On the Analysis of Ait-Sahalia-Type Model for Rough Volatility Modelling

Abstract

Fractional Brownian motion with Hurst parameter $H<\frac{1}{2}$ is used widely, for instance, to describe ‘rough’ volatility data in finance. In this paper, we examine a generalised Ait-Sahalia-type model driven by a fractional Brownian motion with $H<\frac{1}{2}$ and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.

1 Introduction

Over the years, SDEs driven by noise with $\alpha ^-$-Hölder continuous random paths for $\alpha \in [\frac{1}{2},1)$ have been applied to model the dynamical behaviour of volatility of asset prices in finance. See, for example, [1,2,3] and the references therein. However, in recent years, empirical evidence (see e.g. [4]) has shown that volatility paths of asset prices are more irregular in the sense of $\alpha ^-$-Hölder continuity for $\alpha \in (0,\frac{1}{2})$ in many instances. This inadequacy actually showed the need for models based on SDEs driven by a noise of low $\alpha ^-$-Hölder regularity with $\alpha \in (0,\frac{1}{2})$ which has been used by researchers and practitioners to describe the volatility dynamics of asset prices. These models are driven by rough signals that can capture well the ‘roughness’ in the volatility process of asset prices. Such rough signals arise, for example, from paths of the fractional Brownian motion (fBm). The fractional Brownian motion is a generalisation of the ordinary Brownian motion. It is a centred self-similar Gaussian process with stationary increments which depends on the Hurst parameter H. The Hurst parameter lies in (0, 1) and controls the regularity of the sample paths in the sense of a.e. (local) $H^-$-Hölder continuity. The smaller the Hurst parameter, the rougher the sample paths and vice versa. For instance, the authors in [5] employ the fractional Brownian motion with $H<\frac{1}{2}$ to model the ‘rough’ volatility process of asset prices and derive a representation of the sensitivity parameter delta for option prices. Similarly, the authors in [6] also consider an asset price model in connection with the sensitivity analysis of option prices whose correlated ‘rough’ volatility dynamics is described by means of an SDE driven by a fractional Brownian motion with $H<\frac{1}{2}$. The reader may consult [7, 8] for the coverage of properties and financial applications of the fractional Brownian motion with $H<\frac{1}{2}$ (see also Appendix).

In the context of interest rate modelling, Ait-Sahalia proposed a new class of highly nonlinear stochastic models in [9] for the evolution of interest rates through time after rejecting existing univariate linear-drift stochastic models based on empirical studies. In this model, (short-term) interest rates $x_t$ have the SDE dynamics

$$\begin{aligned} \textrm{d}x_t=\left( \alpha _{-1}x_t^{-1}-\alpha _{0}+\alpha _{1}x_t-\alpha _{2}x_t^{2}\right) \textrm{d}t+\sigma x_t^{\theta }\textrm{d}B_t \end{aligned}$$

(1)

on $t\ge 0$ with initial value $x_0$, where $\alpha _{-1},\alpha _{0},\alpha _{1}, \alpha _{2}>0$, $\sigma >0$, $\theta >1$ and $B_t$ is a scalar Brownian motion. SDE (1) has been studied by many authors (see e.g. [10, 11]). Besides interest rate modelling, SDE (1) has also been used extensively among academic researchers and market practitioners to describe stochastic volatility and asset price dynamics. For example, in stochastic volatility modelling, the stock price process $S_t$, $t\ge 0$, may be modelled by the Black–Scholes SDE

$$\begin{aligned} \textrm{d}S_t=\mu S_t\textrm{d}t+\sigma _t S_t\textrm{d}B_t, \quad t\ge 0, \end{aligned}$$

(2)

where $\mu \in {\mathbb {R}}$ is the mean return and $\sigma _t>0$, $t\ge 0$, is the volatility process described by the SDE (1). Generally, there are several classes of SDE (1) with parametric restriction. For example, Black–Scholes, Vasicek, Dothan, CIR and CEV models fall under SDE (1).

In the context of ‘rough’ stochastic volatility modelling, we note that SDE (1) may not provide a good fit since the driving noise is a Brownian motion $B_{\bullet }$. In this case, we recognise the need to replace the driving noise $B_{\bullet }$ with a fractional Brownian motion $B^H_{\bullet }$ and consider a ‘rough’ volatility model based on the SDE

$$\begin{aligned} \textrm{d}x_t=\left( \alpha _{-1}x_t^{-1}-\alpha _{0}+\alpha _{1}x_t-\alpha _{2}x_t^{\rho }\right) \textrm{d}t+\sigma x_t^{\theta }d^{\circ }B_t^H \end{aligned}$$

(3)

for $t\ge 0$ and $H\in (0,\frac{1}{2})$, where $\sigma x_t^{\theta }d^{\circ }B_t^H$ stands for a stochastic integral in the sense of Russo and Vallois (see Sect. 5). However, since the expected value of $\sigma x_t^{\theta }d^{\circ }B_t^H$ (if it exists) is not zero, in general, we observe that the SDE (3) does not necessarily yield the Ornstein–Uhlenbeck dynamics as a special case. In other words, SDE (3) may not be used to capture the mean reversion property, which plays an important role in finance. In order to account for the mean reversion property of SDE (3), we may consider instead the following SDE

$$\begin{aligned} \textrm{d}x_t=\left( \alpha _{-1}x_t^{-1}-\alpha _{0}+\alpha _{1}x_t-\alpha _{2}t^{2H-1}x_t^{\rho }\right) \textrm{d}t+\sigma x_t^{\theta }\textrm{d}B_t^H \end{aligned}$$

(4)

for $t\ge 0$ with initial value $x_0$, $t\in (0,1]$, $H\in (0,\frac{1}{2})$ and $\rho >1$. The stochastic integral for the fractional Brownian motion in (4) is defined via an integral concept in [7] and related to a Wick–Itô–Skorohod type of integral (see also Sect. 5). We mention that the mean of the stochastic integral in (4) is zero (provided that the mean exists). Therefore, one obtains from SDE (4) the Ornstein–Uhlenbeck dynamics as a special case if one formally chooses $\alpha _{-1}$, $\alpha _{2}$ and $\theta$ to be zero.

As mentioned before, the original Ait-Sahalia model has been applied to interest rate modelling. However, the Ait-Sahalia model in our setting, (3) and (4) cannot be employed in interest rate modelling since empirical evidence shows that interest rate paths rather exhibit Hölder continuity with an index bigger than $\frac{1}{2}$ (see [12]). This is also the reason why we in this paper apply the extended Ait-Sahalia model to ‘rough’ volatility modelling.

Although we also prove an existence and uniqueness result for solutions to SDE (3) (see Theorem 5.5), we mainly focus in this paper on the study of SDE (4). We emphasise that our mathematical methods employed in this paper differ significantly from those used in [13]. For example, in the case of $H<\frac{1}{2}$, we cannot apply the Itô Lemma as in [13] for $H>\frac{1}{2}$, to prove the existence of higher moments of solutions to SDE (3) or (4) (see Sect. 4) but have to resort to other techniques based on, for example, the Clark–Ocone formula and the concept of rough path integrals in the sense of Russo–Vallois.

Finally, we mention some other works related to our article: Let us point out here that SDEs with explosive drifts driven by Hölder continuous noises in the case of fBm with $H>\frac{1}{2}$ were initially analysed in Hu et al. [3], where the authors address the properties of positivity, existence of moments and the Malliavin differentiability of strong solutions. Other interesting and more recent results related to the SDE (8) can be found in Di Nunno et al. [14], who establish for a large class of unbounded and explosive drift vector fields existence and uniqueness of local and global solutions to SDEs with additive noise, which is merely Hölder continuous and not necessarily Gaussian. Further, the work of Di Nunno et al. [15], which appeared after the completion of our article, also deals with the Malliavin differentiability of solutions with general Gaussian Volterra drivers. In addition, we refer to Kubilius [16] and Kubilius and Medziunas [17], where the Ait-Sahalia model for a parameter $\theta$ is less than 1 and greater than 1 is investigated as an example, when $H>\frac{1}{2}$. We remark here that SDEs of the type (3) or (4), which involve ‘rough path’ integrals in the sense of Russo and Vallois, were not studied in the above-mentioned works.

The remainder of the paper is organised as follows: In Sect. 2, we introduce the fractional Ait-Sahalia-type model for rough volatility modelling. We establish an existence and uniqueness result for solutions to SDE (4) in Sect. 5 by studying the properties of solutions to an associated SDE driven by an additive fractional noise (see Sects. 3 and 4). In addition, we also discuss the alternative model (3) in Sect. 5.

2 The Fractional Ait-Sahalia Model

Throughout this paper unless specified otherwise, we employ the following notation. Let $(\Omega , {\mathcal {F}},{\mathbb {P}})$ be a complete probability space with filtration $\{ {\mathcal {F}}_t\}_{t\ge 0}$ satisfying the usual conditions (i.e. it is increasing and right continuous while ${\mathcal {F}}_0$ contains all ${\mathbb {P}}$ null sets). Denote ${\mathbb {E}}$ as the expectation corresponding to ${\mathbb {P}}$. Suppose that $B^H_t$, $0\le t\le 1$, is a scalar fractional Brownian motion (fBm) with Hurst parameter $H\in (0,\frac{1}{2})$ and $B_t$, $0\le t\le 1$, is a scalar Brownian motion defined on this probability space.

