# On Local Linear Approximations to Diffusion Processes

Authors: X. L. Duan; Z. M. Qian; W. A. Zheng
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## 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.


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## Body

## 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(1.1)dXt=A(Xt,θ)dt+σ(Xt,θ)dWt,
where (Wt)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 forθ 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 that(1.2)σ2T=lim⁡∑j=1n(X(j-1)T/n-XjT/n)2inprob.
when σ is a constant. Therefore we will limit ourselves on diffusion models with constant volatility: (1.3)dXt=A(Xt,θ)dt+dWt.
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 distributionμTX of (Xt)t≥0 over a finite time interval [0,T] has a density with respect to the Wiener measure μTW (the law of the Brownian motion W), given by the Cameron-Martin formula: (1.4)dμTXdμTW=exp⁡[∫0TA(Xt,θ)dXt-12∫0TA(Xt,θ)2dt],
which is in turn the likelihood function with continuous observation. In practice, only discrete values Xt0,…,Xtn may be observed over the duration [0,T], where 0=t0<t1<⋯<tn=T and ti-ti-1=δ. The corresponding likelihood function L̃(θ) is the conditional expectation under Wiener measure:(1.5)E{L̃(θ)∣Xt0,…,Xtn}=∏j=1npθ(δ,Xtj-1,Xtj)∏j=1nG(δ,Xtj-1,Xtj),
where pθ(t,x,y) is the conditional probability density function of Xt 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 (1.6)L(Xt0,…,Xtn)≡∏j=1npθ(δ,Xtj-1,Xtj),
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 (1.7)∇L(Xt0,…,Xtn)=0.The difficulty with this approach is that, unless for a very special drift vector fieldA, an explicit formula for pθ(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 (1.8)X̃tj-X̃tj-1=A(X̃tj-1,θ)δ+ξjδ,
where {ξj} is an i.i.d. sequence with standard normal distribution N(0,1) and X̃0=X0. However, the likelihood function L1(X0,…,Xn) 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 (1.9)L(Xt0,…,Xtn)L1(Xt0,…,Xtn).The second approach is to discretize the likelihood functiondμTX/dμTW for continuous observations. In order to utilize this likelihood function, we need to handle the Itô integral ∫0TA(Xt,θ)dXt 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, (1.10)dμTXdμTW=exp⁡[f(XT,θ)-∫0T(12(Δf)(XT,θ)+12|A(XT,θ)|2)dt],
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 functionpθ(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 pθ(t,x,y), there is no guarantee that the approximate likelihood function ∏jf(t,xj-1,xj) would tend to ∏jpθ(t,xj-1,xj) 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(Xt0,…,Xtn): (1.11)L2(Xt0,…,Xtn)=∏j=1nhj(δ,Xtj-1,Xtj),
where tj=jT/n so that Xtj is a sample with fixed duration δ=tj-tj-1 over [0,T], and hj(t,x,y) is the probability transition density of the following linear diffusion model (1.12)dX̂t=(A(Xtj-1,θ)+A′(Xtj-1,θ)(X̂t-Xtj-1))dt+dWt,
when tj-1≤t<tj and X̂tj-1=Xtj-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 Theorem3.1 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 Section2, 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 parametersa, b, and σ>0 for the linear diffusion model (Mishra and Bishwal [9] discussed a similar model): (2.1)dXt=(b-aXt)dt+σdWt,
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 (2.2)Xt=e-atX0+ba(1-e-at)+σ∫0te-a(t-s)dWs,
(formula (6.8) of Karatzas and Shreve [10], page 354), and therefore (2.3)h(t,x,y)=1πσa1-e-2atexp⁡[-a1-e-2at|y-e-atx-(b/a)(1-e-at)|2σ2].Suppose we have a discrete sample observed over the equal time scale during the period[0,T], XiT/n, i=0,…,n. According to the Markov property, their joint distribution, or the maximum likelihood function (2.4)L(a,b,σ;x0,…,xn)=μ(x0)∏j=1nh(δ,xj-1,xj),
where δ=T/n, and μ(x) is the probability density function of the initial distribution. Therefore the logarithmic of the maximum likelihood function (2.5)l(a,b,σ;{xi})=log⁡μ(x0)+∑j=1nlog⁡h(1n,xj-1,xj)=log⁡μ(x0)-n2log⁡π-nlog⁡σ+n2log⁡a-n2log⁡(1-e-2aT/n)-na1-e-2aT/n1σ2T∑j=1n(xj-ba-(xj-1-ba)e-aT/n)2.
