Proof. The existence and uniqueness of solutions follow from a standard Galerkin approximation scheme combined with energy estimates and weak convergence techniques, as rigorously established in Lemma 5.4 of [21] for BSEEs with jumps. ◻

Proof of Theorem 2. The stability estimate ?? follows from the standard energy inequality derived via Itô’s formula, using the Lipschitz continuity of \(f\), \(g\) and the coercivity of \(A\). This argument parallels Theorem 5.2 in [21], where similar techniques are applied to a simplified case and minor adaptations suffice for our generalized setting. The specific case ?? is obtained by substituting \(\bar{f} = \bar{g} = \bar{\varphi} = 0\) and \((\bar{u}, \bar{q}, \bar{r}) = (0, 0, 0)\) into ?? .

To extend the solvability to the nonlinear case, fix \(f_0 \in {M}_{\mathbb{F}}^2(0, T; V^*)\) and consider the parameterized BSEE for \(\rho \in [0,1)\) \[\label{eq:1467} \begin{align} u(t) &= \varphi + \int_t^T \Big[A(s)u(s) + \rho f(s, u(s), q(s), r(s,\cdot)) + \rho g(s, u(s), q(s), r(s,\cdot)) + f_0(s)\Big]ds \\ &\quad - \int_t^T q(s)dW_s - \int_t^T \int_Z r(s,e)\,\tilde{\mu}(de,ds), \quad t \in [0,T]. \end{align}\tag{4}\] Under Assumption 1, it is clear that the coefficients \((A, \rho f + f_0, \rho g, \varphi)\) satisfy the conditions of Lemma 3 for any \(\rho \in [0,1)\), due to the boundedness and Lipschitz continuity of \(f,g\) and the fact that \(\rho < 1\).

We first assume that for some fixed \(\rho_0 \in [0,1)\), BSEE 4 admits a unique solution \((u, q, r)\) in the space \[\mathcal{H} := S_{\mathbb{F}}^2(0,T;H) \cap M_{\mathbb{F}}^2(0,T;V) \times M_\mathbb{F}^2(0,T;H) \times M_\mathscr{P}^{\nu,2}([0,T]\times Z;H)\] under the norm \[\|(u,q,r)\|_{\mathcal{H}}^2 := E \sup_{0 \leq t \leq T} \| u(t) \|_H^2 + E \int_0^T \| u(t) \|_V^2 \, dt + E \int_0^T \| q(t) \|_H^2 \, dt + E \int_0^T \int_Z \| r(t, e) \|_H^2 \nu(de) \, dt.\] Then we aim to show that for any \(\rho\) sufficiently close to \(\rho_0\), BSEE 4 also admits a unique solution in \(\mathcal{H}\).

To see this, rewriting 4 in terms of \(\rho_0\) yields \[\label{eq:1468} \begin{align} u(t)= &\varphi + \int_t^T \Big[A(s)u(s) + \rho_0 f(s, u(s), q(s), r(s,\cdot)) + \rho_0 g(s, u(s), q(s), r(s,\cdot)) \\ &+ (\rho - \rho_0)f(s, u(s), q(s), r(s,\cdot)) + (\rho - \rho_0)g(s, u(s), q(s), r(s,\cdot)) + f_0(s)\Big]ds \\ &- \int_t^T q(s)dW_s - \int_t^T \int_Z r(s,e)\,\tilde{\mu}(de,ds). \end{align}\tag{5}\] For any given triplet \((\tilde{u}, \tilde{q}, \tilde{r}) \in \mathcal{H}\), define a mapping \(\Phi:(\tilde{u}, \tilde{q}, \tilde{r})\mapsto(u, q, r)\) by BSEE \[\label{eq:1469} \begin{align} u(t) &= \varphi + \int_t^T \Big[A(s)u(s) + \rho_0 f(s, u(s), q(s), r(s,\cdot)) + \rho_0 g(s, u(s), q(s), r(s,\cdot)) \\ &\qquad + (\rho - \rho_0)f(s, \tilde{u}(s), \tilde{q}(s), \tilde{r}(s,\cdot)) + (\rho - \rho_0)g(s, \tilde{u}(s), \tilde{q}(s), \tilde{r}(s,\cdot)) + f_0(s)\Big]ds \\ &\quad - \int_t^T q(s)dW_s - \int_t^T \int_Z r(s,e)\,\tilde{\mu}(de,ds). \end{align}\tag{6}\] Bearing in mind that for \(\rho = \rho_0\) and any \((\tilde{u}, \tilde{q}, \tilde{r}) \in \mathcal{H}\), BSEE admits a unique solution \((u,q,r) \in \mathcal{H}\), we know that \(\Phi\) is a well-defined mapping from \(\mathcal{H}\) to itself.

In fact, for \(\rho\) close to \(\rho_0\), \(\Phi\) is a contraction mapping. We take \((\tilde{u}_i, \tilde{q}_i, \tilde{r}_i) \in \mathcal{H}\), \(i=1,2\), and denote \((u_i, q_i, r_i) = \Phi(\tilde{u}_i, \tilde{q}_i, \tilde{r}_i)\). Using a priori estimate ?? , together with the Lipschitz continuity of \(f,g\), we have \[\begin{align} \|(u_1-u_2,q_1-q_2,r_1-r_2)\|_{\mathcal{H}}^2 \leq C |\rho - \rho_0|^2 \left( \|(\tilde{u}_1, \tilde{q}_1, \tilde{r}_1) - (\tilde{u}_2, \tilde{q}_2, \tilde{r}_2)\|_{\mathcal{H}}^2 \right), \end{align}\] where \(C\) is a constant depending only on \((l, \lambda, \alpha, \kappa, M, T)\) and independent of \(\rho\) and \(\rho_0\). Choose \(\delta := \frac{1}{2\sqrt{C}}\). Then, for any \(\rho\) satisfying \(|\rho - \rho_0| < \delta\), the mapping \(\Phi\) is a contraction in the complete metric space \(\mathcal{H}\). By Banach’s fixed-point theorem, BSEE 4 admits a unique solution in this case. Note that since the constant \(C\) is independent of \(\rho_0\), the step size \(\delta\) stays unchanged throughout the continuation procedure.