In what follows, we are interested to study the SDE

$$\begin{aligned} x_t=x_0+\int _0^t\big (\alpha _{-1}x_s^{-1}-\alpha _0+\alpha _1x_s-\alpha _2s^{2H-1}x_s^{\rho }\big )\textrm{d}s+\int _0^t\sigma x^{\theta }_s\textrm{d}B_s^H, \end{aligned}$$

(5)

$x_0\in (0,\infty )$, $0\le t\le 1$, where $H\in (\frac{1}{3},\frac{1}{2})$, $\tilde{\theta }>0$, $\rho >1+\frac{1}{H\tilde{\theta }}$, $\theta :=\frac{\tilde{\theta }+1}{\tilde{\theta }}$, $\sigma >0$ and $\alpha _i>0$, $i=-1,\ldots ,2$. Here, the stochastic integral term with respect to $B_{\bullet }^H$ in (5) is defined by means of an integral concept introduced by Russo and Vallois [18]. See Sect. 5.

As already mentioned in introduction, solutions to the SDE (5) can be used as a model (fractional Ait-Sahalia model) for the description of the dynamics of (rough) volatility in finance. In fact, in this paper, we aim at establishing the existence and uniqueness of strong solutions $x_t>0$ to SDE (5). In doing so, we show that such solutions can be obtained as transformations of solutions to the SDE

$$\begin{aligned} y_t=x+\int _0^t {\tilde{f}}(s,y_s)\textrm{d}s-\tilde{\sigma } B_t^H, \quad 0\le t\le 1, \quad H\in \left( 0,\frac{1}{2}\right) , \end{aligned}$$

(6)

where

$$\begin{aligned} {\tilde{f}}(s,y)&=\alpha _{-1}\left( -\tilde{\theta } y^{2\tilde{\theta }+1}\right) +\alpha _0y^{\tilde{\theta }+1}-\alpha _1\frac{y}{\tilde{\theta }}\nonumber \\&\quad +\alpha _2 s^{2H-1}\frac{1}{\tilde{\theta }^{\rho }} y^{-\tilde{\theta }\rho +\tilde{\theta }+1}-\tilde{\sigma }Hs^{2H-1}y^{-1}(\tilde{\theta }+1), \end{aligned}$$

(7)

where $\tilde{\sigma }>0$, $0< s\le 1$, $0<y<\infty$. However, after having applied the transformation, we have to restrict $H\in (1/3,1/2)$ to make sense of the stochastic integral in SDE (5). See Sect. 5 for further details.

In the sequel, we want to prove the following new properties for solutions to SDE (6):

• Existence and uniqueness of positive strong solutions (Corollary 3.1),

• Regularity of solutions in the sense of Malliavin differentiability (Theorem 4.2),

• Existence of higher moments (Theorem 4.1).

3 Existence and Uniqueness of Solutions to Singular SDEs with Additive Fractional Noise for $H< \frac{1}{2}$

In this section, we wish to analyse the following generalisation of the SDE (6) given by

$$\begin{aligned} x_t=x_0+\int ^t_0b(s,x_s)ds+\sigma B^H_t,\quad 0\le t\le 1,\quad H\in \left( 0,\frac{1}{2}\right) ,\quad \sigma >0. \end{aligned}$$

(8)

We require the following conditions

• (A1) $b\in C\big ((0,1)\times (0,\infty )\big )$ and has a continuous spatial derivative $b^{\prime }:=\frac{\partial }{\partial x}b$ such that

  $$\begin{aligned} b'(t,x)\le K_t, \quad 0< t< 1, \quad x\in (0,\infty ), \end{aligned}$$

  where $K_t:=t^{2H-1}K$ for some $K\ge 0$.

• (A2) There exist $\kappa _1>0$, $\alpha >\frac{1}{H}-1$ and $h_1>0$ such that $b(t,x)\ge h_1t^{2H-1}x^{-\alpha }$, $t\in (0,1]$, $x\le \kappa _1$.

• (A3) There are $\kappa _2>0$ and $h_2>0$ such that $b(t,x)\le h_2t^{2H-1}(x+1)$, $t\in (0,1]$, $x\ge \kappa _2$.

Theorem 3.1

Suppose that (A1–A3) hold. Then, for all $x_0>0$ the SDE (8) has a unique strong positive solution $x_t$, $0\le t\le 1$.

Proof

Without loss of generality, let $\sigma =1$. We are required to establish the following analytical properties.

• (i) Uniqueness: Suppose $x_{\bullet }$ and $y_{\bullet }$ are two solutions to (8). Then,

  $$\begin{aligned} x_t-y_t=\int _0^t\big (b(s,x_s)-b(s,y_s)\big )\textrm{d}s. \end{aligned}$$

  So, using the product rule, the mean value theorem and (A1), we get

  $$\begin{aligned} (x_t-y_t)^2&=2\int _0^t\big (b(s,x_s)-b(s,y_s)\big )(x_s-y_s)\textrm{d}s\\&\le 2\int _0^tK_s(x_s-y_s)^2\textrm{d}s. \end{aligned}$$

  Hence, Gronwall’s lemma implies that

  $$\begin{aligned} x_t-y_t=0,\quad 0\le t\le 1. \end{aligned}$$

• (ii) Existence: Let $x_0>0$. Because of the regularity assumptions imposed on b, we know that Eq. (8) has (path-by-path) local solutions. Define the stopping times

  $$\begin{aligned} \tau _0:=\inf \{t\in [0,1]:x_t=0\}\quad \text {and}\quad \tau _n:=\inf \{t\in [0,1]:x_t\ge n\}, \end{aligned}$$

  where $\inf \emptyset :=1^+$. Just as in [13], we want to prove that $\tau _0=1^+$ and $\lim _{n\rightarrow \infty }\tau _n=1^+$. Here, $1^+$ stands for an artificially added element larger than 1. Suppose that $\tau _0\le 1$. Then, there is a $\hat{\tau }_0\in (0,\tau _0]$ such that $x_t\le \kappa _1$ for all $(\hat{\tau }_0,\tau _0]$. By (A2), we know that $b(t,x)>0$ for $x\in (0,\kappa _1)$ and $t>0$. Hence,

  $$\begin{aligned} 0=x_{\tau _0}=x_t+\int _t^{\tau _0}b(s,x_s)\textrm{d}s+B^H_{\tau _0}-B^H_t,\quad t\in (\hat{\tau }_0,\tau _0]. \end{aligned}$$

  (9)

  This implies

  $$\begin{aligned} x_t\le \left| B^H_{\tau _0}-B^H_t\right| \le \vert \vert B_{\bullet }^H\vert \vert _{\beta }(\tau _0-t)^{\beta },\ t\in (\hat{\tau }_0,\tau _0] \text { for } \beta \in (0,H). \end{aligned}$$

  (10)

  Here, $\vert \vert \cdot \vert \vert _{\beta }$ denotes the Hölder-seminorm given by

  $$\begin{aligned} \vert \vert f\vert \vert _{\beta }=\sup _{0\le s<t\le 1}\frac{\vert f(s)-f(t)\vert }{(t-s)^{\beta }} \end{aligned}$$

  for $\beta$-Hölder continuous functions f. So, we also obtain that

  $$\begin{aligned} \vert \vert B_{\bullet }^H\vert \vert _{\beta }(\tau _0-t)^{\beta }&\ge \left| B^H_{\tau _0}-B^H_t\right| \ge \int ^{\tau _0}_tb(s,x_s)\textrm{d}s\\&\ge h_1\int ^{\tau _0}_t s^{2H-1}x_s^{-\alpha }\textrm{d}s\ge \frac{h_1}{\vert \vert B_{\bullet }^H\vert \vert _{\beta }^{\alpha }}\int ^{\tau _0}_t s^{2H-1}\frac{1}{(\tau _0-s)^{\alpha \beta }}\textrm{d}s\\&\ge \frac{h_1}{\vert \vert B_{\bullet }^H\vert \vert _{\beta }^{\alpha }}\tau _0^{2H-1}\int ^{\tau _0}_t \frac{1}{(\tau _0-s)^{\alpha \beta }}\textrm{d}s. \end{aligned}$$

If $\alpha \beta \ge 1$, we get a contradiction. For $\alpha \beta < 1$, we find that

$$\begin{aligned} \vert \vert B_{\bullet }^H\vert \vert _{\beta }(\tau _0-t)^{\beta }&\ge \frac{h_1}{\vert \vert B_{\bullet }^H\vert \vert _{\beta }^{\alpha }}\tau _0^{2H-1} \frac{(\tau _0-t)^{1-\alpha \beta }}{1-\alpha \beta },\quad t\in (\hat{\tau }_0,\tau _0]. \end{aligned}$$

Hence,

$$\begin{aligned} 0=\lim _{t\rightarrow \tau _0}\vert \vert B_{\bullet }^H\vert \vert _{\beta }(\tau _0-t)^{\beta +\alpha \beta -1}\ge \frac{h_1\tau _0^{2H-1}}{\vert \vert B_{\bullet }^H\vert \vert _{\beta }^{\alpha }(1-\alpha \beta )}>0. \end{aligned}$$

So, $\tau _0=1^+$. Assume now that

$$\begin{aligned} \tau _{\infty }:=\lim _{n\rightarrow \infty }\tau _n\le 1. \end{aligned}$$

Then (compare [13]), we can distinguish between the following two cases:

• 1. Case: There is $\widehat{\tau }_1$ such that $x_{\widehat{\tau }_1}=\kappa _2+x_0$ and $x_t\ge \kappa _2+x_0$ for all $t\in (\widehat{\tau }_1,\tau _{\infty })$.

