A general HJM framework for multiple yield curve modelling

Abstract

We propose a general framework for modelling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor’s length. When the driving semimartingale is an affine process, we obtain a flexible and tractable Markovian structure. Finally, we show that the proposed framework allows unifying and extending several recent approaches to multiple yield curve modelling.

Appendices

Appendix A: Pricing under collateral and FRA rates

Let us here briefly review pricing under perfect collateralization for general derivatives which we then apply to the pricing of FRAs. For a more detailed discussion on general valuation with collateralization and funding costs, we refer to the growing literature on this topic, e.g. [3] and the references therein. We here follow closely [23, Sect. 2.2].

Throughout, let $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t\geq0}, \mathbb {P})$ be a filtered probability space, where ℙ stands for the statistical/historical probability measure. We consider OIS zero coupon bonds as basic traded instruments, which play the role of risk-free zero coupon bonds in the classical setting. In order to guarantee no arbitrage, we assume that

• (i) There exists an OIS (risk-free) bank account denoted by $(B_{t})_{t\geq0}$ and such that $B_{t}=\exp(\int_{0}^{t} r_{s} \,ds)$, where $r$ denotes the OIS short rate;

• (ii) There exists an equivalent probability measure ℚ such that the OIS bonds for all maturities are ℚ-martingales when denominated in units of the OIS bank account.⁷

Let now $X$ be an $\mathcal{F}_{T}$-measurable payoff of some derivative security. We assume here a perfect collateral agreement where 100 % of the derivative’s present value $V_{t}$ is posted in the collateral at any time $t < T$. The receiver of the collateral can invest it at the risk-free rate $r$, corresponding to the OIS short rate, and has to pay an agreed collateral rate $r^{c}_{t}$ to the poster of the collateral. Applying risk neutral pricing, we obtain for the present value of the collateralized transaction the expression

$$V_{t}=\mathbb{E}^{\mathbb{Q}}\bigg[ e^{-\int_{t}^{T} r_{s} \,ds}X+\int_{t}^{T} e^{-\int_{t}^{s} r_{u} \,du} (r_{s} -r^{c}_{s})V_{s} \,ds \bigg| \mathcal{F}_{t}\bigg]. $$

As shown in [23, Appendix A], this formula is equivalent to

$$V_{t}=\mathbb{E}^{\mathbb{Q}}\Big[ e^{-\int_{t}^{T} r^{c}_{s} \,ds}X \Big| \mathcal {F}_{t}\Big]. $$

Assuming that the collateral rate $r^{c}$ corresponds to the OIS short rate $r$, which is usually the case, we obtain the classical risk neutral valuation formula.

Since market quotes of FRAs correspond to perfectly collateralized contracts, where the collateral rate $r^{c}$ is assumed to be the OIS short rate $r$, the above pricing approach is applied for the definition of FRA rates. As in classical interest rate theory, the FRA rate, denoted by $L_{t}(T,T+\delta)$, is the rate $K$ fixed at time $t$ such that the value of the FRA contract, whose payoff at time $T+\delta$ is given by $\delta(L_{T}(T,T+\delta)- K)$, is 0. Therefore, it holds that, for all $t\in[0,T]$ and $T\geq0$,

$$\mathbb{E}^{\mathbb{Q}}\big[e^{-\int_{t}^{T+\delta} r_{s} \,ds}\big(L_{T}(T,T+\delta)-K\big) \big| \mathcal{F}_{t}\big] \stackrel{!}{=}0. $$

Hence by Bayes’ formula,

$$\begin{aligned} L_{t}(T,T+\delta)=\mathbb{E}^{\mathbb{Q}^{T+\delta}}[L_{T}(T,T+\delta) | \mathcal{F}_{t}], \end{aligned}$$

where $\mathbb{Q}^{T+\delta}$ denotes the $(T+\delta)$-forward measure associated with the numéraire $B(\cdot,T+\delta)$ and density process $\frac{d\mathbb{Q}^{T+\delta}}{d\mathbb {Q}}|_{\mathcal {F}_{t}}=\frac{B(t,T+\delta)}{B_{t} B(0,T+\delta)}$. In particular, this provides a rigorous justification for the market practice of taking the expression (2.1) as the definition of fair FRA rates.

Appendix B: Foreign exchange analogy

For simplicity of presentation, let us consider a fixed tenor $\delta$ and define artificial “risky” bond prices $B^{\delta}(t,T)$ at time $t$ and maturity $T$ for the tenor $\delta$ by the relation, for all $t\leq T$ and $T\geq0$,

$$L_{t}(T,T+\delta)=:\frac{1}{\delta} \left(\frac{B^{\delta }(t,T)}{B^{\delta }(t,T+\delta)}-1\right). $$

While the family $\{(B(t,T))_{t\in[0,T]},T\geq0\}$ represents prices of domestic risk-free bonds (in units of the domestic currency), $\{ (B^{\delta}(t,T))_{t\in[0,T]},T\geq0\}$ can be thought of as representing prices of zero-coupon bonds of a foreign “risky” economy, expressed in units of the foreign currency.

According to this foreign exchange analogy, one is naturally led to look at the ratio

$$ R^{\delta}(t,T):=\frac{B(t,T)}{B^{\delta}(t,T)}, $$

for $t\leq T$ and $T\geq0$, where $B^{\delta}(t,T)$ (resp., $B(t,T)$) has here to be seen as the discount factor for the foreign (resp., domestic) economy.⁸ Note also that $R^{\delta}(T,T)=1$, for all $T\geq0$. Following the presentation in [56, Sect. 4.2.1], the quantity $R^{\delta}(t,T)$ corresponds to the forward exchange premium between the domestic and the foreign currency over the time interval $[t,T]$. Indeed, in standard foreign exchange markets, there is the following no arbitrage relation between the spot exchange rate $Q_{t}$ (domestic price of one unit of the foreign currency) and the forward exchange rate $F(t,T)$ (forward price in domestic currency of one unit of the foreign currency paid at time $T$): we have

$$\frac{Q_{t}}{F(t,T)}=\frac{B(t,T)}{B^{\delta}(t,T)}=R^{\delta}(t,T). $$

The multiplicative spread $S^{\delta}(t,T)$ introduced in (2.2) corresponds now to

$$S^{\delta}(t,T) = \frac{1+\delta L_{t}(T,T+\delta)}{1+\delta L^{D}_{t}(T,T+\delta)} =\frac{B^{\delta}(t,T)}{B(t,T)}\frac{B(t,T+\delta)}{B^{\delta }(t,T+\delta)} =\frac{R^{\delta}(t,T+\delta)}{R^{\delta}(t,T)}, $$

for all $t\leq T$ and $T\geq0$, while the spot multiplicative spread is simply given by

$$S^{\delta}(T,T)=R^{\delta}(T,T+\delta), $$

for all $T\geq0$. In particular, note that $R^{\delta}(T,T+\delta)$ corresponds exactly to the quantity $Q^{\delta}_{T}$ considered in Sect. 2.1 which thus has the interpretation of a foreign exchange premium over $[T,T+\delta]$.