Finally, since BSEE 4 is solvable at \(\rho = 0\) due to Lemma 3, we construct a finite sequence \(0 = \rho_0 < \rho_1 < \cdots < \rho_n = 1\) such that \(|\rho_{i+1} - \rho_i| < \delta\) for all \(i=0,1,\cdots,n-1\), and successively solve 4 for each \(\rho_i\) based on the contraction principle. This allows us to reach \(\rho = 1\) in finite steps, and in this way the existence of solution to BSEE 1 follows.

The uniqueness of solution to BSEE 1 is an immediate result from a priori estimate ?? . Indeed, if \((u_1, q_1, r_1)\) and \((u_2, q_2, r_2)\) are two solutions of 1 , then applying ?? with \(\bar{f} = f\), \(\bar{g} = g\), \(\bar{\varphi} = \varphi\) yields \[\|(u_1-u_2,q_1-q_2,r_1-r_2)\|_{\mathcal{H}}^2 \leq 0,\] which implies that the two solutions coincide with each other. ◻

Proof. We utilize the Gelfand triple framework and take \((V, H, V^*) = \big(H^1, H^0, H^{-1})\). For any \(t\in[0,T]\) and \((u, q, r) \in H^1 \times H^0 \times M^{\nu,2}(Z; H^0)\), we define some operators: \[\begin{align} &A(t)u:= D_{i}\big(a^{ij}(t)D_{j}u\big) + b^i(t)D_{i}u + c(t)u + \int_Z \big[u(\cdot + k(t,e,\cdot)) - u - k^i(t,e,\cdot)D_{i}u\big] \nu(de), \\ &f(t,u,q,r):= D_{i}\big(\sigma^i(t)q\big) + v(t)q + f_0(t), \\ &g(t,u,q,r):= \int_Z \big[r(e,\cdot + k(t,e,\cdot)) - r(e,\cdot)\big] \nu(de). \end{align}\] Actually, BSPDIE 7 is a specific type of BSEE 1 with above \((A, f, g)\). Next we verify that all conditions in Assumption 1 for \((A, f, g)\) in BSPDIE 7 are satisfied.

Step 1: Verification of Conditions (A.1)–(A.3)

The measurable condition (A.1), integrable condition (A.2), and uniform bounded condition (A.3) for the random mapping \(A\) can be directly verified by the conditions in Theorem 4.

Step 2: Coercivity of \(A\) (Condition A.4) For any \(\eta \in C_0^\infty(\mathbb{R}^d)\), we divide the estimate of the bilinear form \(\langle A(t)\eta, \eta \rangle\) for \(t \in [0,T]\) into three parts.

a. Principal term: by the uniform ellipticity of \(a^{ij}\) and integration by parts, we have \[\int_{\mathbb{R}^d} -a^{ij}(t,x) D_{i}\eta D_{j}\eta \, dx \leq -\alpha \|\nabla \eta\|_{L^2}^2 = -\alpha \|\eta\|_{H^1}^2 + \alpha \|\eta\|_{L^2}^2.\]

b. Lower-order terms: by the boundedness of \(b^i, c\) and Young’s inequality, for arbitrary \(\varepsilon>0\), we have \[\int_{\mathbb{R}^d} \big(b^i D_{i}\eta \cdot \eta + c \eta^2\big) dx \leq \varepsilon \|\eta\|_{H^1}^2 + C(\varepsilon, L) \|\eta\|_{L^2}^2.\]