• 2. Case: For all $n\in {\mathbb {N}}$ with $n>\kappa _2+x_0$ and $\epsilon >0$, one can find an interval $(\widehat{\tau }_1,\widehat{\tau }_2)\subset (\tau _{\infty }-\epsilon ,\tau _{\infty })$ such that $x_{\widehat{\tau }_1}=\kappa _2+x_0$ and

  $$\begin{aligned} \kappa _2+x_0\le \inf _{t\in (\widehat{\tau }_1,\widehat{\tau }_2)}x_t\le n\le \sup _{t\in (\widehat{\tau }_1,\widehat{\tau }_2)}x_t. \end{aligned}$$

Thus, by using (A3), we obtain that

$$\begin{aligned} x_t\le \kappa _2+x_0+\vert \vert B_{\bullet }^H\vert \vert _{\beta }\tau ^{\beta }_{\infty }+h_2\tau ^{2H}_{\infty }(2H)^{-1}+h_2\int _{\hat{\tau }_1}^t s^{2H-1}x_s\textrm{d}s. \end{aligned}$$

So, by letting

$$\begin{aligned} \alpha = \kappa _2+x_0+\vert \vert B_{\bullet }^H\vert \vert _{\beta }\tau ^{\beta }_{\infty }+h_2\tau ^{2H}_{\infty }(2H)^{-1}, \end{aligned}$$

it follows from Gronwall’s lemma that

$$\begin{aligned} x_t&\le \alpha +\int _{\hat{\tau }_1}^t \alpha h_2s^{2H-1}\exp \left( \int _s^th_2u^{2H-1}\textrm{d}u\right) \textrm{d}s\\&\le \gamma +\int _0^1 \gamma h_2s^{2H-1}\exp \left( \int _s^1 h_2u^{2H-1}\textrm{d}u\right) \textrm{d}s, \end{aligned}$$

where $\gamma :=\kappa _2+x_0+\vert \vert B_{\bullet }^H\vert \vert _{\beta }+\frac{h_2}{2H}$. The latter estimate leads to a contradiction.

$\square$

As a consequence of Theorem 3.1, we obtain the following result:

Corollary 3.2

Suppose that $x\in (0,\infty )$ and $\rho >\frac{1}{H\tilde{\theta }}+1$, where $\rho$ and $\tilde{\theta }$ are parameters of ${\tilde{f}}$ in (7). Then, there exists a unique strong solution $y_t>0$ to SDE (6).

Proof

Let $\epsilon =\frac{H}{2}$. Then,

$$\begin{aligned} {\tilde{f}}(s,y)={\tilde{g}}_1(s,y)+{\tilde{g}}_2(s,y), \end{aligned}$$

where

$$\begin{aligned} {\tilde{g}}_1(s,y):=\alpha _{-1}\left( -\tilde{\theta } y^{2\tilde{\theta }+1}\right) +\alpha _0y^{\tilde{\theta }+1}-\alpha _1\frac{y}{\tilde{\theta }}+\epsilon \tilde{\sigma }s^{2H-1}y^{-1}(\tilde{\theta }+1), \end{aligned}$$

and

$$\begin{aligned} {\tilde{g}}_2(s,y):=\alpha _2 s^{2H-1}\frac{1}{\tilde{\theta }^{\rho }} y^{-\tilde{\theta }\rho +\tilde{\theta }+1}-(H+\epsilon )\tilde{\sigma }s^{2H-1}y^{-1}(\tilde{\theta }+1). \end{aligned}$$

We see that

$$\begin{aligned} {\tilde{g}}_1(s,y)&\ge \alpha _{-1}(-\tilde{\theta } y^{2\tilde{\theta }+1})+\alpha _0y^{\tilde{\theta }+1}-\alpha _1\frac{y}{\tilde{\theta }}+\epsilon \tilde{\sigma }y^{-1}(\tilde{\theta }+1)\\&\ge 0 \end{aligned}$$

for all $s\in (0,1]$ and $y\in (0,y_0)$ for some $y_0>0$. Since

$$\begin{aligned} -\tilde{\theta }\rho +\tilde{\theta }+1<-\frac{1}{H}+1<-1, \end{aligned}$$

we also find some $y_1>0$ such that

$$\begin{aligned} {\tilde{g}}_2(s,y)&=s^{2H-1} y^{-\tilde{\theta }\rho +\tilde{\theta }+1}\left( \alpha _2\frac{1}{\tilde{\theta }^{\rho }}-(H+\epsilon )\tilde{\sigma }(\tilde{\theta }+1)y^{\tilde{\theta }\rho -\tilde{\theta }-2}\right) \\&\ge h_1s^{2H-1}y^{-\alpha } \end{aligned}$$

for all $s\in (0,1]$ and $y\in (0,y_1]$, where $h_1>0$ and $\alpha :=\tilde{\theta }\rho -\tilde{\theta }-1$. So,

$$\begin{aligned} {\tilde{f}}(s,y)\ge h_1s^{2H-1}y^{-\alpha } \end{aligned}$$

for all $s\in (0,1]$, $y\in (0,y_1)$ for some $y_1>0$, which shows that ${\tilde{f}}$ satisfies (A2). As for (A3), we see that there exists some $y_2\ge 1$ such that

$$\begin{aligned} {\tilde{f}}(s,y)&\le s^{2H-1}\left( \alpha _2\frac{1}{\tilde{\theta }^{\rho }}y^{-\tilde{\theta }\rho +\tilde{\theta }+1}-H\tilde{\sigma }y^{-1}(\tilde{\theta }+1)\right) \le h_2s^{2H-1}(1+y) \end{aligned}$$

for all $s\in (0,1]$, $y\in (y_2,\infty )$ and some $h_2>0$. We have that

$$\begin{aligned} {\tilde{f}}^{\prime }(s,y)=f_1(s,y)+f_2(s,y), \end{aligned}$$

where

$$\begin{aligned} f_1(s,y):=-\alpha _{-1}\tilde{\theta }(2\tilde{\theta }+1)y^{2\tilde{\theta }}+\alpha _0(\tilde{\theta }+1)y^{\tilde{\theta }}-\frac{\alpha _1}{\tilde{\theta }} \end{aligned}$$

and

$$\begin{aligned} f_2(s,y):= s^{2H-1}\left( \alpha _2\frac{1}{\tilde{\theta }^{\rho }}(-\tilde{\theta }\rho +\tilde{\theta }+1)y^{-\tilde{\theta }\rho +\tilde{\theta }}+H\tilde{\sigma }(\tilde{\theta }+1)y^{-2}\right) . \end{aligned}$$

So, there exist $y_1,y_2>0$ such that

$$\begin{aligned} {\tilde{f}}'(s,y)\le f_1(s,y)\le K\le s^{2H-1}K=K_s \end{aligned}$$

for all $s\in (0,1]$, $y\in (0,y_1)$ as well as

$$\begin{aligned} {\tilde{f}}'(s,y)\le f_2(s,y)\le s^{2H-1}K=K_s \end{aligned}$$

for all $s\in (0,1]$, $y\in (y_2,\infty )$ and some $K>0$. On the other hand, we see that

$$\begin{aligned} {\tilde{f}}'(s,y)\le K_1+s^{2H-1}K_2\le s^{2H-1}K=K_s \end{aligned}$$

for all $s\in (0,1]$, $y_0\in [y_1,y_2]$ for some $K_1,K_2,K>0$. Altogether, we see that ${\tilde{f}}$ also satisfies (A1). Since $-B^H_{\bullet }$ is a fractional Brownian motion, the proof follows.$\square$

4 Malliavin Differentiability and Existence of Higher Moments of Solutions

In this section, we want to show that the solution x to the SDE

$$\begin{aligned} x_t=x+\int _0^t {\tilde{f}}(s,x_s)ds-\tilde{\sigma }B_t^H, \quad 0\le t\le 1,\quad x>0, \end{aligned}$$

(11)

is Malliavin differentiable in the direction of $B_{\bullet }^H$ for $H\in (0,\frac{1}{2})$ and where $\tilde{\sigma }>0$ is an arbitrary constant. Furthermore, we verify that solutions $x_t$ to (11) belong to $L^q$ for all $q\ge 1$. For this purpose, let ${\tilde{f}}_n:(0,1]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, $n\ge 1$ be a sequence of bounded, globally Lipschitz continuous (and smooth) functions such that

• (i) ${\tilde{f}}\mid _{[\frac{1}{n},n]}={\tilde{f}}\mid _{(0,1]\times [\frac{1}{n},n]}$ for all $n\ge 1$,

• (ii) ${\tilde{f}}'_n(s,x)\le K_s$ for all $(s,x)\in (0,1]\times {\mathbb {R}}$, $n\ge 1$, where $K_s$ is defined in (A1).