Since Libor rates reflect the overall credit risk of the Libor panel, the exchange rate premium $R^{\delta}(t,T+\delta)$ can be seen as a market valuation (at time $t$) of the riskiness of the foreign economy, i.e. of the credit and liquidity quality of the current Libor panel over the period $[t,T+\delta]$. Moreover, according to the same interpretation, the quantity

$$S^{\delta}(t,T)=R^{\delta}(t,T+\delta)/R^{\delta}(t,T)=\mathbb {E}_{\mathbb{Q}^{T}}[R(T,T+\delta)|\mathcal{F}_{t}] $$

is thus an expectation of the riskiness of the future Libor panel over the future time period $[T,T+\delta]$, as seen from the market at time $t$ (calculated as the relative riskiness of the current Libor panel over the period $[t,T+\delta]$ relative to the one over $[t,T]$). For instance, a large value of $S^{\delta}(t,T)$ would mean that the market anticipates a worsening of the credit quality of the Libor panel on $[T,T+\delta]$ compared to the credit quality on $[t,T]$ as seen at time $t$.

In this sense, the multiplicative spread $S^{\delta}(t,T)$ is a rather natural quantity to model in a multiple curve setting, because it represents the market’s expectation at time $t$ (being computed from financial instruments traded at date $t$) of the credit and liquidity quality of the Libor panel over $[T,T+\delta]$.

Appendix C: Local independence and semimartingale decomposition

In this section, we let $(X,Y)$ be a general Itô semimartingale taking values in $\mathbb{R}^{d+n}$ and denote by $\varPsi^{X,Y}$ its local exponent and by $\varPsi^{X}$ and $\varPsi^{Y}$ the local exponents of $X$ and $Y$, respectively, and let $\mathcal{U}^{X,Y}$ be defined as in Definition 3.2. In view of [39, Lemma A.11], the following definition is equivalent to the notion of local independence as given in [39, Definition A.10].

Definition C.1

We say that $X$ and $Y$ are locally independent if outside a $d\mathbb{Q}\otimes dt$-null set, it holds that

$$\varPsi_{t}^{X,Y}(u_{t},v_{t})(\omega)=\varPsi_{t}^{X}(u_{t})(\omega)+\varPsi _{t}^{Y}(v_{t})(\omega), \quad\text{for all }(u,v)\in\mathcal{U}^{(X,Y)}. $$

Following [39, Appendix A.3], let us recall the notion of semimartingale decomposition of $Y$ relative to $X$. We denote by $c^{Y,X}$ and $c^{X}$ the second local characteristic of $(Y,X)$ and $X$, respectively, and by $K^{Y,X}$ and $K^{X}$ the third local characteristic of $(Y,X)$ and $X$, respectively. Denote also by $\mu^{Y,X}$ the jump measure of $(Y,X)$. Supposing that $1\in\mathcal{U}^{Y}$ (i.e. $Y$ is exponentially special, see Proposition 3.3), let

$$\begin{aligned} Y^{\parallel,i} := \log\mathcal{E}\bigg(&\int_{0}^{\cdot}\bigl(c_{t}^{Y^{i},X}(c_{t}^{X})^{-1}\bigr)\,dX^{c}_{t} \\ &{}+\int_{0}^{\cdot}\int (e^{y^{i}}-1)\textbf{1}_{\{ x\neq0\}}\bigl(\mu^{Y,X}(dy,dx,dt)-K^{Y,X}_{t}(dy,dx)\,dt\bigr)\bigg), \end{aligned}$$

(C.1)

for $i=1,\ldots,n$, where $(c^{X})^{-1}$ denotes the pseudoinverse of the matrix $c^{X}$ and $X^{c}$ is the continuous local martingale part of $X$ (see [36, Proposition I.4.27]). We call $Y^{\parallel }:=(Y^{\parallel,1},\ldots,Y^{\parallel,n})^{\top}$ the dependent part of $Y$ relative to $X$ and $Y^{\perp}:=Y-Y^{\parallel}$ the independent part of $Y$ relative to $X$. The following lemma corresponds to [39, Lemma A.22 and Lemma A.23].

Lemma C.2

Let $(X,Y)$ be an $\mathbb{R}^{d\times n}$-valued Itô semimartingale and suppose that $1\in\mathcal{U}^{Y}$. Then the following hold:

• (i) $Y\mapsto Y^{\parallel}$ is a projection, in the sense that $(Y^{\parallel})^{\parallel}=Y^{\parallel}$.

• (ii) If $Z$ is an Itô semimartingale locally independent of $X$, then $(Z+Y)^{\parallel}=Y^{\parallel}$.

• (iii) $\exp(Y^{\parallel,i})$ is a local martingale, for all $i=1,\ldots,n$.

• (iv) $Y^{\perp}$ and $(Y^{\parallel},X)$ are locally independent semimartingales.

Appendix D: Proofs of the results of Sect. 4

Proof of Proposition 4.4

Let us fix any $i\in\{0,1,\ldots,m\}$. By the same argument as in [19, Corollary 5.12], it can be shown that $\kappa_{4}(H^{\lambda}_{m+1}) \subseteq H^{\lambda,0}_{1}$ and $\kappa_{j}^{i}(H^{\lambda}_{m+1}) \subseteq H^{\lambda,0}_{1}$ for $j=1,2$. In the sequel, $C$ always denotes a positive constant which can vary from line to line. The following estimates can be derived similarly as in the proof of [22, Proposition 3.2] (to which we refer the reader for more details), by using Assumption 4.3, the Hölder inequality and [22, Theorem 2.1].