c. Jump term: by the absolute value inequality, we have \[\begin{align} &\int_{\mathbb{R}^{d}}\int_{Z}[ \eta(x + k(t,e,x)) - \eta(x) - k^{i}(t,e,x) D_{i}\eta(x) ] \eta(x) \nu (de) dx\\ \leq& \int_{\mathbb{R}^{d}}\int_{Z}\eta(x + k(t,e,x))\nu(de)\eta(x) dx + \int_{Z}\int_{\mathbb{R}^{d}}\vert\eta(x)\vert ^{2}dx\nu (de) \\ &+\int_{Z}\int_{\mathbb{R}^{d}}\vert k^{i}(t,e,x) D_{i}\eta(x) \eta(x)\vert dx \nu (de). \end{align}\] For arbitrary \(\varepsilon>0\), it follows from Hölder’s inequality and Young’s inequality that \[\begin{align} &\int_{\mathbb{R}^{d}}\int_{Z}\eta(x + k(t,e,x))\nu(de)\eta(x) dx + \int_{Z}\int_{\mathbb{R}^{d}}|\eta(x)|^{2}dx\nu (de) \\ &+\int_{Z}\int_{\mathbb{R}^{d}}|k^{i}(t,e,x) D_{i}\eta(x) \eta(x)| dx \nu (de) \\ \leq &\frac{1}{2}\int_{\mathbb{R}^{d}}\left|\int_{Z}\eta(x + k(t,e,x)) \nu(de)\right|^{2}dx + \varepsilon\|\eta\|_1^2 + C(L,\varepsilon)\|\eta\|_0^2. \end{align}\] Based on the property of the measure \(\nu\) on \(Z\), we have \[\begin{align} &\frac{1}{2}\int_{\mathbb{R}^{d}}\vert \int_{Z}\eta(x + k(t,e,x)) \nu(de)\vert ^{2}dx+\varepsilon\|\eta\|_1^2+C(L,\varepsilon)\|\eta\|_0^2\\ \leq& \frac{1}{2}\nu(Z)\int_{Z}\int_{\mathbb{R}^{d}}\vert\eta(x + k(t,e,x))\vert ^{2}dx\nu(de)+\varepsilon\|\eta\|_1^2+C(L,\varepsilon)\|\eta\|_0^2. \end{align}\] Set \(y = x + k(t,e,x)\), and we do the change of variables. Noticing the Jacobian determinant \(\left|\det\left(I + \partial_x k(t,e,x)\right)\right| \geq \delta>0\), we have \[\begin{align} &\frac{1}{2}\nu(Z)\int_{Z}\int_{\mathbb{R}^{d}}\vert\eta(x + k(t,e,x))\vert ^{2}dx\nu(de)+\varepsilon\|\eta\|_1^2+C(L,\varepsilon)\|\eta\|_0^2\\ =&\frac{1}{2}\nu(Z)\int_{Z}\int_{\mathbb{R}^{d}}\vert\eta(y)\vert ^{2}\vert\det (I + \partial_{x}k(t,e,x))\vert^{-1}dy\nu(de)+\varepsilon\|\eta\|_1^2+C(L,\delta,\varepsilon)\|\eta\|_0^2\\ \leq&\frac{1}{2}\frac{\nu(Z) ^{2}}{\delta }\|\eta\|_0^2+\varepsilon\|\eta\|_1^2+C(L,\delta,\varepsilon)\|\eta\|_0^2, \end{align}\] which implies \[\begin{align} &\int_{\mathbb{R}^{d}}\int_{Z}\big[ \eta(x + k(t,e,x)) - \eta(x) - k^{i}(t,e,x) D_{i}\eta(x) \big] \eta(x) \nu(de) \, dx \notag \\ \leq\; &\frac{1}{2}\frac{\nu(Z)^2}{\delta} \|\eta\|_0^2 + \varepsilon \|\eta\|_1^2 + C(L,\delta,\varepsilon)\|\eta\|_0^2. \label{eq:346131} \end{align}\tag{11}\]

In conclusion, choosing \(\varepsilon = {\alpha\over4}\) and combining estimates in above \(\boldsymbol{a,b,c}\), we have the desired coercivity \[\langle A(t)\eta, \eta \rangle \leq -\frac{\alpha}{2} \|\eta\|_1^2 + C \|\eta\|_0^2.\]

Step 3: Lipschitz Continuity of \(f\) and \(g\) (Condition A.5)

Take any \((u_1, q_1, r_1), (u_2, q_2, r_2) \in H^1 \times H^0 \times M^{\nu,2}(Z; H^0)\), and we verify the Lipschitz continuity of \(f\) and \(g\) in turn. For \(f\), due to the bound \((\sigma^{i}(t,x))(\sigma^{i}(t,x))^{*} \leq \kappa^2 I\), for any \(u \in H^1\), it turns out that \[\begin{align} &|\langle f(t,u_1,q_1,r_1) - f(t,u_2,q_2,r_2), u \rangle| \\ =& |\langle D_{i}(\sigma^i(t)(q_1 - q_2)) + v(t)(q_1 - q_2), u \rangle| \\ =& \left| -\int_{\mathbb{R}^d} \sigma^i(t)(q_1 - q_2)(x) \cdot D_{i} u(x) \, dx + \int_{\mathbb{R}^d} v(t)(q_1 - q_2)(x) \cdot u(x) \, dx \right| \\ \leq& \kappa \|q_1 - q_2\|_H \|u\|_V + C \|q_1 - q_2\|_H \|u\|_H. \end{align}\] For \(g\), the Lipschitz continuity follows immediately from \[\begin{align} \label{eq:34615} &&\Vert g( t,u_1,q_1,r_1) -g(t,u_2,q_2,r_2) \Vert _0\nonumber\\ &=&\|\int_{Z}\bigg(\Big(r_1\big(t, e,\cdot+k(t,e,\cdot) \big)-r_2\big(t, e,\cdot+k(t,e,\cdot) \big)\Big) -\big(r_1(t, e,\cdot)-r_2(t, e,\cdot)\big) \bigg)\nu( de)\|_0\nonumber\\ &\leq&\|\int_{Z}\big(r_1(t, e,\cdot)-r_2(t, e,\cdot\big)|\det (I+\partial_{x}k(t,e,x)|^{-1}\nu( de)\|_0\nonumber\\ &&+\|\int_{Z}\big(r_1(t, e,\cdot)-r_2(t, e,\cdot)\big)\nu( de)\|_0\nonumber\\ &=&C(\delta,\nu(Z) ) \Vert r_1(t,\cdot,\cdot)-r_2(t,\cdot,\cdot)\Vert_ {\nu,0}. \end{align}\tag{12}\]

Now we have verified all conditions in Assumption 1 for the operators \((A,f,g)\). By Theorem 2, there exists a unique (weak) solution \((u, q, r) \in \mathbb{S}^2(L^2) \cap \mathbb{H}^1 \times \mathbb{H}^0 \times \mathbb{H}^{\nu,0}\) to BSPDIE 7 satisfying the energy estimate \[\mathbb{E}\left[\sup_{0 \leq t \leq T} \|u(t)\|_0^2\right] + |||u|||_1^2 + |||q|||_0^2 + |||r|||_{\nu,0}^2 \leq C \left(|||f_0|||_{-1}^2 + \mathbb{E}[\|\varphi\|_0^2]\right),\] where \(C = C(L, \kappa, \alpha, \delta, T)\). This completes the proof of Theorem 4. ◻

Proof. The proof is divided into two steps.