So, we see that

$$\begin{aligned} {\tilde{f}}'_n(s,x)\underset{n\rightarrow \infty }{\longrightarrow }{\tilde{f}}(s,x) \end{aligned}$$

for all $(s,x)\in (0,1]\times (0,\infty )$. Denote by $D^H_{\bullet }$ and $D_{\bullet }$, the Malliavin derivative in the direction of $B^H_{\bullet }$ and $W_{\bullet }$, respectively. Here, $W_{\bullet }$ is the Wiener process with respect to the representation

$$\begin{aligned} B^H_t=\int ^t_0K_H(t,s)\textrm{d}W_s,\quad t\ge 0. \end{aligned}$$

(12)

See Appendix. Since $-B^H_{\bullet }$ is a fractional Brownian motion, let us without loss of generality assume in (11) that $\tilde{\sigma }=-1$. Because of the regularity of the functions ${\tilde{f}}_n$, $n\ge 1$, we find that the solutions $x^n_{\bullet }$ to

$$\begin{aligned} x^n_t=x+\int _0^t {\tilde{f}}_n(s,x_s)\textrm{d}s+B_t^H, \quad x>0, \quad 0\le t\le 1 \end{aligned}$$

are Malliavin differentiable with Malliavin derivative $D_u^Hx_t$ satisfying the equation

$$\begin{aligned} D_u^Hx_t^n=\int ^t_u {\tilde{f}}^{\prime }_n(s,x^n_s)D_u^Hx_s^n\textrm{d}s+\chi _{[0,t]}(u). \end{aligned}$$

Hence,

$$\begin{aligned} D_u^Hx^n_t=\chi _{[0,t]}(u)\exp \left( \int _u^t{\tilde{f}}_n^{\prime }(s,x_s^n )\textrm{d}s \right) \quad \lambda \times \text {P-a.e.} \end{aligned}$$

for all $0\le t\le 1$ ( $\lambda$ Lebesgue measure). Further, using the transfer principle between $D^H_{\bullet }$ and $D_{\bullet }$ (see [5, Proposition 5.2.1]), we have that

$$\begin{aligned} K_H^*D^H_{\bullet }x_t=D_{\bullet }x_t \end{aligned}$$

(13)

where $K_H^*:{\mathcal {H}}\rightarrow L^2([0,T])$ is given by

$$\begin{aligned} (K_H^*y)(s)=K_H(T,s)y(s)+\int _s^T(y(t)-y(s))\frac{\partial }{\partial t}K_H(t,s)\textrm{d}t \end{aligned}$$

(14)

for

$$\begin{aligned} \frac{\partial }{\partial t}K_H(t,s)=c_H\left( H-\frac{1}{2}\right) \left( \frac{1}{2}\right) ^{H-\frac{1}{2}}\left( t-s\right) ^{H-\frac{3}{2}}. \end{aligned}$$

(15)

Here ${\mathcal {H}}=I_{T^-}^{\frac{1}{2}-H}(L^2)$. See Appendix. On the other hand, using (13), we also see that

$$\begin{aligned} D_ux^n_t=\int _u^t {\tilde{f}}'_n\left( s,x_s^n\right) D_ux_s^n\textrm{d}s+K_H(t,u) \end{aligned}$$

(16)

in $L^2([0,t]\times \Omega )$ for all $0\le t\le 1$. Set

$$\begin{aligned} Y_t^n(u)=D_ux_t^n-K_H(t,u). \end{aligned}$$

Then,

$$\begin{aligned} Y_t^n(u)=\int _u^t\left\{ {\tilde{f}}'_n\left( s,x_s^n\right) Y^n_s(u)+{\tilde{f}}_n\left( s,x_s^n\right) K_H(s,u) \right\} \textrm{d}s. \end{aligned}$$

Using the fundamental solution of the equation

$$\begin{aligned} \dot{\Phi }(t)={\tilde{f}}'_n(t,x_t^n)\cdot \Phi (t),\quad \Phi (u)=1. \end{aligned}$$

We then obtain that

$$\begin{aligned} Y_t^n(u)=\int _u^t\exp \left( \int _s^t{\tilde{f}}'\left( r,x_r^n\right) \textrm{d}r\right) {\tilde{f}}'_n(s,x_s^n)K_H(s,u)\textrm{d}s. \end{aligned}$$

Hence,

$$\begin{aligned} D_ux_t^n&=\int ^t_u \exp \left( \int _s^t{\tilde{f}}'\left( r,x_r^n\right) \textrm{d}r\right) {\tilde{f}}'_n(s,x_s^n)K_H(s,u)\textrm{d}s+K_H(t,u)\\&= J_1^n(t,u)+J_2^n(t,u)+K_H(t,u), \quad u<t,\quad \lambda \times \text {P-a.e.}, \end{aligned}$$

where

$$\begin{aligned} J_1^n(t,u):=\int _u^t\exp \left( \int _s^t {\tilde{f}}'(r,x_r)\textrm{d}r\right) \left( {\tilde{f}}'_n(s,x_s^n)-K_s\right) K_H(s,u)ds \end{aligned}$$

and

$$\begin{aligned} J_2^n(t,u):=\int _u^t\exp \left( \int _s^t {\tilde{f}}'(r,x_r)\textrm{d}r\right) K_s\cdot K_H(s,u)\textrm{d}s. \end{aligned}$$

Without loss of generality, let $T=t=1$. Then,

$$\begin{aligned} \int _0^1(D_ux_1^n)^2\textrm{d}u{} & {} \le C\left\{ \int _0^1(J^n_1(1,u))^2\textrm{d}u+\int _0^1(J^n_2(1,u))^2\textrm{d}u\right. \nonumber \\{} & {} \quad \left. +\int _0^1(K_H(1,u))^2\textrm{d}u \right\} . \end{aligned}$$

(17)

Using Fubini’s theorem, we get that

$$\begin{aligned}&\int _0^1(J^n_1(1,u))^2\textrm{d}u\\&\quad =\int _0^1\left( \int _0^1\chi _{_{[u,1]}}(s)\exp \left( \int _s^t {\tilde{f}}'(r,x_r)\textrm{d}r\right) \left( {\tilde{f}}'_n(s,x_s^n)-K_s\right) K_H(s,u)\textrm{d}s\right) ^2\textrm{d}u\\&\quad =\int _0^1\int _0^1 \left\{ \exp \left( \int _{s_1}^1{\tilde{f}}'(r,x_r)\textrm{d}r\right) \left( {\tilde{f}}'_n(s_1,x_{s_1}^n)-K_{s_1}\right) \right. \\&\qquad \left. \times \exp \left( \int _{s_2}^1{\tilde{f}}'(r,x_r)\textrm{d}r\right) \left( {\tilde{f}}'_n(s_2,x_{s_2}^n)-K_{s_2}\right) \int _0^{s_1\wedge s_2} K_H(s_1,u)K_H(s_2,u)\textrm{d}u\right\} \textrm{d}s_1\textrm{d}s_2. \end{aligned}$$

From (12), we see for the covariance function

$$\begin{aligned} R_H(s_1,s_2)={\mathbb {E}}\left[ B_{s_1}^H\cdot B_{s_2}^H\right] \end{aligned}$$

that

$$\begin{aligned} R_H(s_1,s_2)=\int _0^{s_1\wedge s_2}K_H(s_1,u)K_H(s_2,u)\textrm{d}u. \end{aligned}$$

Since

$$\begin{aligned} 0\le R_H(s_1,s_2)=\frac{1}{2}\left( s_1^{2H}+s_2^{2H}-\vert s_1-s_2\vert ^{2H}\right) \le 1,\quad H< \frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \left( {\tilde{f}}'_n(s_1,x_{s_1}^n)-K_{s_1}\right) \cdot \left( {\tilde{f}}'_n(s_2,x_{s_2}^n)-K_{s_2}\right) \ge 0 \end{aligned}$$

for $0< s_1,s_2\le 1$, we find that

$$\begin{aligned} \int _0^1(J^n_1(1,u))^2\textrm{d}u&\le \left( \int _0^1\left( \exp \left( \int _s^t {\tilde{f}}'(r,x_r)\textrm{d}r\right) \left( {\tilde{f}}'_n(s,x_s^n)-K_s\right) \textrm{d}s\right) ^2\right. \\&=\left\{ \left. -\exp \left( \int _s^1{\tilde{f}}'(r,x_r)\textrm{d}r\right) \right| _{s=0}^1-\int _0^1K_s\exp \left( \int _s^1{\tilde{f}}'(r,x_r)\textrm{d}r\right) \textrm{d}s\right\} ^2\\&\le \left( \exp \left( \int _0^1K_r\textrm{d}r\right) +\int _0^1K_s\textrm{d}s\cdot \exp \left( \int _0^1K_r\textrm{d}r\right) \right) ^2. \end{aligned}$$

