Admissibility. Let us verify that the pair
\(({\widehat{u}},{\widehat{z}})\) given in the statement above belongs to
\({\mathcal {A}}\). We start from condition (i) in Definition
2.1. First,
\({\widehat{u}}, {\widehat{z}}\) are trivially progressively measurable and real valued. We need to check
\({\widehat{z}}_t > -1\), which is equivalent to
\(\frac{\sigma ^2}{g}D(t) > -1\). We distinguish two cases: if
\(a \le \frac{2\lambda ^2a^4}{\kappa }\) we have
\(D(t) \ge 0\) (this is a consequence of Remark
2.2 or, alternatively, the explicit formula (
2.12)), hence
\({\widehat{z}}_t >-1\) for all
\(t \in [0,T]\). On the other hand, if
\(a > \frac{2\lambda ^2a^4}{\kappa }\), it follows from expression (
2.12) that
D(
t) is nondecreasing with
\(D(T)=0\). Therefore, it suffices to check that
\(D(0)> -g/\sigma ^2\), where
$$\begin{aligned} D(0)= -\frac{2 \left( a - \frac{2\lambda ^2a^4}{\kappa }\right) }{\theta \coth (\frac{\theta T}{2}) + \frac{4\lambda a^2}{\kappa }}. \end{aligned}$$
After some computation, we obtain that
\(D(0)> -g/\sigma ^2\) if and only if
$$\begin{aligned} \coth \left( \frac{\theta T}{2}\right) > \frac{2\sigma ^2 a}{\theta g} H, \end{aligned}$$
where
H is the constant defined in the statement. Now, if
\(H <0\) the inequality above is always satisfied as the LHS above is nonnegative. If
\(H>0\), the inequality above is guaranteed by the condition
\(T < T_{max}\). We can conclude that even in this second case, provided
\(T< T_{max}\), we have
\({\widehat{z}}_t >-1\) for all
\(t\in [0,T]\). Regarding the integrability properties, we verify now that
$$\begin{aligned} {\mathbb {E}}\left[ \int _0 ^T ({\widehat{u}}_t ^2 + {\widehat{z}}^2 _t)dt\right]< \infty , \quad {\mathbb {E}}\left[ \int _0 ^T {\widehat{q}}_t ^2 (1 + {\widehat{z}}_t) dt\right] < \infty , \end{aligned}$$
(2.20)
where
\({\widehat{q}} = q^{{\widehat{u}},{\widehat{z}}}\). Since
\({\widehat{u}}\) is affine in
\({\widehat{q}}\) with continuous time-dependent coefficients and
\({\widehat{z}}\) is deterministic and continuous in
t, checking the properties above boils down to show
$$\begin{aligned} {\mathbb {E}}\left[ \int _0 ^T {\widehat{q}}_t ^2 dt\right] < \infty . \end{aligned}$$
First, we use Fubini’s theorem to get
\({\mathbb {E}} [\int _0 ^T {\widehat{q}}_t ^2 dt ] = \int _0 ^T {\mathbb {E}}[{\widehat{q}}_t ^2] dt\). Moreover, since
\({\widehat{q}}_t\) is a Gaussian random variable for any fixed
t (see Remark
2.3 below), the function
\(t \mapsto {\mathbb {E}}[{\widehat{q}}_t ^2]\) is continuous over [0,
T], so its integral is finite. Regarding condition (ii), we need to show that there exists a unique EMM
\(\widehat{{\mathbb {Q}}} = {\mathbb {Q}}^{{\widehat{u}},{\widehat{z}}}\) for the production process
\({\widehat{q}}\). Let us recall that
$$\begin{aligned} \frac{d{\widehat{{\mathbb {Q}}}}}{d{\mathbb {P}}} = \exp \left\{ -\int _0 ^T {\widehat{\delta }}_t dW_t - \frac{1}{2}\int _0 ^T {\widehat{\delta }}_t ^2 dt \right\} , \quad {\widehat{\delta }}_t = \frac{{\widehat{u}}_t}{\sigma \sqrt{1+ {\widehat{z}}_t}}. \end{aligned}$$