Step 1. In this step, we consider an even more simple version of BSPDIE 8 with all lower-order coefficients vanish, i.e., \[b = 0, \quad c = 0, \quad v = 0, \quad k = 0.\]

In this case, BSPDIE 8 reduces to \[\label{eq:3467} \begin{align} u(t,x) = &\varphi(x) + \int_t^T \bigg( a^{ij}(s)D_{ij} u(s,x) + \sigma^i(t) D_{i} q(s,x) + f_0(s,x) \bigg) ds \\ & - \int_t^T q(s,x)dW_s - \int_t^T \int_Z r(s,e,x)\tilde{\mu}(de,ds). \end{align}\tag{13}\] We apply Theorem 2 with the Gelfand triple \((V, H, V^*) =(H^2, H^1, L^2)\).

This setting is well suited for second-order equations. The operator \(A(t)\), \(t\in[0,T]\), is a mapping from \(H^2\) to \(L^2\), and the forcing term \(f_0(t)\), \(t\in[0,T]\), lies in \(L^2\), which allows us to define the variational formulation in the dual space \(V^*=L^2\). Moreover, the energy estimates can be carried out within the \(H^1\) framework, where the main solution resides.

For any \(t\in[0,T]\) and \((u, q, r) \in H^2 \times H^1 \times M^{\nu,2}(Z; H^1)\), we define related operators: \[A(t)u := a^{ij}(t) D_{ij} u, \quad f(t,u,q,r) := \sigma^i(t) D_{i} q + f_0(t), \quad g(t,u,q,r) := 0.\] Then for any \(\eta\in H^2\), the dual pairings of operators are \[\langle A(t)u, \eta \rangle = -\left( a^{ij}(t) D_{i} u, D_{j} \eta \right)_1\;\;\;\text{and}\;\;\; \langle f(t,u,q,r), \eta \rangle = -\left( \sigma^i(t) q, D_{i} \eta \right)_1 + \left( f_0(t), (I - \Delta)\eta \right)_0.\] Here the use of the operator \(I - \Delta\) ensures that the \(L^2\)-function \(f_0(t)\) can be paired with the second-order test function \(\eta\).

We verify that \(A\) satisfies the coercivity condition (A.4) in Assumption 1. By integration by parts and the uniform ellipticity of \(a^{ij}\), we have \[\begin{align} \langle A(t)\eta, \eta \rangle &= -\int a^{ij}(t) \, D_{i} \eta(x) \, D_{j} \eta(x) \, dx - \int a^{ij}(t) \, D_{i\ell} \eta(x) \, D_{j\ell} \eta(x) \, dx \\ &\leq -\alpha \| D \eta \|_0^2 - \alpha \sum_{\ell=1}^d \| D(D_{\ell} \eta) \|_0^2 = -\alpha \left( \| \eta \|_1^2 + \sum_{\ell=1}^d \| D(D_{\ell} \eta) \|_0^2 \right) = -\alpha \| \eta \|_2^2 + \alpha \| \eta \|_1^2. \end{align}\]

Let’s turn to the Lipschitz continuity condition of non-homogeneous term \(f\). For any \(q_1, q_2 \in H^1\) and \(\eta \in H^2\), since \(\sigma^i\) is independent of the spatial variable \(x\), we have \[\begin{align} \langle f(t, u, q_1, r) - f(t, u, q_2, r), \eta \rangle &= - \left( \sigma^i(t)(q_1 - q_2), D_{i} \eta \right)_1\\ &= - \int \sigma^i(t)(q_1 - q_2) D_{i} \eta \, dx - \int D[\sigma^i(t)(q_1 - q_2)] \cdot D (D_{i} \eta)dx\\ &= - \sigma^i(t) \left[ \int (q_1 - q_2) D_{i} \eta \, dx + \int D(q_1 - q_2) \cdot D (D_{i} \eta)dx \right]\\ &\leq \kappa \left( \| q_1 - q_2 \|_0 \cdot \| D \eta\|_0 + \| D (q_1 - q_2) \|_0 \cdot \| D^2 \eta \|_0 \right). \end{align}\] Notice \[\| q_1 - q_2 \|_0 \leq \| q_1 - q_2 \|_1 \quad \text{and} \quad \| D \eta \|_0 \leq \| \eta \|_2.\] We conclude \[|\langle f(t, u, q_1, r) - f(t, u, q_2, r), u \rangle| \leq 2\kappa \| q_1 - q_2 \|_1 \cdot \| \eta \|_2,\] which implies the Lipschitz continuity of \(f(t, u, q, r)\) with respect to \(q\).