Similarly, we also obtain that

$$\begin{aligned} \int _0^1\left( J^n_2(1,u)\right) ^2\textrm{d}u&\le C(K,H) \end{aligned}$$

for a constant $C(K,H)<\infty$. We also have that

$$\begin{aligned} \int _0^1(K_H(1,u))^2\textrm{d}u={\mathbb {E}}\left[ \left( B_1^H\right) ^2\right] =1. \end{aligned}$$

Altogether, we get that

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^1\left( D_ux_1^n\right) ^2\textrm{d}u\right] \le C(K,H) \end{aligned}$$

(18)

for all $n\ge 1$ for a constant $C(K,H)<\infty$. Define now the stopping times $\tau _n$ by

$$\begin{aligned} \tau _n=\inf \left\{ 0\le t\le 1; x_t\notin \left[ \frac{1}{n},n\right] \right\} \quad (\inf \emptyset =\infty ) \end{aligned}$$

Then, we know from the proof of the existence of solutions in the previous section that $\tau _n\nearrow \infty$ for $n\rightarrow \infty$. So,

$$\begin{aligned} x^n_{t\wedge \tau _n}-x_{t\wedge \tau _n}&=\int _0^{t\wedge \tau _n}\left\{ {\tilde{f}}_n\left( s,x_s^n\right) -{\tilde{f}}(s,x_s)\right\} \textrm{d}s\\&=\int _0^t \chi _{_{[0,\tau _n)}}(s)\left\{ {\tilde{f}}_n\left( s,x_{s\wedge \tau _n}^n\right) -{\tilde{f}}_n(s,x_{s\wedge \tau _n})\right\} \textrm{d}s. \end{aligned}$$

Hence,

$$\begin{aligned} \left| x^n_{t\wedge \tau _n}-x_{t\wedge \tau _n}\right|&\le K_n\int ^t_0\left| x^n_{s\wedge \tau _n}-x_{s\wedge \tau _n}\right| \textrm{d}s \end{aligned}$$

for a Lipschitz constant $K_n$. Then, Gronwall’s lemma implies that

$$\begin{aligned} x^n_{t\wedge \tau _n}=x_{t\wedge \tau _n} \end{aligned}$$

for all t, n P-a.e. Since $\tau _n\nearrow \infty$ for $n\rightarrow \infty$ a.e., we have that

$$\begin{aligned} x_t^n\underset{n\rightarrow \infty }{\rightarrow }\ x_t \end{aligned}$$

(19)

for all t P-a.e. Using the Clark–Ocone formula (see [7]), we get that

$$\begin{aligned} x^n_1={\mathbb {E}}\left[ x^n_1\right] +\int _0^1{\mathbb {E}}\left[ D_sx^n_1\vert {\mathcal {F}}_s\right] \textrm{d}W_s, \end{aligned}$$

where $\{{\mathcal {F}}\}_{0\le t\le 1}$ is the filtration generated by $W_{\bullet }$. It follows that

$$\begin{aligned} {\mathbb {E}}[(x_1^n-{\mathbb {E}}[x^n_1])^2]&={\mathbb {E}}\left[ \int _0^1\left( {\mathbb {E}}[D_sx^n_1\vert {\mathcal {F}}_s]\right) ^2\textrm{d}s\right] \\&\le {\mathbb {E}}\left[ \int _0^1{\mathbb {E}}[(D_sx^n_1)^2\vert {\mathcal {F}}_s]\textrm{d}s\right] =\int _0^1{\mathbb {E}}\left[ \left( D_sx^n_1\right) ^2\right] \textrm{d}s. \end{aligned}$$

So, we see from (18) that

$$\begin{aligned} {\mathbb {E}}\left[ \left( x_1^n-{\mathbb {E}}\left[ x^n_1\right] \right) ^2\right] \le C(K,H)< \infty . \end{aligned}$$

for all $n\ge 1$. We also have that

$$\begin{aligned} \left| \left| x_1^n-{\mathbb {E}}[x^n_1]\right| -\left| x_1-{\mathbb {E}}[x^n_1]\right| \right| \le \left| x^n_1-x_1\right| \underset{n\rightarrow \infty }{\longrightarrow }\ 0 \end{aligned}$$

because of (19). So,

$$\begin{aligned} \underset{n\rightarrow \infty }{\varliminf }\left| x_1^n-{\mathbb {E}}\left[ x^n_1\right] \right| =\underset{n\rightarrow \infty }{\varliminf }\left| x_1-{\mathbb {E}}\left[ x^n_1\right] \right| . \end{aligned}$$

Suppose that ${\mathbb {E}}[x_1^n]$, $n\ge 1$ is unbounded. Then, there exists a subsequence $n_k$, $k\ge 1$ such that

$$\begin{aligned} \left| {\mathbb {E}}\left[ x^{n_k}_1\right] \right| \underset{n\rightarrow \infty }{\longrightarrow }\ \infty . \end{aligned}$$

It follows from the lemma of Fatou and the positivity of $x_t$ that

$$\begin{aligned} \infty&={\mathbb {E}}\left[ \underset{k\rightarrow \infty }{\varliminf }\left( \big \vert x_1-\vert {\mathbb {E}}[x^{n_k}_1]\vert \big \vert \right) ^2\right] \\&\le {\mathbb {E}}\left[ \underset{k\rightarrow \infty }{\varliminf }\left( \big \vert x_1-{\mathbb {E}}[x^{n_k}_1]\big \vert \right) ^2\right] \\&={\mathbb {E}}\left[ \underset{k\rightarrow \infty }{\varliminf }\left( \big \vert x^{n_k}_1-{\mathbb {E}}[x^{n_k}_1]\big \vert \right) ^2\right] \\&\le \underset{k\rightarrow \infty }{\varliminf }{\mathbb {E}}\left[ \big \vert x^{n_k}_1-{\mathbb {E}}[x^{n_k}_1]\big \vert ^2\right] \le C<\infty , \end{aligned}$$

which is a contradiction. Hence,

$$\begin{aligned} \sup _{n\ge 1}\vert {\mathbb {E}}[x_1^n]\vert <\infty . \end{aligned}$$

Further, we also obtain from the Burkholder–Davis–Gundy inequality and (18) that

$$\begin{aligned} {\mathbb {E}}[\vert x_1^n\vert ^{2p}]&<C_p\left( \vert {\mathbb {E}}[x_1^n]\vert ^{2p}+{\mathbb {E}}\left[ \left( \int _0^1 {\mathbb {E}}[D_sx_1^n\vert {\mathcal {F}}_s]dW_s\right) ^{2p}\right] \right) \nonumber \\&\le C_p\left( \vert {\mathbb {E}}[x_1^n]\vert ^{2p}+{\mathbb {E}}\left[ \left( \sup _{0\le u\le 1}\big \vert \int _0^u {\mathbb {E}}[D_sx_1^n\vert {\mathcal {F}}_s]dW_s\big \vert \right) ^{2p}\right] \right) \nonumber \\&\le C_p\left( \vert {\mathbb {E}}[x_1^n]\vert ^{2p}+m_p{\mathbb {E}}\left[ \left( \int _0^1 {\mathbb {E}}[D_sx_1^n\vert {\mathcal {F}}_s]^2ds\right) ^{p}\right] \right) \nonumber \\&\le C(p,K,H) \end{aligned}$$

(20)

for $n\ge 1$. So, it follows from (19) and the lemma of Fatou that

$$\begin{aligned} {\mathbb {E}}[\vert x_1\vert ^{2p}]\le \underset{n\rightarrow \infty }{\varliminf }{\mathbb {E}}[\vert x_1^n\vert ]^{2p}\le C(p,K,H)<\infty \end{aligned}$$

for all $p\ge 1$. So, we obtain the following result:

Theorem 4.1

Let $x_t, 0\le t\le 1$ be the solution to (11). Then, $x_t\in L^q(\Omega )$ for all $q\ge 1$ and $0\le t\le 1$.

In addition, we obtain from Lemma 1.2.3 in [7] in connection with estimate (20) that $x_1$ is Malliavin differentiable in the direction of $W_{\bullet }$. The latter, in combination with (13), also entails the Malliavin differentiability of $x_1$ with respect to $B^H_{\bullet }.$ Thus, we have also shown the following result:

Theorem 4.2

The positive unique strong solution $x_t$ to (11) is Malliavin differentiable in the direction of $B^H_{\bullet }$ and $W_{\bullet }$ for all $0\le t\le 1$.

