A new discretization scheme for one dimensional stochastic differential equations using time change method

We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of an SDE using a time-changed Brownian motion, which dates back to Doeblin (1940). In cases where the diffusion coefﬁcient is bounded and is β -Hölder continuous with 0 < β ≤ 1 , we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat SDEs with no strong solution. Our scheme is the ﬁrst to achieve strong convergence for the case of 0 < β < 1 / 2


Introduction
In this article, we provide a numerical method of approximating a weak solution of a one dimensional stochastic differential equation. There are many studies about numerical approximation for SDEs which converges strongly to the solution. A variety of applications includes path-dependent option pricing in financial engineering. Here we focus on the following one-dimensional SDE; dX t =σ(t, X t )dW t .
(1.1) be reduced to (1.1); time homogeneous one-dimensional SDEs can be transformed to ones without drift term by scale function in a pathwise sense, and time inhomogeneous SDEs also can be transformed to martingales using the Girsanov-Maruyama transformation in a sense of law.
In order to study numerical scheme of SDEs (1.1), we must discuss the conditions where the existence and uniqueness of the solution hold in some different senses; strong uniqueness, pathwise uniqueness and uniqueness in the sense of probability law. Many researchers have been studied the unique existence of the solution to SDEs for a long time. The most famous condition for the strong unique existence of the solution to SDE is Lipschitz continuity and linear growth of the drift and diffusion coefficient (see [6]).
Bru and Yor discussed in [2] about this issue. According to [2], W. Doeblin has written a paper about this issue before the many facts about the structure of martingale were found. He showed that a diffusion process can be represented by some stochastic process which is driven by a time changed Brownian motion. Although this Doeblin's work in 1940 became public only after 2000, the idea was rediscovered and extended in stochastic calculus; already in a textbook [10] by Ikeda and Watanabe in 1984, it is shown that a one dimensional SDE of the form (1.1) in a certain class has a unique solution represented as a time changed Brownian motion, where the time change is given as a solution of a random ordinary differential equation, as we see in the next section in more detail. We use this representation to construct a new approximation scheme for one dimensional SDEs. For time homogeneous case, i.e., σ(t, x) = σ(x) in (1.1), Engelbert and Schmidt [3] gave an an equivalent condition for the weak existence and uniqueness in the sense of probability law, under which the weak solution is represented as the time-changed Brownian motion. For time homogeneous SDEs, an excellent survey [9] about the existence and uniqueness of SDEs is available.
The most famous numerical scheme for SDEs is the Euler-Maruyama Scheme. This method approximates a solution of SDEs in a very similar way to the Euler scheme for ordinary differential equations. It is well known that the Euler-Maruyama approximation converges to the strong solution of SDE uniformly in the sense of L p with convergence rate n −1/2 when the diffusion coefficient is Lipschitz continuous [7]. Under the β-Hölder continuity of the diffusion coefficient σ(t, x), where 1/2 ≤ β ≤ 1, Gyöngy and Rásonyi [5] showed that for any T > 0 there exists a constant C > 0 such that for any n ≥ 2, where X t is the strong solution of SDE(1.1) and X (n) t is its Euler-Maruyama approximation of step size 1/n. When β < 1/2, the existence of a strong solution is lost in general [1] and no numerical scheme is available so far.
In Section 2, we propose a new method of approximating SDE (1.1). In Section 3, we provide convergence rates of our method under the β-Hölder condition with 0 < β ≤ 1, and some smoothness condition of diffusion coefficient. An advantage of our approach is that we approximate the weak solution, which enable us to treat a SDE with no strong solution. Our scheme is the first to achieve the strong convergence for 0 < β < 1/2, and provides a better convergence rate than in [5] for 1/2 ≤ β < 2/3.

Discretization with time change
Let (Ω, F , P, {F t } t≥0 ) be a filtered probability space and {B t } be a {F t }-Brownian motion. Throughout this paper, we consider one-dimensional SDE(1.1) under the following condition.
Our method is based on the following theorem from [10].
satisfies Condition 1 and there exists a process of time change ϕ such that holds and if such a ϕ is unique, (i.e., ψ is another process of time change satisfying (2.1), then ϕ t ≡ ψ t a.s.), then the solution of (1.1) with initial value X 0 exists and is unique. Moreover, if we denote τ(t) := ϕ −1 (t), inverse function of t → ϕ(t), then the solution is given by X t = ξ τ(t) .

Remark 1. A sufficient condition for the ODE (2.1) to be well-posed is that σ(y, x) is locally Lipschitz continuous in y and satisfies the inequality
for all y ∈ [0, ∞) and x ∈ R, where a(x) and b(x) are some continuous non-negative functions of x; see [4]. In our setting, the local Lipschitz continuity of σ(t, x) in t is sufficient because the condition (2.2) follows from the boundedness of σ −2 (t, x).
The main goal of this paper is to build a numerical approximation of solution {X t } of the SDE (1.1) using Theorem 1. In order to approximate this timechanged Brownian motion, we will first make an approximation of Brownian motion {ξ t } by {ξ (n) t } which is a linear interpolation of a random walk generated by normal distributed random variables, that is ξ (n) where (ξ (⌊nt⌋+1)/n − ξ ⌊nt⌋/n ) ∼ N(0, 1/n). Second, we approximate {ϕ(t)} by ϕ n (t), Euler Method for ordinary differential equation,i.e., where t ∈ ϕ n ( k n ), ϕ n ( k+1 n ) . We can easily make sure that τ n (t) is inverse function of ϕ n (t) by its definition.
The concrete algorithm of this method is given as below.
STEP2 At the first instant ϕ n (t j ) passes over t, calculate τ n (t) using the formula Thus we can obtain a path of ξ . The main result of this paper is discretization error of {ξ (n) τ n (t) } in the sense of L p under Hölder condition of σ(t, s), which will be provided in next section.