By Theorem 2, we know that BSPDIE 13 admits a unique solution \((u, q, r) \in \mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\) and a priori estimate ?? holds true. In particular, for any test function \(\eta \in C_0^\infty(\mathbb{R}^d)\), the triple \((u,q,r)\) satisfies the variational identity \[\begin{align} (u(t), \eta)_1 &= (\varphi, \eta)_1 + \int_t^T \Big[ - \big( a^{ij}(s) D_{i} u(s), D_{j} \eta \big)_1 - \big( \sigma^i(s) q(s), D_{i} \eta \big)_1 + \big( f_0(s), (I - \Delta)\eta \big)_0 \Big] ds \\ &\quad - \int_t^T (q(s), \eta)_1 \, dW_s - \int_t^T \int_Z (r(s,e), \eta)_1 \, \tilde{\mu}(de,ds). \end{align}\] For the convenience of subsequent arguments, we take the test function in the form \(\eta = (I - \Delta)^{-1} \eta\) instead. Then the variational identity becomes \[\begin{align} (u(t), (I - \Delta)^{-1} \eta)_1 =& (\varphi, (I - \Delta)^{-1} \eta)_1 + \int_t^T \Big[ - \big( a^{ij}(s) D_{i} u(s), (I - \Delta)^{-1} D_{j} \eta \big)_1 \\ &- \big( \sigma^i(s) q(s), (I - \Delta)^{-1} D_{i} \eta \big)_1 + \big( f_0(s), \eta \big)_0 \Big] ds \\ &- \int_t^T (q(s), (I - \Delta)^{-1} \eta)_1 \, dW_s - \int_t^T \int_Z (r(s,e), (I - \Delta)^{-1} \eta)_1 \, \tilde{\mu}(de,ds). \end{align}\] By Green’s formula, the above variational form can be equivalently rewritten in the standard \(L^2\) framework, i.e., for any \(t \in [0,T]\) and a.e. \(x \in \mathbb{R}^d\), \[\begin{align} (u(t), \eta)_0 &= (\varphi, \eta)_0 + \int_t^T \Big[ - \big( a^{ij}(s) D_{ij} u(s), \eta \big)_0 - \big( \sigma^i(s) D_{i} q(s), \eta \big)_0 + \big( f_0(s), \eta \big)_0 \Big] ds \\ &\quad - \int_t^T (q(s), \eta)_0 \, dW_s - \int_t^T \int_Z (r(s,e), \eta)_0 \, \tilde{\mu}(de,ds). \end{align}\] This confirms that the triple \((u, q, r)\) is a strong solution to BSPDIE 13 .

Moreover, according to Definitions 2 and 3, a strong solution is also a weak solution, which together with the uniqueness of weak solution in Theorem 2 leads to the uniqueness of strong solution to BSPDIE 13 .

Step 2. We remove the additional assumptions imposed in Step 1.

Since the coefficients \(a\) and \(\sigma\) are independent of the spatial variable \(x\), BSPDIE 8 can be rewritten in a divergence form \[\label{twbxqovz} \begin{align} u(t,x) =& \varphi(x) + \int_t^T \Bigg( a^{ij}(s)\,D_{ij}u(s,x) + b^i(s,x)\,D_{i}u(s,x) + c(s,x)\,u(s,x) \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \sigma^i(s)\,D_{i}q(s,x) + v(s,x)\,q(s,x) + f_0(s,x) \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \int_{Z}\left[u(s,x+k(s,e,x)) - u(s,x) - k^i(s,e,x)\,D_{i}u(s,x)\right] \nu(de) \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \int_Z \left[ r(s,e,x+k(s,e,x)) - r(s,e,x) \right] \nu(de) \Bigg) ds \\ & - \int_t^T q(s,x)dW_s - \int_t^T \int_Z r(s,e,x)\tilde{\mu}(de,ds). \end{align}\tag{14}\] By Theorem 4, BSPDIE 9 admits a unique weak solution \((u, q, r) \in \mathbb{S}^2(H^0) \cap \mathbb{H}^1 \times \mathbb{H}^0 \times \mathbb{H}^{\nu,0}.\)

Define \[\label{eq:tilde-F} \begin{align} \tilde{F}(t,x) :=\;& b^i(t,x)\,D_{i} u(t,x) + c(t,x)\,u(t,x) + v(t,x)\,q(t,x) + f_0(t,x) \\ & + \int_Z \left[ u(t,x+k(t,e,x)) - u(t,x) - k^i(t,e,x)\,D_{i} u(t,x) \right] \nu(de) \\ & + \int_Z \left[ r(t,e,x+k(t,e,x)) - r(t,e,x) \right] \nu(de). \end{align}\tag{15}\] This \(\tilde{F}\) is completely determined by the weak solution \((u,q,r)\), and obviously belongs to \(\mathbb{H}^0\). Then BSPDIE 9 can be regarded as an equation without all lower-order coefficients \[\label{eq:tilde-eq} \begin{align} \tilde{u}(t,x) =&\varphi(x) + \int_t^T \bigg[ a^{ij}(s)D_{ij} \tilde{u}(s,x) + \sigma^i(s)D_{i} \tilde{q}(s,x) + \tilde{F}(s,x) \bigg] ds \\ & - \int_t^T\tilde{q}(s,x)dW_s - \int_t^T\int_Z \tilde{r}(s,e,x)\tilde{\mu}(de,ds). \end{align}\tag{16}\] By the result in Step 1, BSPDIE 16 admits a unique strong solution \((\tilde{u}, \tilde{q}, \tilde{r})\) in \(\mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\), and ?? holds for \((\tilde{u}, \tilde{q}, \tilde{r})\) with \(f_0\) replaced by \(\tilde{F}\).