5 Application

In this section, we aim at applying the results of the previous section to obtain a unique strong solution $x_t$ to the SDE

$$\begin{aligned} x_t=x_0+\int _0^t\big (\alpha _{-1}x^{-1}_s-\alpha _0+\alpha _1x_s-\alpha _2s^{2H-1}x_s^{\rho }\big )\textrm{d}s+\int _0^t\sigma x_s^{\theta }\textrm{d}B_s^H, \end{aligned}$$

(21)

$0\le t\le 1$, for $H\in (\frac{1}{3},\frac{1}{2})$, $\tilde{\theta }>0$, $\rho >1+\frac{1}{H\tilde{\theta }}$, $\sigma >0$ and $\theta :=\frac{\tilde{\theta }+1}{\tilde{\theta }}$. Here, the stochastic integral with respect to $B_{\bullet }^H$ is defined by

$$\begin{aligned} \int _0^tg(x_s)\textrm{d}B^H_s=\int _0^t-Hs^{2H-1}g^{\prime }(x_s)\textrm{d}s+\int _0^tg(x_s)d^{\circ }B_s^H \end{aligned}$$

(22)

for functions $g\in {\mathcal {C}}^3((0,\infty );{\mathbb {R}})$. See also the second Remark 5.3. The stochastic integral on the right-hand side of (22) is the symmetric integral with respect to $B^H_{\bullet }$ introduced by Russo and Vallois. See, for example, [18] and the references therein. Such an integral denoted by

$$\begin{aligned} \int _0^tY_sd^{\circ }X_s, \quad t\in [0,1] \end{aligned}$$

(23)

for continuous process $X_{\bullet }$, $Y_{\bullet }$ is defined as

$$\begin{aligned} \lim _{\epsilon \searrow 0}\frac{1}{2\epsilon }\int _0^tY_s(X_{s+\epsilon }-X_s)ds, \end{aligned}$$

provided this limit exists in the ucp-topology. In order to construct a solution to (21), we need a version of the Itô formula for processes $Y_{\bullet }$, which have a finite cubic variation. A continuous process is said to have a finite strong cubic variation (or 3-variation), denoted by [Y, Y, Y], if

$$\begin{aligned}{}[Y,Y,Y]:=\lim _{\epsilon \searrow 0}\frac{1}{\epsilon }\int _0^t(Y_{s+\epsilon }-Y_s)^3ds \end{aligned}$$

exists in ucp as well as

$$\begin{aligned} \sup _{0< \epsilon \le 1}\frac{1}{\epsilon }\int _0^1\vert Y_{s+\epsilon }-Y_s\vert ^3ds< \infty \quad \text {a.e.} \end{aligned}$$

See [18]. Using the concept of finite strong cubic variation, one can show the following Itô formula (see [18]).

Theorem 5.1

Assume that $Y_{\bullet }$ is a real valued process with finite strong cubic variation and $g\in {\mathcal {C}}^3((0,\infty );{\mathbb {R}})$. Then,

$$\begin{aligned} g(Y_t)=g(Y_0)+\int _0^tg^{\prime }(Y_s)d^{\circ }Y_s-\frac{1}{12}\int _0^tg^{\prime \prime \prime }(Y_s)d[Y,Y,Y]_s, \quad 0\le t\le 1. \end{aligned}$$

Remark 5.2

The last term on the right-hand side of the equation is a Lebesgue–Stieltjes integral with respect to the bounded variation process [Y, Y, Y].

Remark 5.3

• We mention that for $Y_{\bullet }=B_{\bullet }^H$, $H\in (\frac{1}{3},\frac{1}{2})$, $[B_{\bullet }^H,B_{\bullet }^H,B_{\bullet }^H]$ is zero a.e.

• If $X_{\bullet }=B_{\bullet }^H$ in (22), then it follows from Theorem 6.3.1 in [8] that our stochastic integral in (22) equals the Wick–Itô–Skorohod integral. The latter also gives a justification for the definition of the stochastic integral in (22) in the general case.

Theorem 5.4

Suppose that $H\in (\frac{1}{3},\frac{1}{2})$, $\tilde{\theta }>0$, $\sigma >0$ and $\rho >1+\frac{1}{H\tilde{\theta }}$. Let $\theta =\frac{\tilde{\theta }+1}{\tilde{\theta }}$. Then, there exists a unique strong and positive solution to the SDE (21).

Proof

Let $y_{\bullet }$ be the unique strong and positive solution to

$$\begin{aligned} y_t=x+\int _0^t{\tilde{f}}(s,y_s)ds-\tilde{\sigma }B_t^H, \quad 0\le t\le 1,\quad x>0, \end{aligned}$$

where ${\tilde{f}}$ is defined as in Sect. 2. Define $g\in {\mathcal {C}}^3((0,\infty );{\mathbb {R}})$ by $g(y)=\frac{y^{-\tilde{\theta }}}{\tilde{\theta }}$. Then, a modification of Theorem 5.1 (see Lemma 6.1) entails that

$$\begin{aligned} x_t:=g(y_t)=\frac{x^{-\tilde{\theta }}}{\tilde{\theta }}+\int _0^t(-1)y_s^{-(\tilde{\theta }+1)}d^{\circ }y_s-\frac{1}{12}\int _0^tg^{\prime \prime \prime }(y_s)d[y,y,y]_s. \end{aligned}$$

Since $[B_{\bullet }^H,B_{\bullet }^H,B_{\bullet }^H]$ is zero a.e. (see Remark 5.3), we observe that [y, y, y] is zero a.e. So,

$$\begin{aligned} x_t&=\frac{x^{-\tilde{\theta }}}{\tilde{\theta }}+\int _0^t(-1)y_s^{-(\tilde{\theta }+1)}d^{\circ }y_s\\&=\frac{x^{-\tilde{\theta }}}{\tilde{\theta }}+\int _0^t(-1)y_s^{-(\tilde{\theta }+1)}{\tilde{f}}(s,y_s)\textrm{d}s+\int _0^t\tilde{\sigma } y_s^{-(\tilde{\theta }+1)}d^{\circ } B_s^H\\&=\frac{x^{-\tilde{\theta }}}{\tilde{\theta }}-\int _0^t\Big \{(-1)y_s^{-(\tilde{\theta }+1)}{\tilde{f}}(s,y_s)-H\tilde{\sigma } s^{2H-1}y_s^{-(\tilde{\theta }+2)}(\tilde{\theta }+1)\Big \}\textrm{d}s\\&\quad +\int _0^t\tilde{\sigma }y_s^{-(\tilde{\theta }+1)}\textrm{d}B_s^H. \end{aligned}$$

Since we can write $(y_s)^{-(\tilde{\theta }+1)}=\tilde{\theta }^{\theta }\left( \frac{y_s^{-\tilde{\theta }}}{\tilde{\theta }}\right) ^{\theta }$, we now have

$$\begin{aligned} x_t&=\frac{x^{-\tilde{\theta }}}{\tilde{\theta }}+\int _0^tf\left( s,\frac{y_s^{-\tilde{\theta }}}{\tilde{\theta }}\right) \textrm{d}s+\int _0^t\tilde{\sigma } y_s^{-(\tilde{\theta }+1)}\textrm{d}B_s^H\\&=\frac{x^{-\tilde{\theta }}}{\tilde{\theta }}+\int _0^tf\left( s,\frac{y_s^{-\tilde{\theta }}}{\tilde{\theta }}\right) \textrm{d}s+\int _0^t\tilde{\sigma } \tilde{\theta }^{\theta }(x_s)^{\theta }\textrm{d}B_s^H, \end{aligned}$$

where $f(s,y):=\alpha _{-1}y^{-1}-\alpha _0+\alpha _1y-\alpha _2s^{2H-1}y^{\rho }$, $s\in (0,1]$, $y\in (0,\infty )$. So $x_{\bullet }$ satisfies the SDE (21) if we choose $\tilde{\sigma }=\tilde{\theta }^{-\theta }\sigma$ for $\sigma >0$. In order to show the uniqueness of solutions to SDE (21), one can apply the Itô formula in Theorem 5.1 to the inverse function $g^{-1}$ given by $g^{-1}(y)=\big (\tilde{\theta }\big )^{-\frac{1}{\tilde{\theta }}} y^{-\frac{1}{\tilde{\theta }}}$ by using the fact that $[B_{\bullet }^H,B_{\bullet }^H,B_{\bullet }^H]=0$ a.e. for $H\in (\frac{1}{3},\frac{1}{2})$. $\square$

Finally, using the same arguments as in the proof of Theorem 5.4, we also get the following result for the alternative Ait-Sahalia model (3):

Theorem 5.5

Retain the conditions of Theorem 5.4 with respect to $H,\tilde{\theta }, \theta$ and $\rho$. Then, there exists a unique strong solution $x_t>0$ to SDE (3).