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
(2.6)â=nTlog⁡∑j=1nxj-12-(1/n)(∑j=1nxj-1)2∑j=1nxj-1xj-(1/n)∑j=1nxj-1∑j=1nxj,β̂=1n∑j=1nxj+1nρ̂1-ρ̂(xn-x0),σ̂2=1T2â1-ρ̂2∑j=1n(xj-ρ̂xj-1-(1-ρ̂)β̂)2.As an interesting consequence we have the following.Corollary 2.2.
The maximum likelihood estimators(â,b̂,σ̂) to the linear diffusion model (2.1) are not sufficient statistics while  (â,b̂,σ̂,X0,Xn) 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,…,xn} over the time scale δ in the time interval [0,T]. Then, up to a constant factor, its maximum likelihood function (3.1)L(x0,…,xn)=∏j=1np(δ,xj-1,xj),
where p(t,z,y) is the transition probability density of (Xt) (we have dropped the subscript θ for simplicity). The approximation maximum likelihood function, proposed in  [6], is given by (3.2)L2(x0,…,xn)=∏j=1nhj(δ,xj-1,xj),
where hj(t,x,y) is the transition density function to the linear diffusion model (3.3)dXt=(A(xj-1,θ)+A′(xj-1,θ)(Xt-xj-1))dt+dWt,
which is the first-order approximation to (1.3).In what follows we assume thatA has bounded first and second derivatives and (3.4)|A′(x,θ)|,|A′′(x,θ)|≤2C0,
for some constant C0>0 independent of parameters θ.The main result of the paper is follows.Theorem 3.1.
Assume thatA′(·,θ) and  A′′(·,θ) are bounded uniformly in  θ. Let  T>0 be a fixed time and  C>0 be a constant. Suppose  {xjn}j≤n (n=1,2,…) is a family of discrete samples such that
(3.5)|xjn-xj-1n|2≤Cδ(n),|xjn|≤C,
for all pair (j,n) such that j≤n, n=1,2,…, where δ(n)=T/n. Then
(3.6)lim⁡n→∞L(x0n,…,xnn)L2(x0n,…,xnn)=1,
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 |xjn-xj-1n|2≤Cδ(n) and |xjn|≤C are reasonable. The first condition, that is |xjn-xj-1n|2≤Cδ(n), just says the “variance” of the sample cannot be too big. Since (3.7)∑j|XjT/n-X(j-1)T/n|2⟶Tinprobability,
so that on average we should have |xjn-xj-1n|2≤Cδ(n). Since (Xt) has continuous sample paths, so that {X(ω)t:t∈[0,T]} for a fixed sample point ω is bounded. Since xjn are sampled from the fixed duration [0,T], thus we can assume that {xjn} is bounded, though here we have a countable many samples. It is possible to relax this constraint, for example, we may impose that |xjn|≤Cnα 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 functionp(t,x,y), it is easy to see that (3.8)lim⁡δ→0p(δ,xj-1,xj)hj(δ,xj-1,xj)=1,
for each j, while, as our observation {x0,…,xn} 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 xj, 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 thatT=1. Let Xj/n be a discrete observation of the diffusion model (1.3) at tj=j/n (j=0,…,n). For simplicity, write Xj/n as Xj if no confusion may arise. Consider the family of linear diffusions (4.1)dXtj=(A(Xj-1,θ)+A′(Xj-1,θ)(Xtj-Xj-1))dt+dWt,
with X(j-1)/nj=Xj-1. Let (4.2)bj=A(Xj-1,θ)-A′(Xj-1,θ)Xj-1,aj=A′(Xj-1,θ).
Then (4.3)Xtj=eaj(t-tj-1)Xj-1+eaj(t-tj-1)-1ajbj+∫tj-1teaj(t-s)dWs,
so that (4.4)hj(δ,Xj-1,Xj)=P(Xtjj=Xj∣Xtj-1j=Xj-1)=2aje2ajδ-112πe-(aj/(e2ajδ-1))(Xj-Xj-1-((eajδ-1)/aj)(bj+ajXj-1))2,
where δ=1/n. The approximating likelihood function is (4.5)L2(δ)=∏j=1nhj(δ,Xj-1,Xj).
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 (4.6)L̂2(δ)=∏j=1nhj(δ,Xj-1,Xj)G(δ,Xj-1,Xj),
where (4.7)G(δ,x,y)=12πδexp⁡(-|x-y|22δ).