Moreover, \((\tilde{u}, \tilde{q}, \tilde{r})\) is also a weak solution to BSPDIE 16 . But we have known that \((u, q, r)\) is a weak solution to 9 , and hence a weak solution to BSPDIE 16 . By the uniqueness of weak solutions to BSPDIE 16 , we conclude that \((\tilde{u}, \tilde{q}, \tilde{r}) = (u, q, r)\) a.s., and the triple \((u, q, r)\) is indeed the unique strong solution to BSPDIE 9 , or equivalently BSPDIE 8 with \(a, \sigma\) independent of \(x\). Consequently, ?? holds for \((u, q, r)\) but with \(f_0\) replaced by \(\tilde{F}\). Noticing the uniform boundedness of \(b, c, v, k\) and Assumption 3 (iii), we get from the definition of \(\tilde{F}\) that \(|||\tilde{F} |||_0^2\leq ||| f_0 |||_0^2\), which implies that ?? holds. ◻

Proof. For given \((\bar{u}, \bar{q}, \bar{r}) \in \mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\), consider BSPDIE \[\label{eq:d2} \begin{align} u(t,x)=& \varphi(x) + \int_t^T \Big[ a_0^{ij}(s) D_{ij} u(s,x) + \sigma_0^i(s) D_{i} q(s,x)+ \big(a^{ij}(s,x) - a_0^{ij}(s)\big) D_{ij} \bar{u}(s,x) \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \big(\sigma^i(s,x) - \sigma_0^i(s)\big) D_{i} \bar{q}(s,x) + f_0(s,x) \Big]ds \\ &- \int_t^T q(s,x)dW_s - \int_t^T \int_Z r(s,e,x)\tilde{\mu}(de,ds). \end{align}\tag{17}\] By Proposition 6, the above BSPDIE admits a unique strong solution \((u,q,r)\in \mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\). Construct a mapping \(\Psi\) from \(\mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\) into itself by defining \(\Psi(\bar{u}, \bar{q}, \bar{r}) := (u, q, r)\). For \((\bar{u}_i, \bar{q}_i, \bar{r}_i) \in \mathbb{S}^2 (H^1)\cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\), set \((u_i, q_i, r_i)= \Psi(\bar{u}_i, \bar{q}_i, \bar{r}_i)\), \(i = 1, 2\). Then applying the estimate ?? to BSPDIE 17 , we have \[E\left[\sup_{0 \leq t \leq T} \| u_1(t) - u_2(t) \|_{H^1}^2 \right] + ||| u_1 - u_2 |||_2^2 + ||| q_1 - q_2 |||_1^2 + ||| r_1 - r_2 |||_{\nu,1}^2\] \[\leq \epsilon C \left( E\left[\sup_{0 \leq t \leq T} \| \bar{u}_1(t) - \bar{u}_2(t) \|_{H^1}^2 \right] + ||| \bar{u}_1 - \bar{u}_2 |||_2^2 + ||| \bar{q}_1 - \bar{q}_2 |||_1^2 + ||| \bar{r}_1 - \bar{r}_2 |||_{\nu,1}^2 \right),\] where \(C = C(\kappa, \alpha, T)\) is a constant. Take \(\delta= \frac{1}{2C}\), and then \(\Psi\) is a contraction mapping on the Banach space \(\mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\). By the fixed point theorem, it follows that BSPDIE 8 admits a unique strong solution \((u, q, r)\in \mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\).

Then applying the estimate ?? again to the BSPDIE 17 with \((\bar u,\bar q,\bar r)\) replaced by \((u,q,r)\), we have \[\begin{align} &\mathbb{E}\left[\sup_{0 \leq t \leq T} \|u\|_{H^1}^2 + \int_0^T \left( \|u\|_{H^2}^2 + \|q\|_{H^1}^2 \right) dt + \int_0^T \int_Z \|r\|_{H^1}^2 \nu(de) dt \right] \\ \leq &\epsilon C\mathbb{E}\left[\sup_{0 \leq t \leq T} \|u\|_{H^1}^2 + \int_0^T \left( \|u\|_{H^2}^2 + \|q\|_{H^1}^2 \right) dt + \int_0^T \int_Z \|r\|_{H^1}^2 \nu(de) dt \right] \\ &+ C\left( \mathbb{E} \int_0^T \|f_0\|_{L^2}^2 dt + \mathbb{E} \|\varphi\|_{H^1}^2 \right). \end{align}\] Noticing \(\epsilon C (2C)^{-1}<1\), we get the desired a priori estimate ?? for the strong solution \((u, q, r)\). ◻