For $\kappa_{3}^{i}(h)$ and $\kappa_{5}^{i}(h)$, we have for all $h \in H_{m+1}^{\lambda}$, $\xi\in\mathbb{R}^{d}$ and $s \in\mathbb{R}_{+}$ that

$$ |(\zeta^{i0}(h)(s))^{\top}\xi|\leq C \| \zeta^{i0}(h)\|_{\lambda ,d}\| \xi\|_{d} \quad\text{and}\quad |(\zeta^{0}(h)(s))^{\top}\xi| \leq C\| \zeta^{0}(h)\|_{\lambda,d}\| \xi\|_{d}, $$

and for all $\xi\in\mathbb{R}^{d}$, $\hat{\xi}\in\mathbb{R}^{n}$ and $s \in\mathbb {R}_{+}$ that

$$\begin{aligned} \bigl|e^{u_{i}^{\top} \hat{\xi}+ (Z^{i0}(h)(s))^{\top}\xi}-1\bigr|&\leq Ce^{\|u_{i}\|_{n}\|\hat{\xi}\|_{n}+(C_{0}+C_{i}) \|\xi\|_{d}} \big(\|u_{i}\|_{n}\| \hat{\xi}\| _{n}+\| \zeta^{i0}(h)\|_{\lambda,d}\| \xi\|_{d}\big),\\ \bigl|e^{-(Z^{0} (h)(s))^{\top}\xi}-1\bigr|&\leq Ce^{C_{0} \|\xi\|_{d}}\| \zeta^{0}(h)\|_{\lambda,d}\| \xi\|_{d}. \end{aligned}$$

Together with (4.9) and (4.10), this shows that we have $\lim_{s \to\infty}\kappa_{3}^{i}(h)(s)=0$ and $\lim_{s \to\infty}\kappa_{5}^{i}(h)(s)=0$. Moreover, we have

$$\begin{aligned} &\int_{\mathbb{R_{+}}} \bigg(\int\Big(\big(\zeta^{i0}(h)(s)\big)^{\top }\xi\Big)^{2}\bigl(e^{u_{i}^{\top} \hat{\xi}+ (Z^{i0}(h)(s))^{\top }\xi}\bigr)K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)^{2} e^{\lambda s} \,ds \\ & \quad\leq C (M_{0}+M_{i})^{4}K_{i}^{2},\\ &\int_{\mathbb{R_{+}}} \bigg( \int\Big(\big(\zeta^{0}(h)(s)\big)^{\top}\xi \Big)^{2} e^{-(Z^{0} (h)(s))^{\top}\xi}F(d\xi)\bigg)^{2} e^{\lambda s} \,ds \leq CM_{0}^{4}K_{0}^{2},\\ &\int_{\mathbb{R_{+}}} \bigg(\int\frac{d}{ds}\big(\zeta ^{i0}(h)(s)\big)^{\top}\xi\bigl(e^{u_{i}^{\top} \hat{\xi}+ (Z^{i0}(h)(s))^{\top }\xi}-1\bigr)K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)^{2}e^{\lambda s} \,ds \\ & \quad\leq C (M_{0}+M_{i})^{2}K_{i},\\ &\int_{\mathbb{R_{+}}} \left(\int\frac{d}{ds}\bigl(\zeta ^{0}(h)(s)\bigr)^{\top}\xi\bigl(e^{-(Z^{0} (h))^{\top}\xi}-1\bigr)F(d\xi)\right)^{2} e^{\lambda s} \,ds \leq CM_{0}^{4}K_{0}^{2}. \end{aligned}$$

In view of the form of $\partial_{s} \kappa^{i}_{3}(h)$ and $\partial_{s} \kappa _{5}(h)$ as given in (4.11), (4.12), this implies that $\kappa_{j}^{i}(H^{\lambda}_{m+1}) \subseteq H^{\lambda,0}_{1}$ for $j=3,5$. We have thus shown that

$$\kappa^{i}(H^{\lambda}_{m+1}) \subseteq H^{\lambda,0}_{1}. $$

For $h_{1}, h_{2} \in H^{\lambda}_{m+1}$ we obtain

$$\begin{aligned} \| \kappa^{i}_{1}(h_{1})-\kappa^{i}_{1}(h_{2})\|_{\lambda,1}&\leq C(L_{0}+L_{i})\| h_{1}-h_{2}\|_{\lambda, m+1},\\ \| \kappa^{i}_{2}(h_{1})-\kappa^{i}_{2}(h_{2})\|_{\lambda,1}& \leq C(2M_{i}+2M_{0})(L_{0}+L_{i})\|h_{1}-h_{2}\|_{\lambda, m+1},\\ \| \kappa_{4}(h_{1})-\kappa_{4}(h_{2})\|_{\lambda,1}& \leq C2M_{0}L_{0}\|h_{1}-h_{2}\|_{\lambda, m+1}. \end{aligned}$$

Furthermore, due to (4.11) and (4.12), we can estimate

$$\begin{aligned} \| \kappa^{i}_{3}(h_{1})-\kappa^{i}_{3}(h_{2})\|_{\lambda,1}^{2}& \leq4 (I^{i}_{1}+I^{i}_{2}+I^{i}_{3}+I^{i}_{4}),\\ \| \kappa_{5}(h_{1})-\kappa_{5}(h_{2})\|_{\lambda,1}^{2}& \leq4 (J_{1}+J_{2}+J_{3}+J_{4}), \end{aligned}$$