We use [
23, Theorem 2.1] to prove that under our assumptions the probability
\({\widehat{{\mathbb {Q}}}}\) is well-defined (see also [
22] for more general results of the same type). According to that results, we need to check Assumption 2.2 in [
23], which in our case is satisfied as long as
\(\sigma ^2 (1+{\widehat{z}}_t) > 0\) for all
\(t\in [0,T]\). By the same arguments used for condition (i), we get the result. A standard application of Girsanov theorem, together with the integrability properties in (
2.20), yields immediately that
\({\widehat{q}}\) is a martingale under
\({\widehat{{\mathbb {Q}}}}\). To end checking condition (ii), we have to show
\(h_T \in L^1({\mathbb {P}}) \cap L^1({\widehat{{\mathbb {Q}}}})\). Now,
\(h_T = (s_0 -a{\widehat{q}}_T)^2\), hence quadratic in
\({\widehat{q}}_T\). Since under both probability measures
\({\widehat{q}}_T\) is a Gaussian random variable, we have
\(q_T ^2 \in L^1({\mathbb {P}}) \cap L^1({\widehat{{\mathbb {Q}}}})\), which gives the desired property. We pass to condition (iii) in Definition
2.1. First,
\({\widehat{\Delta }}\) is trivially a progressively measurable process with real values. For the integrability property, since both
\({\widehat{u}}\) and
\({\widehat{\Delta }}\) are linear in
\({\widehat{q}}_t\), we are again reduced to the square integrability
\({\mathbb {E}}[\int _0 ^T {\widehat{q}}_t ^2 dt]<\infty \), which has been proved just before. To conclude this part of the proof, it remains to check that, given
\(({\widehat{u}}, {\widehat{z}})\) as above,
\({\widehat{h}}_t := h^{{\widehat{u}},{\widehat{z}}}_t = {\mathbb {E}}^{{\widehat{{\mathbb {Q}}}}} [ h_T| {\mathcal {F}}_t] = {\mathbb {E}}^{{\widehat{{\mathbb {Q}}}}} [ h_T] + \int _0 ^t {\widehat{\Delta }}_s dq_s\) a.s. under
\({\widehat{{\mathbb {Q}}}}\), for all
\(t \in [0,T]\). This can be done by direct computation as follows: applying Itô’s formula to
\({\widehat{h}}_t\) in (
2.17) we get
$$\begin{aligned} d{\widehat{h}}_t = -2a(s_0 -a{\widehat{q}}_t)d{\widehat{q}}_t \end{aligned}$$
whence, in integral form,
$$\begin{aligned} {\widehat{h}}_t = {\widehat{h}}_0 -2a \int _0 ^t (s_0 -a{\widehat{q}}_s)d{\widehat{q}}_s = {\widehat{h}}_0 + \int _0 ^t {\widehat{\Delta }}_s d{\widehat{q}}_s. \end{aligned}$$
Moreover, one easily find
$$\begin{aligned} \widehat{{\mathbb {E}}} [h_T] = \widehat{{\mathbb {E}}} [(s_0 -a{\widehat{q}}_T)^2] = s_0^2 - 2as_0 q_0 + a^2 \widehat{{\mathbb {E}}}[{\widehat{q}}_T ^2], \end{aligned}$$
where
\(\widehat{{\mathbb {E}}}\) denotes the expectation with respect to the measure
\({\widehat{{\mathbb {Q}}}}\). Using Itô’s isometry, we also have
$$\begin{aligned} \widehat{{\mathbb {E}}}[{\widehat{q}}_T ^2] = q_0 ^2 + \int _0 ^T \sigma ^2 \left( 1+\frac{\sigma ^2}{g}D(t)\right) dt, \end{aligned}$$
which leads to the remaining property in (iii).