Proof. Step 1: energy estimate via Itô’s formula. According to the infinite-dimensional Itô’s formula with jumps, we have \[\begin{align} \|u(t,\cdot)\|^2_0=&\|\phi\|^2 _0+ 2\int_t^T\int_{\mathbb{R}^d}u(s,x)\big[a^{ij}(s,x)D_{ij}u(s,x) +b^i(s,x)D_{i}u(s,x)+c(s,x)u(s,x)\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\int_Z\left(u(s,x + k(s,e,x))-u(s,x)-k^i(s,e,x)D_{i}u(s,x)\right)\nu(de)\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sigma^i(s,x)D_{i}q(s,x)+v(s,x)q(s,x)+f_0(s,x)\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\int_Z\left(r(s,e,x + k(s,e,x))-r(s,e,x)\right)\nu(de)\big]dxds\\ &-\int_t^T\int_{\mathbb{R}^d}|q(s,x)|^2dxds- 2\int_t^T\int_{\mathbb{R}^d}u(s,x)q(s,x)dx dW_s\\ &-\int_t^T\int_Z\int_{\mathbb{R}^d}\left(- 2u(s,x)r(s,e,x)-\|r(s,e,x)\|^2\right)dx\tilde{\mu}(de,ds)\\ &-\int_t^T\int_Z\int_{\mathbb{R}^d}|r(s,e,x)|^2dx\nu(de)ds. \end{align}\] Taking the expectation of both sides and applying Cauchy-Schwarz inequality, we further have \[\begin{align} \label{eq:energy-ineq} \begin{aligned} &\|u(t,\cdot)\|^2_0+||| q |||_0^2 + ||| r |||_{\nu,0}^2\\ \leq& \|\phi\|_0^2 + 2\mathbb{E} \big[ \int_t^T \int_{\mathbb{R}^d} u(s,x) \bigg( a^{ij}(s,x)D_{ij}u(s,x) + b^i(s,x)D_{i} u(s,x)+ c(s,x) u(s,x) \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \int_Z \left( u(s,x + k(s,e,x)) - u(s,x) - k^i(s,e,x)D_{i} u(s,x) \right) \nu(de)\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \sigma^i(s,x)D_{i} q(s,x)+ v(s,x) q(s,x) + f_0(s,x) \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \int_Z \left( r(s,e,x + k(s,e,x)) - r(s,e,x) \right) \nu(de) \bigg) dx\, ds \big]\\ \leq\;& \|\phi\|_0^2 + \varepsilon \left( ||| u |||_2^2 + ||| q |||_1^2 + ||| r |||_{\nu,0}^2 \right) + C(\varepsilon, K)\, ||| u |||_0^2 + ||| f_0 |||_0^2, \end{aligned} \end{align}\tag{18}\] where \(\varepsilon\) is a small positive number to be specified later.

Step 2: localization and regularization.

Due to Assumption 5, we can choose a sufficiently small constant \(\rho>0\) such that for any \((\omega,t,x,y)\in\Omega\times[0,T]\times\mathbb{R}^d\times\mathbb{R}^d\) with \(|x-y|\leq4\rho\), \[|a(t,x)-a(t,y)|\leq\delta,\quad |\sigma(t,x)-\sigma(t,y)|\leq\delta,\] where \(\delta\) is the constant in Lemma 7 depending on \(\kappa,\alpha,T\).

For any fixed point \(z\in\mathbb{R}^d\), denote the open ball \[B_l(z):=\big\{x\in\mathbb{R}^d: |x-z|<l\big\}.\] Choose a non-negative cut-off function \(\zeta\in C_0^\infty(\mathbb{R}^d)\) satisfying \[\operatorname{supp}(\zeta)\subset B_{2\rho}(0)\;\;\;\text{and}\;\;\;\zeta(x)=1\;\;\text{for}\;|x|\leq\rho.\] Define the localized cut-off functions \[\zeta^z(x):=\zeta(x-z)\;\;\;\text{and}\;\;\;\eta^z(x):=\zeta\left(\frac{x-z}{\rho}\right),\] and the localized solution \[u^z:=u\zeta^z,\quad q^z:=q\zeta^z,\quad r^z:=r\zeta^z.\] By Itô’s formula, it is not difficult to verify that \((u^z, q^z, r^z)\) satisfies a localized BSPDIE \[\begin{align} \label{eq:local-BSPDIE} \mathrm{d} u^z(t,x) = -\Big[\tilde{a}^{ij}(t,x)D_{ij}u^z + \tilde{\sigma}^i(t,x)D_{i}q^z + \tilde{F}(t,x)\Big]\mathrm{d} t+ q^z(t,x)\mathrm{d}W_t + \int_Z r^z(t,e,x)\tilde{\mu}(\mathrm{d} e,\mathrm{d} t),\nonumber \end{align}\tag{19}\] where the coefficients ahead of highest-order terms are defined as \[\begin{align} \tilde{a}^{ij}(t,x) &:= a^{ij}(t,x)\eta^z(x) + a^{ij}(t,z)\big(1-\eta^z(x)\big),\\ \tilde{\sigma}^i(t,x) &:= \sigma^i(t,x)\eta^z(x) + \sigma^i(t,z)\big(1-\eta^z(x)\big), \end{align}\] and the inhomogeneous term \(\tilde{F}\) is defined as \[\begin{align} \tilde{F}(t,x) =\, & f_0(t,x)\zeta^z(x) + \Big( b^i(t,x)\zeta^z(x)-2a^{ij}(t,x)D_{j}\zeta^z(x) \Big)D_{i}u(t,x)\\ &+ \Big( v^k(t,x)\zeta^z(x)-\sigma^{ik}(t,x)D_{i}\zeta^z(x) \Big)q^k(t,x)- \Big( c(t,x)\zeta^z(x)+a^{ij}(t,x)D_{ij}\zeta^z(x) \Big)u(t,x)\\ &+ \int_Z \zeta^z(x) \Big[ r(t,e,x+k(t,e,x))-r(t,e,x) \Big]\nu(\mathrm{d} e)\\ &+ \int_Z\zeta^z(x) \Big[ u(t,x+k(t,e,x))-u(t,x)-k^i(t,e,x)D_{i}u(t,x) \Big]\nu(\mathrm{d} e). \end{align}\] Then all conditions of Lemma 7 are satisfied with \(a_0(t)=a^{ij}(t,z)\) and \(\sigma_0(t)=\sigma^i(t,z)\), due to the choice of sufficiently small \(\rho\) and the fact that \(\tilde{F}\in\mathbb{H}^0\).