where

$$\begin{aligned} I^{i}_{1} &=\int_{\mathbb{R}_{+}}\bigg(\int\Big(\big(\zeta^{i0}(h_{1})(s)\big)^{\top }\xi\Big)^{2}e^{u_{i}^{\top} \hat{\xi}}\\ &\phantom{I^{i}_{1}=\int_{\mathbb{R}_{+}}\bigg(\int}{}\times\bigl(e^{(Z^{i0}(h_{1})(s))^{\top}\xi} -e^{(Z^{i0}(h_{2})(s))^{\top}\xi}\bigr)K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)^{2} e^{\lambda s} \,ds, \\ I^{i}_{2}&=\int_{\mathbb{R}_{+}}\bigg(\int e^{u_{i}^{\top} \hat{\xi}+ (Z^{i0}(h_{2})(s))^{\top}\xi}\bigg(\Big(\big(\zeta^{i0}(h_{1})(s)\big)^{\top }\xi\Big)^{2}\\ &\phantom{I^{i}_{2}=\int_{\mathbb{R}_{+}}\bigg(\int e^{u_{i}^{\top} \hat {\xi}+ (Z^{i0}(h_{2})(s))^{\top}\xi}}{} -\Big(\big(\zeta^{i0}(h_{2})(s)\big)^{\top}\xi\Big)^{2}\bigg) K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)^{2}e^{\lambda s} \,ds, \\ I^{i}_{3}&=\int_{\mathbb{R}_{+}}\bigg(\int\frac{d}{ds}\bigl(\zeta ^{i0}(h_{1})(s)\bigr)^{\top}\xi e^{u_{i}^{\top} \hat{\xi}}\\ &\phantom{I^{i}_{3}=\int_{\mathbb{R}_{+}}\bigg(\int}{}\times\bigl(e^{(Z^{i0}(h_{1})(s))^{\top}\xi} -e^{(Z^{i0}(h_{2})(s))^{\top}\xi}\bigr)K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)e^{\lambda s} \,ds,\\ I^{i}_{4}&=\int_{\mathbb{R}_{+}}\bigg(\int\Bigl(e^{u_{i}^{\top} \hat{\xi}+ (Z^{i0}(h_{2})(s))^{\top}\xi}-1\Bigr) \bigg(\frac{d}{ds}\bigl(\zeta ^{i0}(h_{1})(s)\bigr)^{\top}\xi\\ &\phantom{I^{i}_{4}=\int_{\mathbb{R}_{+}}\bigg(\int}{} -\frac{d}{ds}\big(\zeta^{i0}(h_{2})(s)\big)^{\top}\xi\bigg)K^{\hat{Y} ,X}(d\hat{\xi}, d\xi)\bigg)^{2}e^{\lambda s} \,ds, \\ J_{1}&=\int_{\mathbb{R}_{+}}\bigg(\int\Big(\big(\zeta^{0}(h_{1})(s)\big)^{\top }\Big)^{2} \\ &\phantom{J_{1}=\int_{\mathbb{R}_{+}}\bigg(}{}\times\bigl(e^{-(Z^{0} (h_{1})(s))^{\top}\xi} -e^{-(Z^{0} (h_{2})(s))^{\top}\xi}\big)F(d\xi)\bigg)^{2}e^{\lambda s} \,ds,\\ J_{2}&=\int_{\mathbb{R}_{+}}\bigg(\int e^{-(Z^{0} (h_{2})(s))^{\top}\xi}\\ &\phantom{J_{2}=\int_{\mathbb{R}_{+}}\bigg(}{}\times\bigg(\Big(\big(\zeta ^{0}(h_{1})(s)\big)^{\top}\xi\Big)^{2} -\Big(\big(\zeta^{0}(h_{2})(s)\big)^{\top}\xi\Big)^{2}\bigg)F(d\xi )\bigg)^{2} e^{\lambda s} \,ds,\\ J_{3}&=\int_{\mathbb{R}_{+}}\bigg(\int\frac{d}{ds}\big(\zeta ^{0}(h_{1})(s)\big)^{\top}\xi\\ &\phantom{J_{3}=\int_{\mathbb{R}_{+}}\bigg(}{}\times\big(e^{-(Z^{0} (h_{1})(s))^{\top}\xi} -e^{-(Z^{0} (h_{2})(s))^{\top}\xi}\big)F(d\xi)\bigg)^{2}e^{\lambda s} \,ds,\\ J_{4}&=\int_{\mathbb{R}_{+}}\bigg(\int\big(e^{-(Z^{0} (h_{2})(s))^{\top }\xi }-1\big)\\ &\phantom{J_{4}=\int_{\mathbb{R}_{+}}\bigg(}{}\times\bigg(\frac {d}{ds}\big(\zeta^{0}(h_{1})(s)\big)^{\top}\xi -\frac{d}{ds}\big(\zeta^{0}(h_{2})(s)\big)^{\top}\xi\bigg)F(d\xi )\bigg)^{2}e^{\lambda s} \,ds. \end{aligned}$$

We get for all $\xi\in\mathbb{R}^{d}$, $s \in\mathbb{R}_{+}$ that

$$\begin{aligned} \bigl|e^{(Z^{i0}(h_{1})(s))^{\top}\xi}-e^{(Z^{i0}(h_{2}(s)))^{\top}\xi }\bigr| \leq& Ce^{(C_{0}+C_{i})\| \xi\|_{d}}\bigl( \|\zeta^{i}(h_{1})-\zeta^{i}(h_{2})\|_{\lambda,d}\bigr. \\ &\bigl.{}+\|\zeta^{0}(h_{1})-\zeta^{0}(h_{2})\|_{\lambda,d}\bigr)\| \xi \| _{d},\\ \bigl|e^{-(Z^{0} (h_{1}))^{\top}\xi}-e^{-(Z^{0} (h_{2}))^{\top}\xi}\bigr| \leq& Ce^{C_{0}\| \xi\|_{d}}\|\zeta^{0}(h_{1})-\zeta^{0}(h_{2})\|_{\lambda ,d}\| \xi\|_{d}. \end{aligned}$$

Therefore,

$$\begin{aligned} I^{i}_{1}&\leq C\int_{\mathbb{R}_{+}}\bigg(\int\Big(\big(\zeta ^{i0}(h_{1})(s)\big)^{\top}\xi\Big)^{2}e^{\|u_{i}\|_{n} \|\hat{\xi}\| _{n}+(C_{0}+C_{i})\| \xi\|_{d}}\\ &\quad{}\times\bigl( \|\zeta^{i}(h_{1})-\zeta^{i}(h_{2})\|_{\delta,d}+\|\zeta^{0}(h_{1})-\zeta ^{0}(h_{2})\|_{\lambda,d}\bigr)\| \xi\|_{d} K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)^{2} e^{\lambda s} \,ds\\ & \leq CK^{2}_{i}(M_{i}^{4}+M_{0}^{4})(L^{2}_{i}+L_{0}^{2})\|h_{1}-h_{2}\|^{2}_{\lambda,m+1},\\ J_{1}&\leq C\int_{\mathbb{R}_{+}}\bigg(\int \Big(\big(\zeta ^{0}(h_{1})(s)\big)^{\top}\xi\Big)^{2} e^{C_{0}\| \xi\|_{d}}\|\zeta ^{0}(h_{1})-\zeta ^{0}(h_{2})\|_{\lambda,d}\| \xi\|_{d}F(d\xi)\bigg)^{2}e^{\lambda s} \,ds\\ &\leq C K^{2}_{0}M_{0}^{4}L_{0}^{2}\|h_{1}-h_{2}\|^{2}_{\lambda,m+1}. \end{aligned}$$