Hence its logarithmic (4.8)l̂2(δ)=12∑jlog⁡2δaje2ajδ-1+12δ∑j|Xj-Xj-1|2-12∑j2aje2ajδ-1(Xj-Xj-1-eajδ-1aj(bj+ajXj-1))2.Proposition 4.1.
One has(4.9)lim⁡δ↓0l̂2(δ)=l
uniformly in  θ, in probability with respect to the Wiener measure, where  l is the log of the Cameron-Martin density (1.4).Proof.
LetDj=Xj-Xj-1. Then
(4.10)l̂2(δ)=12∑jlog⁡2δaje2ajδ-1+12δ∑j(1-2δaje2ajδ-1)Dj2-∑j1eajδ+1(bj+ajXj-1)2eajδ-1aj+2∑j1eajδ+1(bj+ajXj-1)Dj.
Since bj=A(Xj-1,θ)-ajXj-1 and aj=A′(Xj-1,θ), so that
(4.11)l̂2(δ)=12∑jlog⁡2δaje2ajδ-1+12δ∑j(1-2δaje2ajδ-1)Dj2-∑j1eajδ+1A(Xj-1,θ)2eajδ-1aj+2∑j1eajδ+1A(Xj-1,θ)Dj.
However,
(4.12)lim⁡δ↓02∑j1eajδ+1A(Xj-1,θ)Dj=∫01A(Xt,θ)dXt,lim⁡δ↓0∑j1eajδ+1A(Xj-1,θ)2eajδ-1aj=12∫01|A|2(Xt,θ)dt,lim⁡δ↓012∑jlog⁡2δaje2ajδ-1=-12∫01A′(Xt,θ)dt,lim⁡δ↓012δ∑j(1-2δaje2ajδ-1)Dj2=12∫01A′(Xt,θ)dt
in probability. The claim thus follows immediately.


## 5. A Representation Formula

From this section, we develop necessary estimates in order to prove Theorem3.1. 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.Letx∈ℝ. Consider the linear diffusion (5.1)dXt=(A(x,θ)+A′(x,θ)(Xt-x))dt+dWt,
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 (5.2)Xt=X0+σ(a,t)2(aX0+b)+∫0tea(t-s)dWs,
so that (5.3)h(t,z,y)=12πσ(2a,t)exp⁡{-12σ(2a,t)2(y-z-σ(a,t)2(b+az))2},
where b=A(x,θ)-xA′(x,θ), a=A′(x,θ) and σ(a,t)=(eat-1)/a.Observe that for anya∈ℝ, t→σ(a,t) is increasing, and (5.4)lim⁡t↓0σ(a,t)t=1foranya.
We will also use the fact that (5.5)σ(2a,t)=e2at-12a=eat+12σ(a,t).Lemma 5.1.
Forx∈ℝ, and  C(z)=A(z,θ)-A(x,θ)-A′(x,θ)(z-x). Then
(5.6)|C(z)|≤C0min⁡{2|z-x|,12|z-x|2}.Our main tool is a representation formula (5.7) discovered in [8]. Let (Xt,ℙx) be the solution to the linear stochastic differential equation (5.1).Proposition 5.2.
Forx,y∈ℝ and  T>0 one has
(5.7)p(T,x,y)h(T,x,y)=1+∫0TPx{UtC(Xt)∇h(T-t,Xt,y)h(T,x,y)}dt,
where
(5.8)Ut=exp⁡{∫0tC(Xs)dWs-12∫0t|C|2(Xs)ds},
which is a martingale under the probability ℙx.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 (5.9)It=Px{UtC(Xt)∇h(T-t,Xt,y)h(T,x,y)},
which can be achieved since we know the precise form h(T,x,y). Of course, if we knew the joint distribution of (Ut,Xt), our task would be easy, but unfortunately it is rarely the case.  Our arguments are based on the fact that (Ut) is a martingale under ℙx, together with some delicate estimates for the functional integral (5.10)Px{|∇h(T-t,Xt,y)h(T,x,y)|p},
which will be done in the next section.


