6.58) Then (6.53) and (6.54) imply that 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≤1+^2𝑝2^2𝑝−1e^2𝑎𝑇−1^1/4𝑝𝑇02𝑎e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}−1^{1−(1/4𝑝)}𝐷𝑝_𝑡𝐵𝑝_𝑡d𝑡. (6.59)

Lemma 6.9. One has 𝐵𝑝_𝑡≤e^{(𝑝−1)𝛽2𝑝(𝑡)2((e2𝛼𝑇−e2𝛼(𝑇−𝑡))/(e2𝑎𝑇−1))}×^𝑝𝜀_𝑝e^2𝑎𝑇−e^{2𝑎(𝑇−𝑡)}+2𝑎e^{2𝑎(𝑇−𝑡)}−1e^2𝑎𝑇𝑆(𝑇).−1 (6.60) In particular 𝐵𝑝_𝑡≤^𝑝𝜀_𝑝e^2𝑎𝑇−1(𝑇),𝑝2𝑎+𝑆𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≤1+^2𝑝2^2𝑝−1e^2𝑎𝑇−1^1/4𝑝𝑝𝜀_𝑝e^2𝑎𝑇−1(𝑇)×2𝑎+𝑆_𝑇02𝑎e^{𝑎(𝑇−𝑡)}e^{(𝑝−1)𝛽2𝑝(𝑡)2((e2𝛼𝑇−e2𝛼(𝑇−𝑡))/(e2𝑎𝑇−1))}e^{2𝑎(𝑇−𝑡)}−1^{1−(1/4𝑝)}𝐷𝑝_𝑡d𝑡. (6.61)

Proof. Let 𝐺𝑝_𝑡=2𝑎𝑒^{𝑎(𝑇−𝑡)}𝑒^{2𝑎(𝑇−𝑡)}−1𝑒^2𝑎𝑇−1𝑒^{2𝑎(𝑇−𝑡)}−1^2𝑝2^2𝑝−1𝛼_2𝑝(𝑡)^1+(1/2𝑝). (6.62) Then 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≤1+^𝑇₀𝐺𝑝_𝑡𝐷𝑝_𝑡𝐵𝑝_𝑡d𝑡. (6.63) Since 𝐺𝑝_𝑡=2𝑎𝑒^{𝑎(𝑇−𝑡)}𝑒^{2𝑎(𝑇−𝑡)}−1^{1−(1/4𝑝)}𝑒^2𝑎𝑇−1^1/4𝑝2𝑝2^2𝑝−1×𝑒^2𝑎𝑇−12𝑝e^2𝑎𝑇−1−2𝑝−1e^{2𝑎(𝑇−𝑡)}−1^1+(1/2𝑝)≤2𝑎𝑒^{𝑎(𝑇−𝑡)2𝑝}2^2𝑝−1𝑒^{2𝑎(𝑇−𝑡)}−1^{1−(1/4𝑝)}𝑒^2𝑎𝑇−1^1/4𝑝, (6.64) which implies the required estimate.

By collecting all estimates we have established, we may obtain the following.

Proposition 6.10. There is a constant 𝐶₂>0 depending only on 𝜁 and 𝐶_𝑑 such that 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≤1+𝐶₂e^|𝑎|𝑇e^{(𝑝−1)(𝑎/(𝑒2𝑎𝑇−1))𝑆(𝑇)2}e^𝑎𝑇+12+(𝑎𝑥+𝑏)²×e^𝑎𝑇−1𝑎^𝑝𝜀_𝑝e^2𝑎𝑇−1(𝑇),2𝑎+𝑆 (6.65) where 𝑆(𝑇)=e^𝑎𝑇−1𝑎(𝑏+𝑎𝑥)−𝑦−𝑥. (6.66)

Proof. Indeed 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≤1+𝐶₁e^{(𝑝−1)(𝑎/(𝑒2𝑎𝑇−1))𝑆(𝑇)2}e^2𝑎𝑇−1𝑎^{−1/4𝑝𝑝}𝜀_𝑝e^2𝑎𝑇−1(𝑇)×2𝑎+𝑆_𝑇0e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}−1𝑎^{(1/4𝑝)−1}𝜎(2𝑎,𝑡)²+𝜎(𝑎,𝑡)²(𝑎𝑥+𝑏)²d𝑡≤1+𝐶e^{(𝑝−1)(𝑎/(𝑒2𝑎𝑇−1))𝑆(𝑇)2}e^2𝑎𝑇−1𝑎^{−1/4𝑝𝑝}𝜀_𝑝e^2𝑎𝑇−1(𝑇)×e2𝑎+𝑆_𝑎𝑇+12+(𝑎𝑥+𝑏)²×e^𝑎𝑇−1𝑎^𝑇₀e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}−1𝑎^{(1/4𝑝)−1}d𝑡. (6.67) While, ^𝑇₀e^{𝑎(𝑇−𝑡)}e^{2𝑎(𝑇−𝑡)}−1/2𝑎^{1−(1/4𝑝)}d𝑡=^(e2𝑎𝑇₀^−1)/2𝑎1𝑠^{1−(1/4𝑝)}1√1+2𝑎𝑠d𝑠≤4𝑝e^|𝑎|𝑇e^2𝑎𝑇−12𝑎^1/4𝑝, (6.68) and it thus yields our key estimate (6.65).