Moreover, for every $s \in\mathbb{R}_{+}$, we obtain

$$\begin{aligned} &\int e^{\|u_{i}\|_{n} \|\hat{\xi}\|_{n}+ (C_{0}+C_{i})\|\xi\|_{d}} \Big(\big(\zeta^{i0}(h_{1})(s)\big)^{\top}\xi+\big(\zeta ^{i0}(h_{2})(s)\big)^{\top}\xi\Big)^{2} K^{\hat{Y},X}(d\hat{\xi}, d\xi) \\ &\quad\leq2C(M_{i}^{2}+M_{0}^{2})K_{i},\\ &\int e^{C_{0}\|\xi\|_{d}} \Big(\big(\zeta^{0}(h_{1})(s)\big)^{\top}\xi +\big(\zeta^{0}(h_{2})(s)\big)^{\top}\xi\Big)^{2}F(d\xi)\leq2CM_{0}^{2}K_{0}. \end{aligned}$$

Hence,

$$\begin{aligned} I^{i}_{2}&\leq2C(M_{i}+M_{0})K_{i}\int e^{\|u_{i}\|_{n} \|\hat{\xi}\|_{n}+ (C_{0}+C_{i})\|\xi \|_{d}}\\ &\quad{} \times\int_{\mathbb{R}_{+}}\Big(\bigl(\zeta ^{i}(h_{1})(s)-\zeta^{i}(h_{2})(s)\bigr)^{\top}\xi\\ &\phantom{=:\times\int_{\mathbb{R}_{+}}\Big(}{}+\bigl(\zeta ^{0}(h_{2})(s)-\zeta^{0}(h_{1})(s)\bigr)^{\top}\xi\Big)^{2}e^{\lambda s} ds K^{\hat{Y},X}(d\hat{\xi}, d\xi)\\ &\leq2C(M_{i}+M_{0})K_{i}^{2} (L_{0}^{2}+L_{i}^{2})\|h_{1}-h_{2}\|^{2}_{\lambda,m+1},\\ J_{2} &\leq2CM_{0}K_{0}\int e^{C_{0}\|\xi\|_{d}}\int_{\mathbb{R}_{+}}\left (\bigl(\zeta^{0}(h_{2})(s)-\zeta^{0}(h_{1})(s)\bigr)^{\top}\xi\right )^{2}e^{\lambda s} ds F(d\xi)\\ &\leq2CM_{0}K_{0}^{2} L_{0}^{2}\|h_{1}-h_{2}\|^{2}_{\lambda,m+1}. \end{aligned}$$

Moreover,

$$\begin{aligned} I^{i}_{3}&\leq CK_{i}(L^{2}_{i}+L_{0}^{2})\|h_{1}-h_{2}\|^{2}_{\lambda,m+1}\int e^{\|u_{i}\| _{n} \|\hat{\xi}\|_{n}+(C_{0}+C_{i})\| \xi\|_{d}} \\ &\quad{}\times\int_{\mathbb{R}_{+}} \left(\frac{d}{ds}\bigl(\zeta ^{i0}(h_{1})(s)\bigr)^{\top}\xi\right)^{2} e^{\lambda s} \,ds K^{\hat{Y},X}(d\hat{\xi}, d\xi)\\ &\leq CK^{2}_{i}(L^{2}_{i}+L_{0}^{2})(M_{0}+M_{i})\|h_{1}-h_{2}\|^{2}_{\lambda,m+1},\\ J_{3} &\leq CK_{0}L_{0}^{2}\|h_{1}-h_{2}\|^{2}_{\lambda,m+1}\int e^{C_{0}\|\xi\| _{d}}\int _{\mathbb{R}_{+}}\left(\frac{d}{ds}\bigl(\zeta^{0}(h_{1})(s)\bigr)^{\top}\xi \right)^{2} e^{\lambda s} \,ds F(d\xi)\\ &\leq CK^{2}_{0}L_{0}^{2}M_{0}\|h_{1}-h_{2}\|^{2}_{\lambda,m+1}. \end{aligned}$$

Finally, we have

$$\begin{aligned} I^{i}_{4}&\leq C\int_{\mathbb{R}_{+}}\bigg(\int e^{\|u_{i}\|_{n}\|\hat{\xi}\| _{n}+(C_{0}+C_{i}) \|\xi\|_{d}}\bigl(\|u_{i}\|_{n}\|\hat{\xi}\|_{n}+\| \zeta ^{i0}(h)\| _{\lambda,d}\|\xi\|_{d}\bigr)\\ &\phantom{\leq C\int_{\mathbb{R}_{+}}\bigg(}{} \times\left(\frac {d}{ds}\bigl(\zeta^{i0}(h_{1})(s)\bigr)^{\top}\xi-\frac{d}{ds}\bigl(\zeta ^{i0}(h_{2})(s)\bigr)^{\top}\xi\right)K^{\hat{Y},X}(d\hat{\xi}, d\xi)\bigg)^{2}e^{\lambda s} \,ds\\ &\leq C(u_{i}+M_{0}+M_{i})^{2}K_{i}^{2}(L_{0}+L_{i})^{2}\|h_{1}-h_{2}\|^{2}_{\lambda ,m+1},\\ J_{4}&\leq C\int_{\mathbb{R}_{+}}\bigg(\int e^{C_{0} \|\xi\|_{d}}\| \zeta ^{0}(h)\| _{\lambda,d}\| \xi\|_{d} \\ &\phantom{\leq C\int_{\mathbb{R}_{+}}\bigg(}{}\times\bigg(\frac {d}{ds}\bigl(\zeta^{0}(h_{1})(s)\bigr)^{\top}\xi -\frac{d}{ds}\bigl(\zeta^{0}(h_{2})(s)\bigr)^{\top}\xi\bigg)F(d\xi )\bigg)^{2}e^{\lambda s} \,ds,\\ &\leq CM_{0}^{2}K_{0}^{2}L_{0}^{2}\|h_{1}-h_{2}\|^{2}_{\lambda,m+1}. \end{aligned}$$