## 6. Main Estimates

We use the notations established in the previous section. LetT>0, x, y∈ℝ and d=y-x. Then (6.1)∇zh(T-t,z,y)h(T-t,z,y)=2aea(T-t)e2a(T-t)-1(y-ea(T-t)z-σ(a,T-t)2b),
and therefore (6.2)∇zh(T-t,Xt,y)h(T,x,y)=2aea(T-t)e2a(T-t)-1h(T-t,Xt,y)h(T,x,y)×(y-ea(T-t)Xt-σ(a,T-t)2b)=σ(2a,T)ea(T-t)σ(2a,T-t)3e(1/2σ(2a,T)2)S(T)2×(y-ea(T-t)Xt-σ(a,T-t)2b)×e-(1/2σ(2a,T-t)2)|y-ea(T-t)Xt-σ(a,T-t)2b|2,
where (6.3)S(T)=|y-x-σ(a,T)2(ax+b)|.Fort∈(0,T) and p>1 we set (6.4)βp(t)=ap(e2aT-1)-(p-1)(e2a(T-t)-1)|σ(a,T)2(b+ax)-d|,αp(t)=e2a(T-t)-1p(e2aT-1)-(p-1)(e2a(T-t)-1),
for simplicity.Lemma 6.1.
For anyp>1 one has
(6.5)1p|σ(a,T)2(b+ax)-d|≤e2aT-1aβp(t)≤|σ(a,T)2(b+ax)-d|,1pe2a(T-t)-1e2aT-1≤αp(t)≤e2a(T-t)-1e2aT-1,
for all t∈[0,T].Proof.
The two inequalities follow from the fact that(6.6)e2aT-1p(e2aT-1)-(p-1)(e2a(T-t)-1)
assumes its maximum 1 and minimum 1/p.Since(6.7)βp(t)αp(t)=ae2a(T-t)-1|σ(a,T)2(b+ax)-d|,e2a(T-t)-1e2aT-1β2p(t)2α2p(t)2=ae2aT-1|σ(a,T)2(b+ax)-d|2,
so that (6.8)∇zh(T-t,Xt,y)h(T,x,y)=σ(2a,T)σ(2a,T-t)3e((e2a(T-t)-1)/(e2aT-1))(β2p(t)2/α2p(t)2)+a(T-t)×(y-ea(T-t)Xt-σ(a,T-t)2b)×e-(1/2σ(2a,T-t)2)|y-ea(T-t)Xt-σ(a,T-t)2b|2,
which yield, together with (5.7), the following.Lemma 6.2.
One has(6.9)p(T,x,y)h(T,x,y)=1+∫0T2aea(T-t)e2a(T-t)-1e2aT-1e2a(T-t)-1e((e2a(T-t)-1)/(eaT-1))(β2p(t)2/α2p(t)2)×Px{UtC(Xt)K(t)}dt,
where
(6.10)K(t)=(y-ea(T-t)Xt-ea(T-t)-1ab)×e-(a/(e2a(T-t)-1))|y-ea(T-t)Xt-((ea(T-t)-1)/a)b|2.Let(6.11)J(p)t=Px(|K(t)|p)pD(p)t=Px(e(2p/(2p-1))∫0t|C|2(Xs)ds|C(Xt)|2p)2p.Lemma 6.3.
Chooseζ>0 such that
(6.12)Pexp⁡(ζsup⁡[0,1]|Wt|2)=c<+∞.
Then for any T>0 and λ>0, such that
(6.13)4λC0eaTe2aT-1a≤ζ,
one has
(6.14)Px(eλ∫0t|C|2(Xs)ds)≤c∀t≤T.Proof.
Let(6.15)Yt=Xt-x-σ(a,t)2(ax+b)=∫0tea(t-s)dWs.
Then
(6.16)|C|2(Xs)≤4C0|Xs-x|2≤4C0|Ys+σ(a,s)2(ax+b)|2≤4C0σ(a,s)4|(ax+b)|2+4C0C2|Ys|2,
and therefore
(6.17)Px(eλ∫0t|C|2(Xs)ds)≤e4λC0σ(a,t)4|(ax+b)|2tPx(e4λC0∫0t|Ys|2ds).
On the other hand
(6.18)Ys=eas∫0se-ardWr,
so that Ms≡e-asYs is a martingale with
(6.19)〈M〉s=∫0se-2ardr=e2as-12a.