Proof

Just as in the proof of Theorem 5.4, we can consider the SDE (6), where the vector field ${\tilde{f}}$ now is given by

$$\begin{aligned} {\tilde{f}}(s,y)&=\alpha _{-1}(-\tilde{\theta } y^{2\tilde{\theta }+1})+\alpha _0y^{\tilde{\theta }+1}-\alpha _1\frac{y}{\tilde{\theta }}+\alpha _2 \frac{1}{\tilde{\theta }^{\rho }} y^{-\tilde{\theta }\rho +\tilde{\theta }+1} \end{aligned}$$

(24)

for $0<y < \infty$. Then, as in the proof of Corollary (3.2) one immediately verifies that ${\tilde{f}}$ satisfies the assumptions of Theorem 3.1, which yields a unique strong solution $y_t>0$ to (6) in this case. In exactly the same way, we also obtain the results of Theorem 4.1 and Theorem 4.2 with respect to ${\tilde{f}}$ in (24). Finally, we can apply the Itô formula as in the proof of Theorem 5.4 and construct a unique strong solution $x_t>0$ to (3) based on $y_{\bullet }$. $\square$

Appendix

Lemma 6.1

Let $y_t$, $0\le t\le 1$, be the positive strong solution of the SDE in the proof of Theorem 5.4 and let $f:(0,\infty )\rightarrow (0,\infty )$; $x\rightarrow x^{-\alpha }$, where $\alpha >0$. Then,

$$\begin{aligned} f(y_t)=f(y_0)+\int _0^tf'(y_s)d^{\circ }y_s,\quad 0\le t\le 1, \text { a.e.} \end{aligned}$$

Proof

The proof is based on the same arguments for the proof of the Itô formula as in [18] and the fact that $\underset{t\in [0,2]}{\min }y_t>0$ a.e. (see the proof of Theorem 3.1) for a solution $y_{\bullet }$ on $[0,2]\supset [0,1]$. Consider the following type of Taylor formula

$$\begin{aligned} f(b)&=f(a)+f'(a)(b-a)+\frac{1}{2}f''(a)(b-a)^2\\&\quad +\frac{1}{6}f^{(3)}(a)(b-a)^3+R(a,b)(b-a)^3 \end{aligned}$$

for $a,b\in (0,\infty )$, where

$$\begin{aligned} R(a,b):=\int _0^1\frac{\phi ^2}{2}(f^{(3)}(\phi a+(1-\phi )b)-f^{(3)}(a))d\phi . \end{aligned}$$

So for $\epsilon \in (0,1)$ and $s\in [0,1]$, we obtain

$$\begin{aligned} f(y_{s+\epsilon })&=f(y_s)+f'(y_s)(y_{s+\epsilon }-y_s)+\frac{1}{2}f''(y_s)(y_{s+\epsilon }-y_s)^2\\&\quad -\frac{1}{6}f^{(3)}(y_s)(y_{s+\epsilon }-y_s)^3+R(y_s,y_{s+\epsilon })(y_{s+\epsilon }-y_s)^3 \end{aligned}$$

and

$$\begin{aligned} f(y_s)&=f(y_{s+\epsilon })-f'(y_{s+\epsilon })(y_{s+\epsilon }-y_s)+\frac{1}{2}f''(y_{s+\epsilon })(y_{s+\epsilon }-y_s)^2\\&\quad -\frac{1}{6}f^{(3)}(y_{s+\epsilon })(y_{s+\epsilon }-y_s)^3-R(y_{s+\epsilon },y_s)(y_{s+\epsilon }-y_s)^3. \end{aligned}$$

The latter entails that

$$\begin{aligned} \frac{1}{\epsilon }\int _0^t(f(y_{s+\epsilon })-f(y_s))\textrm{d}s=\sum _{i=1}^{4}J_{i,\epsilon }(t), \end{aligned}$$

where

$$\begin{aligned} J_{1,\epsilon }(t)&:=\frac{1}{2\epsilon }\int _0^t(f'(y_{s+\epsilon })+f'(y_s))(y_{s+\epsilon }-y_s)\textrm{d}s,\\ J_{2,\epsilon }(t)&:=-\frac{1}{4\epsilon }\int _0^t(f''(y_{s+\epsilon })-f''(y_s))(y_{s+\epsilon }-y_s)^2\textrm{d}s,\\ J_{3,\epsilon }(t)&:=\frac{1}{12\epsilon }\int _0^t(f^{(3)}(y_{s+\epsilon })+f^{(3)}(y_s))(y_{s+\epsilon }-y_s)^3\textrm{d}s,\\ \end{aligned}$$

and

$$\begin{aligned} J_{4,\epsilon }(t):=-\frac{1}{2\epsilon }\int _0^t(R(y_s,y_{s+\epsilon })+R(y_{s+\epsilon },y_s))(y_{s+\epsilon }-y_s)^3\textrm{d}s. \end{aligned}$$

Since the process $f(y_t)$, $0\le t\le 1$, is continuous, we see that

$$\begin{aligned} \frac{1}{\epsilon }\int _0^{\bullet }(f(y_{s+\epsilon })-f(y_s))\textrm{d}s\underset{\epsilon \searrow 0}{\overset{\text {ucp}}{\longrightarrow }}f(y_{\bullet })-f(y_0). \end{aligned}$$

On the other hand, using the mean value theorem, we find that

$$\begin{aligned} J_{2,\epsilon }(t)&=-\frac{1}{4\epsilon }\int _0^t\psi _{\epsilon }(s)(y_{s+\epsilon }-y_s)^3\textrm{d}s\\&=-\frac{1}{4\epsilon }\int _0^t(\psi _{\epsilon }(s)-\psi _0(s))(y_{s+\epsilon }-y_s)^3\textrm{d}s+\frac{1}{4\epsilon }\int _0^t\psi _0(s)(y_{s+\epsilon }-y_s)^3\textrm{d}s, \end{aligned}$$

where

$$\begin{aligned} \psi _{\epsilon }(s):=\int _0^1f^{(3)}(\phi y_s+(1-\phi )y_{s+\epsilon })d\phi \end{aligned}$$

and

$$\begin{aligned} \psi _0(s):=f^{(3)}(y_s). \end{aligned}$$

We have for $s\in [0,1]$ that

$$\begin{aligned}&\vert \psi _{\epsilon }(s)-\psi _0(s)\vert =\big \vert \int _0^1(f^{(3)}(\phi y_s+(1-\phi )y_{s+\epsilon })-f^{(3)}(y_s))d\phi \big \vert \\&\quad \le \alpha (\alpha +1)(\alpha +3)\int _0^1\frac{\vert (y_s)^{\alpha +3}-(\phi y_s+(1-\phi )y_{s+\epsilon })^{\alpha +3} \vert }{(\phi y_s+(1-\phi )y_{s+\epsilon })^{\alpha +3}(y_s)^{\alpha +3}}d\phi \\&\quad \le \alpha (\alpha +1)(\alpha +3)\int _0^1\frac{\vert (y_s)^{\alpha +3}-(\phi y_s+(1-\phi )y_{s+\epsilon })^{\alpha +3} \vert }{\big (\phi \underset{s\in [0,2]}{\min }y_s+(1-\phi )\underset{s\in [0,2]}{\min }y_s \big )^{\alpha +3}(\underset{s\in [0,2]}{\min }y_s)^{\alpha +3}}d\phi \\&\quad \le \alpha (\alpha +1)(\alpha +2)\frac{1}{(\underset{s\in [0,2]}{\min }y_s)}2(\alpha +3)\underset{\phi \in [0,1]}{\sup }\underset{s\in [0,1]}{\sup }\vert (y_s)^{\alpha +3}- (\phi y_s+(1-\phi )y_{s+\epsilon })^{\alpha +3} \vert \\&\quad \underset{\epsilon \searrow 0}{\longrightarrow }0\quad \text {a.e.}, \end{aligned}$$

because of

$$\begin{aligned} \underset{\phi \in [0,1]}{\sup }\underset{s\in [0,1]}{\sup }\vert y_s-\phi y_s-(1-\phi )y_{s+\epsilon } \vert \le \underset{s\in [0,1]}{\sup }\vert y_s-y_{s+\epsilon } \vert \underset{\epsilon \searrow 0}{\longrightarrow }0\quad \text {a.e.} \end{aligned}$$

and uniform continuity. So,

$$\begin{aligned} \underset{s\in [0,1]}{\sup }\vert \psi _{\epsilon }(s)-\psi _0(s)\vert \underset{\epsilon \searrow 0}{\longrightarrow }0\quad \text {a.e.} \end{aligned}$$

Hence,

$$\begin{aligned}&\left| \frac{1}{4\epsilon }\int _0^t (\psi _{\epsilon }(s)-\psi _0(s))( y_{s+\epsilon }-y_s)^3ds\right| \\&\quad \le \underset{s\in [0,1]}{\sup }\vert \psi _{\epsilon }(s)-\psi _0(s)\vert \cdot \underset{\epsilon >0}{\sup }\frac{1}{4\epsilon }\int _0^t\vert y_{s+\epsilon }-y_s\vert ^3ds\underset{\epsilon \searrow 0}{\longrightarrow }0\quad \text {a.e.}, \end{aligned}$$

since $y_{\bullet }$ is of strong 3-variation.