Summing up, we have shown that there exist constants $Q_{i}>0$ such that condition (4.13) is satisfied for all $h_{1},h_{2} \in H_{m+1}^{\lambda}$. □

Proof of Proposition 4.10

For every $(\omega,\omega')\in\widetilde{\varOmega}$ and $t\geq0$, define the process $Y^{\perp}_{t}(\omega,\omega') := y_{0} + J_{t}(\omega')$, for some starting value $y_{0}\in\mathbb{R}_{+}$. Clearly, $Y^{\perp}$ is a pure jump $(\mathcal{G}_{t})$-adapted process. In order to prove the existence of a probability measure $\widetilde {\mathbb{Q}}$ such that the jump measure of $Y^{\perp}$ with respect to the two filtrations $(\mathcal{G}_{t})_{t\geq0}$ and $(\overline {\mathcal{G}}_{t})_{t\geq 0}$ is given by $K_{t}(\omega, Y^{\perp}_{t-}(\omega,\omega'),d\xi)dt$ and $\widetilde{\mathbb{Q}}|_{\mathcal{F}}=\mathbb{Q}$ holds true, we rely on [34, Theorem 3.6].

For all $\omega\in\varOmega$, $y\in\mathbb{R}_{+}$ and $t\geq0$, let us first extend the definition of $K_{t}(\omega,y, d\xi)$ as in (4.15), (4.16) to $y\in\mathbb{R}_{-}$ by requiring that it is supported on $[-|y|, \infty)$ and by setting $p_{t}(y,\omega )=p_{t}(-y, \omega)$ for $y\in\mathbb{R}_{-}$. Due to Assumption 4.8, the measure $K_{t}(\omega,Y^{\perp}_{t-}(\omega ,\omega'), d\xi)dt$ defined via the moment problem (4.15), (4.16) is a positive random measure on $\mathbb{R}_{+} \times\mathbb{R}$. Since $Y^{\perp}$ is càdlàg and $(\mathcal{G}_{t})$-adapted and since $(p_{t}(\cdot,y))_{t\geq0}$ is $(\mathcal{F}_{t})$-predictable and depends in a measurable way on $y$, the process $p_{t}(\omega, Y^{\perp}_{t-}(\omega ,\omega'))$ is $(\mathcal{G}_{t})$-predictable and the same $(\mathcal{G} _{t})$-predictability, and hence $(\overline{\mathcal {G}}_{t})$-predictability, is inherited by $K_{t}(\omega,Y^{\perp}_{t-}(\omega,\omega'), d\xi)$. Let us then define the $(\overline{\mathcal{G}}_{t})$-predictable random measure $\nu$ by

$$\nu(\widetilde{\omega},dt,d\xi)=\left\{ \textstyle\begin{array}{l@{\quad}l} K_{t}\bigl(\omega, Y^{\perp}_{t-}(\omega,\omega'),d\xi\bigr)dt, & t < T_{\infty},\\ 0, & t \geq T_{\infty}. \end{array}\displaystyle \right. $$

[34, Theorem 3.6] implies that there exists a unique probability kernel ℙ from $(\varOmega,\mathcal{F})$ to ℋ such that $\nu$ is the $(\overline{\mathcal{G}}_{t})$-compensator of the random measure $\mu$ associated with the jumps of $J$. On $(\widetilde {\varOmega}, \mathcal{G})$, we then define the probability measure $\widetilde{\mathbb{Q}}$ by setting $\widetilde{\mathbb {Q}}(d\widetilde{\omega})=\mathbb{Q}(d\omega)\mathbb{P}(\omega ,d\omega')$, whose restriction to ℱ is equal to ℚ. Moreover, since $Y^{\perp}$ is $(\mathcal{G}_{t})$-adapted and $K_{t}(\omega ,Y^{\perp}_{t-}(\omega,\omega'), d\xi)$ is $(\mathcal {G}_{t})$-predictable, the random measure $\nu$ is also the $(\mathcal{G}_{t})$-compensator of the jump measure of $J$. Since for every $\omega\in\varOmega$, $y\in\mathbb{R}_{+}$ and $t\geq 0$, the measure $K_{t}(\omega,y,d\xi)$ is supported by $[-y,\infty)$, the process $Y^{\perp}$ takes values in $\mathbb{R}_{+}$.

It remains to show that $Y^{\perp}$ is of finite activity (equivalently, that $T_{\infty}=\infty$ $\widetilde{\mathbb{Q}}$-a.s.). Since $g_{m+1}(\xi)\geq1$ for all $\xi\in\mathbb{R}$ and due to condition (4.16), it holds that

$$\begin{aligned} \mathbb{E}^{\widetilde{\mathbb{Q}}}\bigl[\mu([0,T]\times\mathbb {R})\bigr] &=\mathbb{E}^{\widetilde{\mathbb{Q}}}\bigl[\nu([0,T]\times\mathbb {R})\bigr] =\mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\int_{0}^{T} K_{t}\bigl(\omega, Y_{t-}(\omega,\omega'), \mathbb{R}\bigr)\,dt\right ]\\ &\leq\mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\int_{0}^{T}\int g_{m+1}(\xi)K_{t}\bigl(\omega, Y_{t-}(\omega,\omega'),d\xi\bigr)\,dt\right] \\ &= \mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\int_{0}^{T}p^{m+1}_{t}\bigl(\omega,Y_{t-}(\omega ,\omega')\bigr)\,dt\right] \leq\bar{H}T, \end{aligned}$$

due to the uniform boundedness of the processes $\{(p^{m+1}_{t}(\cdot ,y))_{t\geq0};y\in\mathbb{R}_{+}\}$. This implies that $\mu([0,T]\times\mathbb{R})<\infty$ $\widetilde {\mathbb{Q}}$-a.s. for each $T \geq0$, and as a consequence, $\widetilde{\mathbb{Q}}[T_{\infty}<\infty]=0$. □