Thus Ms is a time change of a standard Brownian motion, and
(6.20)Ms=B(e2as-1/2a),
for some standard Brownian motion (Bt)t≥0. Since
(6.21)2λC01eate2at-12a≤ζ,
so we have
(6.22)Px{e4λC0∫0teas|Ms|2ds}=Pexp⁡(4λC0∫0teas|B(e2as-1)/2a|2ds)=Pexp⁡(4λC0∫0(e2at-1)/2as2as+1|Bs|2ds)≤Pexp⁡(4λC01eat(e2at-12a)2sup⁡[0,((e2at-1)/2a)]|Bs|2)=Pexp⁡(4λC01eate2at-12asup⁡[0,1]|Bs|2)≤c.Corollary 6.4.
Forp>1 and  T>0 to be such that
(6.23)16pC02p-11eaTe2aT-12a≤ζ,
one has
(6.24)Px{e(4p/(2p-1))∫0t|C|2(Xs)ds}≤c∀t≤T.In what follows, we always assume thatT>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.
Letp>1. Then
(6.25)D(p)t≤C1(σ(2a,t)2+σ(a,t)2(ax+b)2),
where the positive constant C1 depends only on p, ζ, and C0.Proof.
Let(6.26)F(t)=Px(e(2p/(2p-1))∫0t|C|2(Xs)ds|C(Xt)|2p).
Then, by the Hölder inequality
(6.27)F(t)=Px(e(4p/(2p-1))∫0t|C|2(Xs)ds)Px(|C(Xt)|4p)≤C0Px(|C(Xt)|4p).
Next we estimate the expectation ℙx|C(Xt)|4p.  Since
(6.28)|C(Xt)|≤C02|Xt-x|2≤C0|Yt|2+C0σ(a,T)2(ax+b)2,
so that
(6.29)|C(Xt)|4p≤24p-1C04p|Yt|8p+24p-1C04pσ(a,T)8p(ax+b)8p.
On the other hand
(6.30)Px|Yt|8p=12πσ(2a,t)∫|z|8pexp⁡(-|z|2σ(2a,t)2)dz=σ(2a,t)8p,
so that
(6.31)Px|C(Xt)|4p4p≤2(4p-1)/4pC0σ(2a,t)2+2(4p-1)/4pC0σ(a,t)2(ax+b)2.Lemma 6.6.
LetT>0 satisfy condition (6.23),  x,y∈ℝ, and  p>1 and  q>1 such that  (1/p)+(1/q)=1. Then
(6.32)p(T,x,y)h(T,x,y)≤1+exp⁡(e2a(T-t)-1eaT-1β2p(t)2α2p(t)2)×∫0T2aea(T-t)e2a(T-t)-1e2aT-1e2a(T-t)-1D(p)tJ(2p)tdt.Proof.
Since(6.33)(Ute-((q-1)/2)∫0t|C|2(Xs)ds)q=exp⁡{∫0tqC(Xs)dWs-12q2∫0t|C|2(Xs)ds}
is a martingale under ℙx, so that
(6.34)Px(Ute-((q-1)/2)∫0t|C|2(Xs)ds)q=1.
By the Hölder inequality we deduce that
(6.35)|Px(UtC(Xt)K(t))|≤D(p)tJ(2p)t.
Equation (6.32), follows from the representation (6.9).Lemma 6.7.
Letp>1. Then
(6.36)J(p)t≤2p-1pαp(t)1+(1/p)e-βp2×{εppe2aT-e2a(T-t)2a+e2a(T-t)-1aβp(t)}.Proof.
We have(6.37)Px{|K(t)|p}≤Px{|y-ea(T-t)Xt-ea(T-t)-1ab|p×e-(pa/(e2a(T-t)-1))|y-ea(T-t)Xt-((ea(T-t)-1)/a)b|2}.
Under the probability ℙx,
(6.38)Zt≡ea(T-t)(Xt-eatx-eat-1ab)=∫0tea(T-s)dWs
is a central normal distribution with variance
(6.39)∫0te2a(T-s)ds=e2aT-e2a(T-t)2a.
In terms of Zt and d=y-x(6.40)Px{|K(t)|p}≤Px{|Zt+eaT-1a(b+ax)-d|p×e-(pa/(e2a(T-t)-1))|Zt+((eaT-1)/a)(b+ax)-d|2eaT-1a}.
Making change of variable
(6.41)N=2ae2aT-e2a(T-t)Zt.
Then, under ℙx, N has the standard normal distribution N(0,1), so that
(6.42)Px|K(t)|p≤P{|e2aT-e2a(T-t)2aN+eaT-1a(b+ax)-d|p×e-(pa/(e2a(T-t)-1))|((e2aT-e2a(T-t))/2a)N+((eaT-1)/a)(b+ax)-d|2|e2aT-e2a(T-t)2aN+eaT-1a(b+ax)-d|pe2aT-e2a(T-t)2a}=12π∫R{|e2aT-e2a(T-t)2az+eaT-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)|e2aT-e2a(T-t)2aN+eaT-1a(b+ax)-d|pe2aT-e2a(T-t)2a}dz.