Similarly we have a lower bound 𝑝𝑇,𝑥,𝑦ℎ𝑇,𝑥,𝑦≥1−𝐶₃e^|𝑎|𝑇e^{(𝑝−1)(𝑎/(𝑒2𝑎𝑇−1))𝑆(𝑇)2}e^𝑎𝑇+12+(𝑎𝑥+𝑏)²×e^𝑎𝑇−1𝑎^𝑝𝜀_𝑝e^2𝑎𝑇−1(𝑇),2𝑎+𝑆 (6.69)

where 𝐶₃ depends only on 𝜁 and 𝐶₀.

7. Proof of Theorem 3.1

We are now in a position to prove Theorem 3.1. We may assume that 𝑇=1, so that 𝛿(𝑛)=1/𝑛. Let 𝑥^𝑛_𝑗 (𝑗=0,1,…,𝑛) be discrete samplings with time scale 𝛿=𝛿(𝑛)=1/𝑛 on [0,1]. By our assumptions, |𝑥^𝑛_𝑗−𝑥^𝑛_𝑗−1|²≤𝐶𝛿(𝑛), and |𝑥^𝑛_𝑗|≤𝐶 for all pair (𝑗,𝑛) such that 0≤𝑗≤𝑛 and 𝑛≥1. For simplicity we write 𝑥_𝑗 for 𝑥^𝑛_𝑗 if no confusion may arise.

In the proof below, we will use 𝐶_𝑖 to denote nonnegative constants which may depend on 𝐶,𝑇(=1) and the bounds of 𝐴^′ and 𝐴^′′ appearing in our diffusion model (1.3), but independent of 𝑛.

Recall that ℎ_𝑗(𝑡,𝑥,𝑦) is the probability transition density function of the diffusion (3.3), that is, d𝑋_𝑡=𝑏_𝑗+𝑎_𝑗𝑋_𝑡d𝑡+d𝑊_𝑡, (7.1) where 𝑏_𝑗=𝐴𝑥_𝑗−1,𝜃−𝑥_𝑗−1𝐴^′𝑥_𝑗−1,𝜃,𝑎_𝑗=𝐴^′𝑥_𝑗−1,𝜃. (7.2)

According to (6.65) we have 𝑝𝛿,𝑥_𝑗−1,𝑥_𝑗ℎ_𝑗𝛿,𝑥_𝑗−1,𝑥_𝑗≤1+𝐶₈e^{|𝑎𝑗|𝛿}e^{((𝑝−1)𝑎𝑗/(e𝑗𝛿2𝑎−1))𝑆2𝑗}e^𝑎𝑗𝛿+12+𝑎_𝑗𝑥_𝑗−1+𝑏_𝑗²×e^{𝑎𝑗−1𝛿}−1𝑎_𝑗−1^𝑝𝜀_𝑝e^{2𝑎𝑗−1𝛿}−12𝑎_𝑗−1+𝑆_𝑗, (7.3) where 𝑆_𝑗=e^𝑎𝑗𝛿−1𝑎_𝑗𝑏_𝑗+𝑎_𝑗𝑥_𝑗−1−𝑥_𝑗−𝑥_𝑗−1. (7.4) Since 𝑎_𝑗 and 𝑥_𝑗 are bounded, 𝑎_𝑗𝑥_𝑗−1+𝑏_𝑗=𝐴𝑥_𝑗−1,𝜃≤𝐶₄1+𝑥_𝑗−1≤𝐶₉, (7.5) so that𝑆_𝑗≤e^𝑎𝑗𝛿−1𝑎_𝑗𝛿𝐶₉𝛿+𝐶√𝛿≤𝐶₁₀√𝛿. (7.6) Thus 𝑝−1𝑎_𝑗e^{2𝑎𝑗𝛿}𝑆−1^2𝑗=𝑝−1𝑎_𝑗𝛿e^{2𝑎𝑗𝛿}𝑆−1𝛿^2𝑗≤𝐶₁₁,e^𝑎𝑗𝛿+12+𝑎_𝑗𝑥_𝑗−1+𝑏_𝑗²≤𝐶₁₂. (7.7) Therefore𝑝𝛿,𝑥_𝑗−1,𝑥_𝑗ℎ_𝑗𝛿,𝑥_𝑗−1,𝑥_𝑗≤1+𝐶₁₃e^{𝑎𝑗−1𝛿}−1𝑎_𝑗−1𝛿^𝑝𝜀_𝑝e^{2𝑎𝑗−1𝛿}−12𝑎_𝑗−1𝛿√𝛿+𝑆_𝑗𝛿≤1+𝐶₁₄√𝛿𝛿=1+𝐶₁₄1𝑛^3/2, (7.8) where we have used (7.6). It follows that lim^𝑛_{𝛿→0𝑗=1}𝑝𝛿,𝑥_𝑗−1,𝑥_𝑗ℎ_𝑗𝛿,𝑥_𝑗−1,𝑥_𝑗≤lim_𝑛→∞1+𝐶₁₄1𝑛^3/2𝑛=1. (7.9) Similarly we have lim^𝑛_{𝑛→∞𝑗=1}𝑝𝛿,𝑥_𝑗−1,𝑥_𝑗ℎ_𝑗𝛿,𝑥_𝑗−1,𝑥_𝑗≥1. (7.10) Thereforelim_𝑛→∞𝐿𝑥^𝑛₀,…,𝑥^𝑛_𝑛𝐿₂𝑥^𝑛₀,…,𝑥^𝑛_𝑛=1, (7.11) and the proof of Theorem 3.1 is complete.