Proof of Lemma 4.11

(i) For $t\geq0$, let $H$ be a bounded $\mathcal{H}_{t}$-measurable random variable, $F$ a bounded $\mathcal{F}_{t}$-measurable random variable and $A\in\mathcal{F}_{\infty}$. As can be deduced from the proof of Proposition 4.10, the $\mathcal{F}_{\infty }$-conditional law of $(\omega'(s))_{s\in[0,t]}$ under $\widetilde {\mathbb{Q}}$ is $\mathcal{F}_{t}$-measurable (compare also with [20, part (iv) of Theorem 5.1]). In particular, this means that $\mathbb{E}^{\widetilde{\mathbb {Q}}}[H|\mathcal{F}_{\infty}]=\mathbb{E} ^{\widetilde{\mathbb{Q}}}[H|\mathcal{F}_{t}]$. In turn, this implies that

$$\begin{aligned} \mathbb{E}^{\widetilde{\mathbb{Q}}}[FH\textbf{1}_{A}] &= \mathbb{E}^{\widetilde{\mathbb{Q}}}\big[F\mathbb{E}^{\widetilde {\mathbb{Q}}}[H|\mathcal{F}_{\infty}]\textbf{1}_{A}\big]\\ &= \mathbb{E}^{\widetilde{\mathbb{Q}}}\big[F\mathbb{E}^{\widetilde {\mathbb{Q}}}[H|\mathcal{F}_{t}]\textbf{1}_{A}\big] = \mathbb{E}^{\widetilde{\mathbb{Q}}}\big[\mathbb{E}^{\widetilde {\mathbb{Q}}}[FH|\mathcal{F}_{t}]\textbf{1}_{A}\big]. \end{aligned}$$

By a monotone class argument, this means that $\mathbb{E}^{\widetilde {\mathbb{Q}}}[G|\mathcal{F} _{\infty}]=\mathbb{E}^{\widetilde{\mathbb{Q}}}[G|\mathcal{F}_{t}]$ for every bounded $\mathcal{G} _{t}$-measurable random variable $G$. It is well known (see e.g. [38, Proposition 5.9.1.1]) that the latter property is equivalent to the fact that all $(\widetilde {\mathbb{Q}},(\mathcal{F} _{t})_{t\geq0})$-martingales are also $(\widetilde{\mathbb{Q}},(\mathcal{G}_{t})_{t\geq 0})$ martingales. Due to the fact that $\widetilde{\mathbb{Q}}|_{\mathcal{F}_{\infty}}=\mathbb{Q}$, this implies that all $(\mathbb{Q},(\mathcal{F} _{t})_{t\geq0})$-local martingales are also $(\widetilde{\mathbb {Q}},(\mathcal{G}_{t})_{t\geq 0})$-local martingales. As a consequence, every $(\mathbb{Q},(\mathcal {F}_{t})_{t\geq 0})$-semimartingale is also a $(\widetilde{\mathbb{Q}},(\mathcal {G}_{t})_{t\geq0})$ semimartingale. Moreover, since semimartingale characteristics can be characterized in terms of local martingales (see e.g. [36, Theorem II.2.21]), this implies that $(X,\hat{Y})$ is a semimartingale with respect to $(\widetilde{\mathbb{Q}},(\mathcal{G}_{t})_{t\geq0})$ with unchanged characteristics.

(ii) Since $Y^{\perp}$ is a pure jump process, in order to prove its local independence with respect to $(X,\hat{Y})$, it suffices to show that $Y^{\perp}$ and $(X,\hat{Y})$ never jump together. In view of (C.1), this reduces to showing that

$$\widetilde{\mathbb{Q}}[\exists t>0 \mid\varDelta Y^{\perp}_{t}\neq 0\text{ and }\varDelta X_{t}\neq0]=0. $$

Let $\mathfrak{T}$ be the set of jump times of $X$. Since $X$ is càdlàg, the set $\mathfrak{T}$ is countable (see e.g. [36, Proposition I.1.32]) and similarly as in [39, Theorem 4.7],

$$\begin{aligned} \widetilde{\mathbb{Q}}[\exists t>0 \mid\varDelta Y^{\perp}_{t}\neq 0\text{ and }\varDelta X_{t}\neq0] &\leq\mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\sum_{t\in \mathfrak{T}}\textbf{1}_{\{\varDelta Y_{t}^{\perp}\neq0\}}\right] \\ &= \mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\sum_{t\in\mathfrak {T}}\mathbb{E}^{\widetilde{\mathbb{Q}}}[\textbf{1}_{\{ \varDelta Y_{t}^{\perp}\neq0\}}|\mathcal{F}_{\infty}]\right]=0, \end{aligned}$$

where the last equality follows from the fact that $\mathbb {E}^{\widetilde{\mathbb{Q}}}[\textbf{1} _{\{\varDelta Y_{t}^{\perp}\neq0\}}|\mathcal{F}_{\infty}]=0$ for all $t>0$ since due to Proposition 4.10, the jump measure of $Y^{\perp}$ with respect to the larger filtration $(\overline{\mathcal {G}}_{t})_{t\geq0}$ (which satisfies $\overline{\mathcal{G}}_{0}=\mathcal{F}_{\infty }\otimes\{\emptyset,\varOmega '\}$) is absolutely continuous with respect to Lebesgue measure, so that $Y^{\perp}$ does not have any fixed time of discontinuity (see e.g. [36, Lemma II.2.54]).

In order to prove the last assertion, note that condition (4.16) implies that condition $I(0,1)$ from [40] is satisfied, since for all $y\in\mathbb{R}_{+}$, $T\geq0$ and $i=1,\ldots,m$,

$$\begin{aligned} &\sup_{t\in[0,T]}\mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\exp \left(\int_{0}^{t}\int\bigl(e^{u_{i}\xi}(u_{i}\xi-1)+1\bigr)K_{s}\bigl(\omega,Y^{\perp }_{s-}(\omega ,\omega'),d\xi\bigr)\,ds\right)\right] \\ &\quad\leq \sup_{t\in[0,T]}\mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\exp \left((1+u_{m})\int_{0}^{t}\int g_{m+1}(\xi)K_{s}\bigl(\omega,Y^{\perp}_{s-}(\omega,\omega'),d\xi \bigr)\,ds\right)\right] \\ &\quad= \sup_{t\in[0,T]}\mathbb{E}^{\widetilde{\mathbb{Q}}}\left[\exp \left((1+u_{m})\int _{0}^{t}p^{m+1}_{s}\bigl(\omega,Y^{\perp}_{s-}(\omega,\omega')\bigr) \,dt\right)\right] \leq e^{(1+u_{m})T\bar{H}}< \infty. \end{aligned}$$