Let us simplify the last integral. Indeed, set
(6.43)η=pe2aT-(p-1)e2a(T-t)-1e2a(T-t)-1z,Et=2pe2aT-e2a(T-t)2(e2a(T-t)-1)(eaT-1a(b+ax)-d)×ape2aT-(p-1)e2a(T-t)-1.
Then we rewrite the term appearing in the exponential in the last line of (6.42)
(6.44)-pae2a(T-t)-1|e2aT-e2a(T-t)2az+eaT-1a(b+ax)-d|2=-12(η2+2Etη)-pae2a(T-t)-1(eaT-1a(b+ax)-d)2=-12(η+Et)2+12Et2-pae2a(T-t)-1(eaT-1a(b+ax)-d)2,
together with
(6.45)12Et2-pae2a(T-t)-1(eaT-1a(b+ax)-d)2=-pape2aT-(p-1)e2a(T-t)-1(eaT-1a(b+ax)-d)2
the inequality (6.42) may be rewritten as follows:
(6.46)Px|K(t)|p≤12πe-(pa/(pe2aT-(p-1)e2a(T-t)-1))(((eaT-1)/a)(b+ax)-d)2×∫R|e2aT-e2a(T-t)2az+eaT-1a(b+ax)-d|p×e-(1/2)(η+Et)2dz.
Making change of variable in the last integral
(6.47)v=η+Et=pe2aT-(p-1)e2a(T-t)-1e2a(T-t)-1z+Et,
so that
(6.48)dz=e2a(T-t)-1pe2aT-(p-1)e2a(T-t)-1dv,e2aT-e2a(T-t)2az+eaT-1a(b+ax)-d=(e2a(T-t)-1)(e2aT-e2a(T-t))2a(pe2aT-(p-1)e2a(T-t)-1)v+e2a(T-t)-1pe2aT-(p-1)e2a(T-t)-1(eaT-1a(b+ax)-d).
Thus (6.46) yields that
(6.49)Px|K(t)|p≤(e2a(T-t)-12a)p+12ape2aT-(p-1)e2a(T-t)-1×e-(pa/(pe2aT-(p-1)e2a(T-t)-1))(((eaT-1)/a)(b+ax)-d)2Qt,
where
(6.50)Qt=∫R|e2aT-e2a(T-t)pe2aT-(p-1)e2a(T-t)-1z+2a(e2a(T-t)-1)(((eaT-1)/a)(b+ax)-d)pe2aT-(p-1)e2a(T-t)-1|pe-(1/2)z22πdz≤2p-1εp|e2aT-e2a(T-t)pe2aT-(p-1)e2a(T-t)-1|p+2p-1|2a(e2a(T-t)-1)(((eaT-1)/a)(b+ax)-d)pe2aT-(p-1)e2a(T-t)-1|pεp=12π∫R|z|pe-(1/2)z2dz.
Therefore
(6.51)J(p)t≤(e2a(T-t)-12a)1+(1/p)(2ape2aT-(p-1)e2a(T-t)-1)1/(2p)×e-(a/(pe2aT-(p-1)e2a(T-t)-1))(((eaT-1)/a)(b+ax)-d)2×{2p-1εpp|e2aT-e2a(T-t)pe2aT-(p-1)e2a(T-t)-1|+2p-1p|2(e2a(T-t)-1)pe2aT-(p-1)e2a(T-t)-1|a|((eaT-1)/a)(b+ax)-d|2pe2aT-(p-1)e2a(T-t)-1},
which is equivalent to the required inequality.Lemma 6.8.
Let(6.52)H(p)t≡exp⁡(e2a(T-t)-1eaT-1β2p(t)2α2p(t)2)J(2p)t.
Then
(6.53)p(T,x,y)h(T,x,y)≤1+∫0T2aea(T-t)e2a(T-t)-1e2aT-1e2a(T-t)-1D(p)tH(p)tdt,(6.54)H(p)t≤22p-12pα2p(t)1+(1/2p)e-(2p-1)((e2aT-e2a(T-t))2/(e2a(T-t)-1)(e2aT-1))β2p(t)2×{εppe2aT-e2a(T-t)2a+e2a(T-t)-1aβ2p(t)}.Proof.