8. The Euler-Maruyama Approximation

Recall that the Euler-Maruyama approximation to (1.3) is a Markov chain given by 𝑋_𝑗=𝑋_𝑗−1+𝐴𝜃,𝑋_𝑗−1𝛿+𝜉_𝑗√𝛿, (8.1) where {𝜉_𝑗} is an i.i.d. random sequence, with standard normal 𝑁(0,1). The conditional distribution of 𝑋_𝑗 given 𝑋_𝑗−1=𝑥_𝑗−1 is Gaussian with mean 𝑥_𝑗−1+𝐴(𝜃,𝑥_𝑗−1)𝛿 and variance 𝛿 so that the likelihood function is given as 𝐿₁𝑥₀,…,𝑥_𝑛=1(2𝜋𝛿)^𝑛𝑛/2_𝑗=1−𝑥exp^𝑗−𝑥_𝑗−1−𝐴𝜃,𝑥_𝑗−1𝛿²2𝜋𝛿. (8.2) Applying the representation formula (5.7) we have the following.

Proposition 8.1. It holds that 𝐿𝑥₀,…,𝑥_𝑛𝐿₁𝑥₀,…,𝑥_𝑛=_𝑗1−√𝛿𝑒^{𝛼2𝑗/2𝛿𝛿}₀ℙ𝑈_𝑗(𝑠)𝑐_𝑗𝑋^𝑗_𝑠𝑊,𝜃^𝑠+𝛼_𝑗(𝛿−𝑠)^3/2𝑒^{−(𝑊𝑠+𝛼𝑗)2/2(𝛿−𝑠)}d𝑠, (8.3) where (𝑊_𝑡)_𝑡≥0 is the standard Brownian motion, 𝑋^𝑗_𝑠=𝑥_𝑗−1+𝜆_𝑗−1𝑠+𝑊_𝑠, 𝑈_𝑗(𝑡)=exp^𝑡₀𝑐_𝑗𝑋^𝑗_𝑠,𝜃d𝑊_𝑠−12^𝑡₀𝑐_𝑗²𝑋^𝑗_𝑠,𝜃d𝑠, (8.4)𝜆_𝑗=𝐴(𝑥_𝑗−1,𝜃),𝛼_𝑗=𝜆_𝑗−1𝛿−𝑥_𝑗+𝑥_𝑗−1, and 𝑐_𝑗(𝑧,𝜃)=𝐴(𝑧,𝜃)−𝐴𝑥_𝑗−1,𝜃. (8.5)

From which we may deduce the following estimate.

Proposition 8.2. If 𝐴(𝑥,𝜃) is bounded and Lipschitz continuous, uniformly in 𝜃, then the maximum likelihood function with discrete sampling is stable, in the sense that _𝑗1−𝐶₀₁𝛿𝑒^{𝛼2𝑗/2𝛿}≤𝐿𝑥₀,…,𝑥_𝑛𝐿₁𝑥₀,…,𝑥_𝑛≤_𝑗1+𝐶₀₂𝛿𝑒^{𝛼2𝑗/2𝛿}, (8.6) for some constant 𝐶₀₁ and 𝐶₀₂, where 𝛼_𝑗=𝐴(𝑥_𝑗−1,𝜃)𝛿−(𝑥_𝑗−𝑥_𝑗−1).

However, this estimate does not lead to the same result as for the local linear approximation. It is not known (to our best knowledge) whether lim_𝑛→∞𝐿𝑥^𝑛₀,…,𝑥^𝑛_𝑛𝐿₁𝑥^𝑛₀,…,𝑥^𝑛_𝑛=1 (8.7) holds or not under similar conditions in Theorem 3.1.

Acknowledgment

The authors thank Dr. Zeng for providing a list of errors contained in the first version of the paper.