Moreover, condition (4.16) can be easily shown to imply that

$$\int_{0}^{T}\int|\xi e^{u_{i}\xi}-\xi|K_{t}\big(\omega,Y^{\perp }_{t-}(\omega ,\omega'),d\xi\big)\,dt< \infty \quad\widetilde{\mathbb{Q}}\text{-a.s.} $$

for all $T\geq0$. Hence, [40, Theorem 3.2] implies that

$$\bigg(\exp\Big(u_{i}Y^{\perp}_{t}-\int_{0}^{t}\varPsi^{Y^{\perp }}_{s}(u_{i})\,ds\Big)\bigg)_{t\in[0,T]} $$

is a uniformly integrable $(\widetilde{\mathbb{Q}},(\overline {\mathcal{G}}_{t})_{t\geq 0})$-martingale, for all $i=1,\ldots,m$. In turn, because $T\geq0$ is arbitrary, this proves the $(\widetilde{\mathbb{Q}},(\overline {\mathcal{G}}_{t})_{t\geq 0})$-martingale property of $(\exp(u_{i}Y^{\perp}_{t}-\int _{0}^{t}\varPsi ^{Y^{\perp}}_{s}(u_{i})\,ds))_{t\geq0}$, for all $i=1,\ldots,m$. Finally, since the latter process is $(\mathcal{G}_{t})$-adapted, it is also a martingale in the smaller filtration $(\mathcal{G}_{t})_{t\geq0}$. □

Proof of Theorem 4.12

Due to Lemma 4.11, the local exponent of $\hat{Y}$ with respect to the extended filtered probability space $(\varOmega,\mathcal {G},(\mathcal{G} _{t})_{t\geq0},\widetilde{\mathbb{Q}})$ is still given by $\varPsi^{\hat {Y}}$ and it holds that $Y^{\parallel}=(\hat{Y})^{\parallel}=\hat{Y}$. Since $Y^{\perp}$ and $Y-Y^{\perp}=\hat{Y}$ are locally independent (see Lemma 4.11), the consistency condition (3.11) directly follows from condition (4.15).

In order to prove the martingale property of the process in (3.10), note first that the $(\widetilde{\mathbb {Q}},(\mathcal{G}_{t})_{t\geq 0})$-martingale property of $(\exp(u_{i}Y^{\perp}_{t}-\int _{0}^{t}\varPsi ^{Y^{\perp}}_{s}(u_{i})ds))_{t\geq0}$ (see Lemma 4.11), condition (iv) in Definition 4.1 and the local independence of $Y^{\perp}$ and $(X,\hat{Y})$ on $(\varOmega,\mathcal{G},(\mathcal{G}_{t})_{t\geq0},\widetilde{\mathbb {Q}})$ imply that the process in (3.10) is a $(\widetilde{\mathbb{Q}},(\mathcal {G}_{t})_{t\geq0})$-local martingale, for all $i=1,\ldots,m$. Being a nonnegative local martingale, it is also a supermartingale by Fatou’s lemma. Hence, to establish the true martingale property, it suffices to observe that since $\overline{\mathcal{G}}_{0}=\mathcal {F}_{\infty}\otimes\{ \emptyset,\varOmega'\}$, it holds for all $T\geq0$ and $i=1,\ldots,m$ that

$$\begin{aligned} &\mathbb{E}^{\widetilde{\mathbb{Q}}}\bigg[\exp\bigg(u_{i}Y_{T}+\int _{0}^{T}\bigl(\varSigma ^{i}_{s}(T)-\widetilde{\varSigma}_{s}(T)\bigr)dX_{s} \\ &\phantom{\mathbb{E}^{\widetilde{\mathbb{Q}}}\bigg[\exp\bigg(}{} -\int_{0}^{T}\varPsi^{Y,X}_{s}\Big(\bigl(u_{i},\varSigma^{i\top }_{s}(T)-\widetilde {\varSigma}^{\top}_{s}(T)\bigr)^{\top}\Big)\,ds\bigg)\bigg] \\ &= \mathbb{E}^{\widetilde{\mathbb{Q}}}\Bigg[\exp\bigg(u_{i}\hat {Y}_{T}+\int_{0}^{T}\bigl(\varSigma ^{i}_{s}(T)-\widetilde{\varSigma}_{s}(T)\bigr)\,dX_{s} \\ &\phantom{= \mathbb{E}^{\widetilde{\mathbb{Q}}}\Bigg[\exp\bigg(}{} -\int_{0}^{T}\varPsi^{\hat{Y},X}_{s}\Big(\bigl(u_{i},\varSigma^{i\top }_{s}(T)-\widetilde {\varSigma}^{\top}_{s}(T)\bigr)^{\top}\Big)\,ds\bigg) \\ &\phantom{= \mathbb{E}^{\widetilde{\mathbb{Q}}}\Bigg[}{}\times \mathbb{E}^{\widetilde{\mathbb{Q}}}\bigg[\exp\bigg(u_{i}Y_{T}^{\perp}-\int_{0}^{T}\varPsi^{Y^{\perp}}_{s}(u_{i})\,ds\bigg)\bigg|\overline {\mathcal{G}}_{0}\bigg]\Bigg] \\ &= \mathbb{E}^{\widetilde{\mathbb{Q}}}\bigg[\exp\bigg(u_{i}\hat {Y}_{T}+\int_{0}^{T}\bigl(\varSigma ^{i}_{s}(T)-\widetilde{\varSigma}_{s}(T)\bigr)\,dX_{s}\\ &\phantom{= \mathbb{E}^{\widetilde{\mathbb{Q}}}\bigg[\exp\bigg(}{} -\int_{0}^{T}\varPsi^{\hat{Y},X}_{s}\Big(\bigl(u_{i},\varSigma^{i\top }_{s}(T)-\widetilde {\varSigma}^{\top}_{s}(T)\bigr)^{\top}\Big)\,ds\bigg)\bigg]e^{u_{i}Y^{\perp}_{0}} \\ &= \mathbb{E}^{\mathbb{Q}}[\exp(u_{i} Y_{0})], \end{aligned}$$

where in the last equality, we have used $\widetilde{\mathbb {Q}}|_{\mathcal{F}}=\mathbb{Q}$ and the $(\mathbb{Q},(\mathcal{F}_{t})_{t\geq0})$-martingale property of (4.1). □