Indeed, by Lemma6.7 we have
(6.55)H(p)t≤22p-12pα2p(t)1+(1/2p)e-β2p2+((e2a(T-t)-1)/(e2aT-1))(β2p(t)2/α2p(t)2)×{εppe2aT-e2a(T-t)2a+e2a(T-t)-1aβp(t)}.
On the other hand
(6.56)-β2p2+e2a(T-t)-1e2aT-1β2p(t)2α2p(t)2=(p-1)β2p(t)2e2αT-e2α(T-t)e2aT-1,
thus
(6.57)H(p)t≤22p-12pα2p(t)1+(1/2p)e(p-1)β2p(t)2((e2αT-e2α(T-t))/(e2aT-1))×{εppe2aT-e2a(T-t)2a+e2a(T-t)-1aβ2p(t)}.Let(6.58)B(p)t≡e(p-1)β2p(t)2((e2αT-e2α(T-t))/(e2aT-1))×{εppe2aT-e2a(T-t)2a+e2a(T-t)-1aβ2p(t)}.
Then (6.53) and (6.54) imply that (6.59)p(T,x,y)h(T,x,y)≤1+22p-12p(e2aT-1)1/4p∫0T2aea(T-t)(e2a(T-t)-1)1-(1/4p)D(p)tB(p)tdt.Lemma 6.9.
One has(6.60)B(p)t≤e(p-1)β2p(t)2((e2αT-e2α(T-t))/(e2aT-1))×{εppe2aT-e2a(T-t)2a+e2a(T-t)-1e2aT-1S(T)}.
In particular
(6.61)B(p)t≤εppe2aT-12a+S(T),p(T,x,y)h(T,x,y)≤1+22p-12p(e2aT-1)1/4p{εppe2aT-12a+S(T)}×∫0T2aea(T-t)e(p-1)β2p(t)2((e2αT-e2α(T-t))/(e2aT-1))(e2a(T-t)-1)1-(1/4p)D(p)tdt.Proof.
Let(6.62)G(p)t=2aea(T-t)e2a(T-t)-1e2aT-1e2a(T-t)-122p-12pα2p(t)1+(1/2p).
Then
(6.63)p(T,x,y)h(T,x,y)≤1+∫0TG(p)tD(p)tB(p)tdt.
Since
(6.64)G(p)t=2aea(T-t)(e2a(T-t)-1)1-(1/4p)(e2aT-1)1/4p22p-12p×(e2aT-12p(e2aT-1)-(2p-1)(e2a(T-t)-1))1+(1/2p)≤2aea(T-t)22p-12p(e2a(T-t)-1)1-(1/4p)(e2aT-1)1/4p,
which implies the required estimate.By collecting all estimates we have established, we may obtain the following.Proposition 6.10.
There is a constantC2>0 depending only on  ζ and  Cd such that
(6.65)p(T,x,y)h(T,x,y)≤1+C2e|a|Te(p-1)(a/(e2aT-1))S(T)2(eaT+12+(ax+b)2)×eaT-1a(εppe2aT-12a+S(T)),
where
(6.66)S(T)=|eaT-1a(b+ax)-(y-x)|.Proof.
Indeed(6.67)p(T,x,y)h(T,x,y)≤1+C1e(p-1)(a/(e2aT-1))S(T)2(e2aT-1a)-1/4p(εppe2aT-12a+S(T))×∫0Tea(T-t)(e2a(T-t)-1a)(1/4p)-1(σ(2a,t)2+σ(a,t)2(ax+b)2)dt≤1+Ce(p-1)(a/(e2aT-1))S(T)2(e2aT-1a)-1/4p(εppe2aT-12a+S(T))×(eaT+12+(ax+b)2)×eaT-1a∫0Tea(T-t)(e2a(T-t)-1a)(1/4p)-1dt.
While,
(6.68)∫0Tea(T-t)((e2a(T-t)-1)/2a)1-(1/4p)dt=∫0(e2aT-1)/2a1s1-(1/4p)11+2asds≤4pe|a|T(e2aT-12a)1/4p,
and it thus yields our key estimate (6.65).Similarly we have a lower bound(6.69)p(T,x,y)h(T,x,y)≥1-C3e|a|Te(p-1)(a/(e2aT-1))S(T)2(eaT+12+(ax+b)2)×eaT-1a(εppe2aT-12a+S(T)),whereC3 depends only on ζ and C0.