Further, since $\psi _0(t)$, $0\le t\le 1$, is a continuous process, it follows from Remark 2.6, (6) in [18] that

$$\begin{aligned} \frac{1}{4\epsilon }\int _0^{\bullet } \psi _0(s)( y_{s+\epsilon }-y_s)^3ds\underset{\epsilon \searrow 0}{\overset{\text {ucp}}{\longrightarrow }} \frac{1}{4}\int _0^{\bullet } \psi _0(s)d[y,y,y]_s=0, \end{aligned}$$

due to $[y,y,y]_s=0$. So,

$$\begin{aligned} J_{2,\epsilon }(\cdot )=\underset{\epsilon \searrow 0}{\overset{\text {ucp}}{\longrightarrow }} 0. \end{aligned}$$

We also see that

$$\begin{aligned} J_{3,\epsilon }(t)&=\frac{1}{12\epsilon }\int _0^t f^{(3)}(y_s)( y_{s+\epsilon }-y_s)^3ds+\frac{1}{12\epsilon }\int _0^t f^{(3)}(y_{s+\epsilon })( y_{s+\epsilon }-y_s)^3ds\\&=J^{(1)}_{3,\epsilon }(t)+J^{(2)}_{3,\epsilon }(t). \end{aligned}$$

Because of Remark 2.6, (6) in [18], we have again

$$\begin{aligned} J^{(1)}_{3,\epsilon }(\cdot )\underset{\epsilon \searrow 0}{\overset{\text {ucp}}{\longrightarrow }} 0. \end{aligned}$$

Further, Remark 2.6, (5) in [18] implies that

$$\begin{aligned} J^{(2)}_{3,\epsilon }(\cdot )\underset{\epsilon \searrow 0}{\overset{\text {ucp}}{\longrightarrow }} 0. \end{aligned}$$

As for the process $J_{4,\epsilon }(\cdot )$, we can use the same arguments as in the case of $J_{2,\epsilon }(\cdot )$ based on uniform continuity and the strong 3-variation of $y_{\bullet }$ and obtain that

$$\begin{aligned} J_{4,\epsilon }(\cdot )\underset{\epsilon \searrow 0}{\overset{\text {ucp}}{\longrightarrow }} 0. \end{aligned}$$

Altogether, we get in connection with Remark 3.2, (1) in [18] that

$$\begin{aligned} \underset{\epsilon \searrow 0}{\lim }J_{1,\epsilon }(\cdot ) \end{aligned}$$

exists in the ucp-topology and must be equal to

$$\begin{aligned} \int _0^{\bullet }f'(y_s)d^{\circ }y_s. \end{aligned}$$

$\square$

For some of the proofs in this article, we need to recall some basic concepts from fractional calculus (see [19, 20]).

Let a,  $b\in {\mathbb {R}}$ with $a<b$. Let $f\in L^{p}([a,b])$ with $p\ge 1$ and $\alpha >0$. Then, the left- and right-sided Riemann–Liouville fractional integrals are defined as

$$\begin{aligned} I_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int _{a}^{x}(x-y)^{\alpha -1}f(y)dy \end{aligned}$$

and

$$\begin{aligned} I_{b^{-}}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int _{x}^{b}(y-x)^{\alpha -1}f(y)dy \end{aligned}$$

for almost all $x\in [a,b]$. Here, $\Gamma$ is the gamma function.

Let $p\ge 1$ and let $I_{a^{+}}^{\alpha }(L^{p})$ (resp. $I_{b^{-}}^{\alpha }(L^{p})$) be the image of $L^{p}([a,b])$ of the operator $I_{a^{+}}^{\alpha }$ (resp. $I_{b^{-}}^{\alpha }$). If $f\in I_{a^{+}}^{\alpha }(L^{p})$ (resp. $f\in I_{b^{-}}^{\alpha }(L^{p})$) and $0<\alpha <1$, then we can define the left- and right-sided Riemann–Liouville fractional derivatives by

$$\begin{aligned} D_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}\int _{a}^{x} \frac{f(y)}{(x-y)^{\alpha }}dy \end{aligned}$$

and

$$\begin{aligned} D_{b^{-}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dx}\int _{x}^{b} \frac{f(y)}{(y-x)^{\alpha }}dy. \end{aligned}$$

The left- and right-sided derivatives of f can be represented as

$$\begin{aligned} D_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\left( \frac{f(x)}{ (x-a)^{\alpha }}+\alpha \int _{a}^{x}\frac{f(x)-f(y)}{(x-y)^{\alpha +1}} dy\right) \end{aligned}$$

and

$$\begin{aligned} D_{b^{-}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\left( \frac{f(x)}{ (b-x)^{\alpha }}+\alpha \int _{x}^{b}\frac{f(x)-f(y)}{(y-x)^{\alpha +1}} dy\right) . \end{aligned}$$

The above definitions imply that

$$\begin{aligned} I_{a^{+}}^{\alpha }\left( D_{a^{+}}^{\alpha }f\right) =f \end{aligned}$$

for all $f\in I_{a^{+}}^{\alpha }(L^{p})$ and

$$\begin{aligned} D_{a^{+}}^{\alpha }\left( I_{a^{+}}^{\alpha }f\right) =f \end{aligned}$$

for all $f\in L^{p}([a,b])$ and similarly for $I_{b^{-}}^{\alpha }$ and $D_{b^{-}}^{\alpha }$.

Denote by $B^{H}=\{B_{t}^{H},t\in [0,T]\}$ a d-dimensional fractional Brownian motion with Hurst parameter $H\in (0,\frac{1}{2})$. The latter means that $B_{\cdot }^{H}$ is a centred Gaussian process with a covariance function given by

$$\begin{aligned} (R_{H}(t,s))_{i,j}:= & {} E\left[ B_{t}^{H,(i)}B_{s}^{H,(j)}\right] \\ {}= & {} \delta _{ij}\frac{1}{2} \left( t^{2H}+s^{2H}-|t-s|^{2H}\right) ,\quad i,j=1,\ldots ,d, \end{aligned}$$

where $\delta _{ij}$ is one, if $i=j$, or zero else.

In the sequel, we also shortly recall the construction of the fractional Brownian motion, which can be found in [7]. For convenience, we restrict ourselves to the case $d=1$.

Denote by ${\mathcal {E}}$ the class of step functions on [0, T], and let ${\mathcal {H}}$ be the Hilbert space which one gets through the completion of ${\mathcal {E}}$ with respect to the inner product

$$\begin{aligned} \langle 1_{[0,t]},1_{[0,s]}\rangle _{{\mathcal {H}}}=R_{H}(t,s). \end{aligned}$$

The latter provides an extension of the mapping $1_{[0,t]}\mapsto B_{t}$ to an isometry between ${\mathcal {H}}$ and a Gaussian subspace of $L^{2}(\Omega )$ with respect to $B^{H}$. Let $\varphi \mapsto B^{H}(\varphi )$ be this isometry.

If $H<\frac{1}{2}$, one finds that the covariance function $R_{H}(t,s)$ can be represented as

$$\begin{aligned} R_{H}(t,s)=\int _{0}^{t\wedge s}K_{H}(t,u)K_{H}(s,u)du, \end{aligned}$$

(25)

where

$$\begin{aligned} K_{H}(t,s)= & {} c_{H}\left[ \left( \frac{t}{s}\right) ^{H-\frac{1}{2}}(t-s)^{H- \frac{1}{2}}+\left( \frac{1}{2}-H\right) s^{\frac{1}{2}-H}\right. \nonumber \\{} & {} \left. \int _{s}^{t}u^{H- \frac{3}{2}}(u-s)^{H-\frac{1}{2}}du\right] . \end{aligned}$$

(26)

Here, $c_{H}=\sqrt{\frac{2H}{(1-2H)\beta (1-2H,H+\frac{1}{2})}}$ and $\beta$ is the beta function. See [7, Proposition 5.1.3].

Using the kernel $K_{H}$, one can obtain via (25) an isometry $K_{H}^{*}$ between ${\mathcal {E}}$ and $L^{2}([0,T])$ such that $(K_{H}^{*}1_{[0,t]})(s)=K_{H}(t,s)1_{[0,t]}(s).$ This isometry allows for an extension to the Hilbert space ${\mathcal {H}}$, which has the following representations in terms of fractional derivatives

$$\begin{aligned} (K_{H}^{*}\varphi )(s)=c_{H}\Gamma \left( H+\frac{1}{2}\right) s^{\frac{1 }{2}-H}\left( D_{T^{-}}^{\frac{1}{2}-H}u^{H-\frac{1}{2}}\varphi (u)\right) (s) \end{aligned}$$

and

$$\begin{aligned} (K_{H}^{*}\varphi )(s)=&\,c_{H}\Gamma \left( H+\frac{1}{2}\right) \left( D_{T^{-}}^{\frac{1}{2}-H}\varphi (s)\right) (s) \\&+c_{H}\left( \frac{1}{2}-H\right) \int _{s}^{T}\varphi (t)(t-s)^{H-\frac{3}{ 2}}\left( 1-\left( \frac{t}{s}\right) ^{H-\frac{1}{2}}\right) dt. \end{aligned}$$

for $\varphi \in {\mathcal {H}}$. One can also prove that ${\mathcal {H}} =I_{T^{-}}^{\frac{1}{2}-H}(L^{2})$. See [21] and [22, Proposition 6].

We know that $K_{H}^{*}$ is an isometry from ${\mathcal {H}}$ into $L^{2}([0,T])$. Thus, the d-dimensional process $W=\{W_{t},t\in [0,T]\}$ defined by

$$\begin{aligned} W_{t}:=B^{H}((K_{H}^{*})^{-1}(1_{[0,t]})) \end{aligned}$$

(27)

is a Wiener process and the process $B^{H}$ has the representation

$$\begin{aligned} B_{t}^{H}=\int _{0}^{t}K_{H}(t,s)dW_{s}. \end{aligned}$$

(28)