The function \( \sigma\) is bounded and \( \inf_{\mathbb R}|\sigma| > 0\) .

Under Assumption 1, for every \( s,t\in [0,T]\) ,

Consider

Assume that \( H\in (1/3,1/2]\) , and let \( x : [0,T]^2\rightarrow\mathbb R\) be a function such that

and

where \( \alpha\in (1 - 2H,H)\) and \( \mathfrak c_x\) is a positive constant. Then, the 2D Young integral of \( x\) with respect to the covariance function \( R\) of the fBm is well-defined, and

Assume that \( H\in (1/3,1/2]\) . Under Assumption 1, if \( b\) is bounded, then

and

Let us add a few remarks on Proposition 3.

1. Since \( \sigma\) , \( 1/\sigma\) , and the derivatives of \( b\) and \( \sigma\) are already assumed to be bounded, if \( b\) is itself bounded, then so are \( \varphi\) and \( \psi\) . This additional boundedness condition on \( b\) , which is not needed when \( H \in (1/2,1)\) , is the price to pay in order to apply Theorem 3.1 of Song and Tindel as well as Proposition 2 in the proof of Proposition 3 (see Section 5).

2. It is worth noting that, after application of Proposition 2, the relationships between Skorokhod and pathwise integrals in the two cases \( H > 1/2\) and \( H\in (1/3,1/2]\) , summarized in Equations (3) and (7) respectively, appear quite similar. However, there is a crucial difference: in Equation (7), the right-hand side cannot be simplified in such a way that the second integral cancels the \( -1\) appearing in the third term, as happens in Equation (3). This is because the map

   \[ (s,t)\in [0,T]^2\longmapsto \varphi(X_t)\exp\left(\theta_0\int_{s}^{t}\psi(X_u)du\right)\mathbf 1_{[0,t)}(s) \]

   does not satisfy the assumptions of Proposition 2.

Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive. Under Assumption 1, if

where \( \mathfrak c\) is a deterministic constant arbitrarily chosen in \( (0,1)\) ,

then \( \Theta_N\) is a contraction from \( \mathbb R_+\) into itself. Therefore, \( R_N\) exists and is unique.

Note that, in particular, the conditions of Proposition 4 on \( (\varphi,\psi)\) imply that it is bounded. Indeed,

leading to

Since \( b'\) and \( \sigma\) are bounded, the ratio \( \sigma'b/\sigma\) is also bounded, and therefore so are \( (\varphi,\psi)\) .

Let us provide examples of drift and volatility functions satisfying the condition \( {\rm (A)}\) :

1. If \( \sigma\) is constant, then \( (\varphi,\psi) = (\sigma^2b',b')\) satisfies \( {\rm (A)}\) if and only if \( b'\leqslant 0\) (as in [60], Proposition 7).

2. Assume that \( b(x) = -x\) , which is quite common. Then,

   \[ ({\rm A})\Longleftrightarrow -1\leqslant\frac{\sigma'(x)}{\sigma(x)}x\leqslant 1. \]

   For instance,

• If \( \sigma(x) =\pi +\arctan(x)\) , then \( \sigma'(x) = (1 + x^2)^{-1}\) , leading to

  \[ \frac{\sigma'(x)}{\sigma(x)}x = \frac{x}{(1 + x^2)(\pi +\arctan(x))}\in [-1,1]. \]

• If \( \sigma(x) = 1 + e^{-x^2}\) , then \( \sigma'(x) = -2xe^{-x^2}\) , leading to

  \[ \frac{\sigma'(x)}{\sigma(x)}x =\frac{-2x^2e^{-x^2}}{1 + e^{-x^2}}\in [-1,1]. \]

Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, \( \theta_0 > 0\) and that

Under Assumption 1, there exists a constant \( \mathfrak c_{5} > 0\) , not depending on \( N\) , such that

Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, and that \( \theta_0 > 0\) . Under Assumption 1, if \( T\) satisfies the condition (12) with \( \mathfrak c = 1/2\) , then

for every \( \alpha\in (0,1)\) , where \( u_{\cdot} =\phi^{-1}(\cdot)\) , \( \phi\) is the standard normal distribution function, and

Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, and that \( \theta_0 > 0\) . Under Assumption 1, if \( T\) satisfies the condition (12), then there exists a constant \( \mathfrak c_{7} > 0\) , not depending on \( N\) , such that

Assume that \( b\) is bounded, and that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive. Under Assumption 1, if \( M_N/D_N\) satisfies the condition (10), then \( \widetilde\Theta_N\) is a contraction from \( \mathbb R_+\) into itself. Therefore, \( R_N\) exists and is unique.

Assume that \( b\) is bounded, \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, and that \( \theta_0\leqslant\theta_{\max}\) with a known \( \theta_{\max} > 0\) . Under Assumption 1, if \( T\) satisfies the condition (12) with \( \mathfrak c = 1/2\) , then

where

Comparing the confidence interval in Proposition 6 with that of Proposition 9, one can notice a difference in the random variables appearing in the widths of the intervals: \( Y_N\) is now replaced by \( \mathfrak Y_N\) . This discrepancy arises because, in the case \( H > 1/2\) , when analyzing the variance of the Skorokhod integral, we could invoke Theorem 3.11.1 in [67], which allows us to propose a Lebesgue integral bounded by \( Y_N\) and converging in probability to the variance of the Skorokhod integral.

For \( H\in (1/3,1/2]\) , this theorem is no longer applicable. Instead, we handle the variance of the Skorokhod integral by directly using the relationship between the Skorokhod and pathwise integrals, as established through our probabilistic tools in Equality (7).

Assume that \( b\) is bounded, and that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive. Under Assumption 1, if \( T\) satisfies the condition (12), then there exists a constant \( \mathfrak c_{10} > 0\) , not depending on \( N\) , such that

For every \( s,t\in [0,T]\) , consider

Under the assumptions of Proposition 3,

1. There exists a deterministic constant \( \mathfrak c_{1} > 0\) such that

   \[ |L(s,t)|\leqslant \mathfrak c_{1}|t - s| \textrm{ \( ;\) }\forall (s,t)\in\Delta_T, \]

   and for every \( \alpha\in (0,H)\) ,

   \[ |L(s,t) - L(u,v)|\leqslant \mathfrak c_{1}(1\vee\|X\|_{\alpha,T})(|t - v|^{\alpha} + |s - u|) \textrm{ \( ;\) }\forall (s,t),(u,v)\in\Delta_T, \]

   where \( \|X\|_{\alpha,T}\) is the \( \alpha\) -H
   specialChar{34}older norm of \( X\) over \( [0,T]\) .
2. For every \( p > 1/H\) , the random variables \( \|\overline L(0,\cdot)\|_{p\textrm{-var},T}\) and \( \|\overline L\|_{p\textrm{-var},[0,T]^2}\) belong to \( \mathbb L^2(\Omega)\) .