## 7. Proof of Theorem3.1

We are now in a position to prove Theorem3.1. We may assume that T=1, so that δ(n)=1/n. Let xjn (j=0,1,…,n) be discrete samplings with time scale δ=δ(n)=1/n on [0,1]. By our assumptions, |xjn-xj-1n|2≤Cδ(n), and |xjn|≤C for all pair (j,n) such that 0≤j≤n and n≥1. For simplicity we write xj for xjn if no confusion may arise.In the proof below, we will useCi 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 thathj(t,x,y) is the probability transition density function of the diffusion (3.3), that is, (7.1)dXt=(bj+ajXt)dt+dWt,
where (7.2)bj=A(xj-1,θ)-xj-1A′(xj-1,θ),aj=A′(xj-1,θ).According to (6.65) we have (7.3)p(δ,xj-1,xj)hj(δ,xj-1,xj)≤1+C8e|aj|δe((p-1)aj/(e2ajδ-1))Sj2(eajδ+12+(ajxj-1+bj)2)×eaj-1δ-1aj-1(εppe2aj-1δ-12aj-1+Sj),
where (7.4)Sj=|eajδ-1aj(bj+ajxj-1)-(xj-xj-1)|.
Since aj and xj are bounded, (7.5)|ajxj-1+bj|=|A(xj-1,θ)|≤C4(1+|xj-1|)≤C9,
so that(7.6)Sj≤eajδ-1ajδC9δ+Cδ≤C10δ.
Thus (7.7)(p-1)aje2ajδ-1Sj2=(p-1)ajδe2ajδ-1Sj2δ≤C11,eajδ+12+(ajxj-1+bj)2≤C12.
Therefore(7.8)p(δ,xj-1,xj)hj(δ,xj-1,xj)≤1+C13eaj-1δ-1aj-1δ(εppe2aj-1δ-12aj-1δδ+Sj)δ≤1+C14δδ=1+C141n3/2,
where we have used (7.6). It follows that (7.9)lim⁡δ→0∏j=1np(δ,xj-1,xj)hj(δ,xj-1,xj)≤lim⁡n→∞(1+C141n3/2)n=1.
Similarly we have (7.10)lim⁡n→∞∏j=1np(δ,xj-1,xj)hj(δ,xj-1,xj)≥1.
Therefore(7.11)lim⁡n→∞L(x0n,…,xnn)L2(x0n,…,xnn)=1,
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 (8.1)Xj=Xj-1+A(θ,Xj-1)δ+ξjδ,
where {ξj} is an i.i.d. random sequence, with standard normal N(0,1). The conditional distribution of Xj given Xj-1=xj-1 is Gaussian with mean xj-1+A(θ,xj-1)δ and variance δ so that the likelihood function is given as (8.2)L1(x0,…,xn)=1(2πδ)n/2∏j=1nexp⁡{-|xj-xj-1-A(θ,xj-1)δ|22πδ}.
Applying the representation formula (5.7) we have the following.Proposition 8.1.
It holds that(8.3)L(x0,…,xn)L1(x0,…,xn)=∏j(1-δeαj2/2δ∫0δP{Uj(s)cj(Xsj,θ)Ws+αj(δ-s)3/2e-(Ws+αj)2/2(δ-s)}ds),
where (Wt)t≥0 is the standard Brownian motion, Xsj=xj-1+λj-1s+Ws,
(8.4)Uj(t)=exp⁡[∫0tcj(Xsj,θ)dWs-12∫0t|cj|2(Xsj,θ)ds],λj=A(xj-1,θ),αj=λj-1δ-xj+xj-1, and
(8.5)cj(z,θ)=A(z,θ)-A(xj-1,θ).From which we may deduce the following estimate.Proposition 8.2.
IfA(x,θ) is bounded and Lipschitz continuous, uniformly in  θ, then the maximum likelihood function with discrete sampling is stable, in the sense that
(8.6)∏j(1-C01δeαj2/2δ)≤L(x0,…,xn)L1(x0,…,xn)≤∏j(1+C02δeαj2/2δ),
for some constant C01 and C02, where αj=A(xj-1,θ)δ-(xj-xj-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(8.7)lim⁡n→∞L(x0n,…,xnn)L1(x0n,…,xnn)=1
holds or not under similar conditions in Theorem 3.1.



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*Source: 906846-2011-09-27.xml*