Rate of convergence
In this section, we provide convergence rates of our approximation scheme. Theorem 2 declares that under the β-Hölder continuity of σ(t, x) with β ∈ (0, 1] our numerical approximation converges towards the exact solution in the sense of L p uniformly and the convergence rate is n −α 2 β , where α is an arbitrary value smaller than 1/2. Theorem 3 provides a more precise convergence rate n −α when σ is sufficiently smooth. Theorem 2. Let σ(t, x) satisfy Condition 1 and suppose that there exist constants C β > 0 and L T > 0 such that for s, t ≤ T, t , τ(t), τ n (t) be defined in the previous section. Then, for any T > 0, p ≥ 1 and α ∈ [0, 1/2), there exists a positive constantK T such that We will use the following lemma that is an immediate consequence of Theorem (2.1) in [8].

Lemma 2.
Let σ(t, x) satisfy Condition 1 and ϕ(t), ϕ n (t) be defined as (2.1) and (2.4). Then for each γ > 0 and T > 0, Proof. It follows from Condition 1 that for t ∈ [ k n , k+1 n ). Therefore, due to the continuity and the strictly increasing property of τ(t), τ n (t) and the bounded property of ϕ(t), ϕ n (t), we get Proof of Theorem 2. First, from Minkowski's inequality, we have where ⌊t⌋ is the largest integer less than t. Since ξ (n) t is the interpolation of the sequence {ξ j/n } j=0,1,2,··· , it follows that Therefore, using Minkowski's inequality again, we obtain Let us provide the desired conclusion by estimating convergence rate of (3.4)- Now we have the rate of convergence of the terms (3.4) and (3.5). It remains to prove that the convergence rate of the term (3.6) is n −α 2 β . From Lemma 1, Lemma 2 and Hölder's inequality, we have We provide the convergence rate of (3.9) by estimating the error function e n (t) := ϕ (n) (t) − ϕ(t). For positive number h, define a function ψ h : [0, ∞) → R as From Lemma 1 and the condition (3.1), for t ≤ T ′ − h and we obtain From Lemma 1, there is a random variable R depending on T ′ which has moments of any order and satisfies that On the other hand, from the definition of ϕ(t), and by the Lipschitz continuity of σ(t, x) over with respect to t, Repeating this calculus and using the fact that 1 Because of the integrable property of the random variable R and Cauchy-Schwartz's inequality, there exists a positive number K depending on T ′ such that Now we complete the proof Remark 2. We now have that our approximation converges to the solution of (1.1) and the rate of convergence is n −α 2 β . Let us compare our result with (1.2) by Gyöngy and Rásonyi [5]. When 1/2 < β < 2/3, it is easily seen that we can take a number α ∈ (0, 1/2) sufficient closed to 1/2 for β such that n −α 2 β < n −(β−1/2)/p . Therefore our method enjoys a better estimate of the convergence rate than that of the Euler-Maruyama scheme for β ∈ (1/2, 2/3). For β ∈ (0, 1/2], the convergence of the Euler-Maruyama approximation is not known. For β ≥ 2/3, (1.2) provides a better rate, while Theorem 3 below implies that the estimated rate in Theorem 2 is not sharp when σ is smooth.
We are going to provide a better estimate of the convergence rate of our scheme when σ is smooth. Denote by L q the class of stochastic process {X t } and q ∈ N such that and by σ t , σ x , σ x,x the partial derivatives of σ : Then for all T > 0, α ∈ (0, 1/2) and p ≥ 1, there exists some constant K T > 0 such that Proof of Theorem 3. Under the assumptions of this theorem, (3.4)-(3.9) remain to hold. Therefore, it only remains to estimate the convergence rate of (3.9). More precisely, it remains to prove that for α ∈ (0, 1/2) and T > 0 there exists a constant K T > 0 such that We will denote by C a generic constant which depends on p, α and T, and may change line by line. Let us write X t := σ −2 (ϕ(t), ξ t ). Since σ(t, x) belongs to C 2,2 , X t is a semimartingale and can be written as where (3.13) and {γ t } and {δ t } are in L q for any q ∈ N. Note that for t ≤ T ′ . Then by Gronwall's lemma, we get (3.19) Using by parts formula for tX t and sX s (s < t), and so we obtain Because of the fact that ⌊nt⌋+1 n ∧ τ − ⌊nt⌋+1 n = 0 for t < ⌊nτ⌋ n , the definition X t = σ −2 (ϕ(t), ξ t ) and Condition 1, we can estimate the second term in (3.21) as follows.
To estimate the first term, we recall the notation (3.13) and obtain Recalling Remark 3, the L 2pα∨1 property of δ t implies that (3.24) Here we have used that On the other hand, using the Burkholder-Davis-Gundy inequality and the L 2pα property of γ t , we obtain that for the second term of (3.23), (3.25) From (3.21) and (3.22)-(3.25), it follows that which concludes the proof.