By Lemma 7, the localized BSPDIE admits a unique strong solution \[(u^z,q^z,r^z)\in \big(\mathbb{S}^2(H^1)\cap\mathbb{H}^2\big)\times \mathbb{H}^1\times \mathbb{H}^{\nu,1},\] and it follows from the uniqueness of solutions that \[u^z=u,\quad q^z=q\quad\text{and}\quad r^z=r\quad \text{on}\;B_{2\rho}(z).\]

Moreover, the a priori estimate in Lemma 7 leads to \[\begin{align} \label{eq:local-est} &\mathbb{E}\int_0^T \Big( \|u(t,\cdot)\|_{2,B_\rho(z)}^2 + \|q(t,\cdot)\|_{1,B_\rho(z)}^2 + \|r(t,\cdot)\|_{\nu,1,B_\rho(z)}^2 \Big)\mathrm{d} t+ \mathbb{E}\sup_{t\leq T}\|u(t,\cdot)\|_{1,B_\rho(z)}^2\nonumber\\ \leq& C\Bigg( \mathbb{E}\|\varphi\|_{2,B_{2\rho}(z)}^2 + \mathbb{E}\int_0^T \|\tilde{F}(t,\cdot)\|_{0,B_{2\rho}(z)}^2\mathrm{d} t \Bigg). \end{align}\tag{20}\] Using a standard finite covering argument for \(\mathbb{R}^d\) with balls \(\{B_\rho(z)\}_{z\in\mathbb{R}^d}\) and integrating the above local estimates for all \(z\in\mathbb{R}^d\), we derive the global a priori estimate \[\begin{align} &\left|\left|\left|u\right|\right|\right|_2^2 + \left|\left|\left|q\right|\right|\right|_1^2 + \left|\left|\left|r\right|\right|\right|_{\nu,1}^2+ \mathbb{E}\Big[\sup_{0\leq t\leq T}\|u(t)\|_{1}^2\Big]\\ \leq& C\Big( \left|\left|\left|f_0\right|\right|\right|_0^2 + \mathbb{E}\|\varphi\|_1^2 + \left|\left|\left|u\right|\right|\right|_1^2 + \left|\left|\left|q\right|\right|\right|_0^2 + \left|\left|\left|r\right|\right|\right|_{\nu,0}^2 \Big). \end{align}\] The above estimate, together with the preliminary energy inequality 18 in Step 1, implies that \[\begin{align} \label{mz1} \left|\left|\left|u\right|\right|\right|_2^2 + \left|\left|\left|q\right|\right|\right|_1^2 + \left|\left|\left|r\right|\right|\right|_{\nu,1}^2+ \mathbb{E}\Big[\sup_{0\leq t\leq T}\|u(t)\|_{1}^2\Big]\leq C\Big( \left|\left|\left|f_0\right|\right|\right|_0^2 + \mathbb{E}\|\varphi\|_1^2 + \left|\left|\left|u\right|\right|\right|_1^2\Big). \end{align}\tag{21}\] Note that for \(t\in[0,T)\), we know \(\mathbb{E}\|u(t)\|_{1}^2\leq C\Big( \left|\left|\left|f_0\right|\right|\right|_0^2 + \mathbb{E}\|\varphi\|_1^2 + \int_t^T\mathbb{E}\left|\left|u(s)\right|\right|_1^2ds\Big)\). An application of Gronwall’s inequality leads to \(\left|\left|\left|u\right|\right|\right|_1^2\leq C\Big( \left|\left|\left|f_0\right|\right|\right|_0^2 + \mathbb{E}\|\varphi\|_1^2\Big)\). Consequently, ?? follows immediately from 21 . ◻

Proof of Theorem 5. The uniqueness of the strong solution to BSPDIE 8 follows immediately from the a priori estimate ?? . We turn to the proof of existence by the method of continuation. First we introduce some operators which interpolate between a simplified version of the original BSPDIE and the concerned equation. For this, define \[\begin{align} \label{mz2} \left\{ \begin{aligned} L_0 u=& a^{ij}(t,0)D_{ij} u + b^i(t,0) D_{i} u + c(t,0) u \\ &+ \int_Z \left[ u(t,x+k(t,e,0)) - u - k^i(t,e,0) D_{i} u \right] v(de), \\ L_1 u=& a^{ij}(t,x)D_{ij} u + b^i(t,x) D_{i} u + c(t,x) u\\ &+ \int_Z \left[ u(t,x+k(t,e,0)) - u - k^i(t,e,0) D_{i} u \right] v(de), \\ B_0 q=& \sigma^i(t,0)D_{i} q + v(t,0) q, \\ B_1 q=& \sigma^i(t,x)D_{i} q + v(t,x) q, \\ M r=& \int_Z \left[ r(t,e,x+k(t,e,x)) - r(t,e,x) \right] \nu(de). \end{aligned} \right. \end{align}\tag{22}\] Then for a parameter \(\lambda \in [0,1]\), the interpolated operators are introduced as follows \[L_\lambda := (1-\lambda)L_0 + \lambda L_1\;\;\;\text{and}\;\;\;B_\lambda := (1-\lambda)B_0 + \lambda B_1.\] Consider a parameterized BSPDIE \[\begin{cases} du = -\left( L_\lambda u + B_\lambda q + M r + f_0 \right)dt + qdW_t + \displaystyle\int_Z r\,\tilde{\mu}(de,dt), \\ u|_{t=T} = \varphi. \end{cases}\] Since for any \(\lambda \in [0,1]\), the coefficients satisfy the conditions of Theorem 5 with unified parameters, the a priori estimate ?? holds with the same constant \(C > 0\) independent of \(\lambda\).

Suppose that the above parameterized BSPDIE admits a unique strong solution for some \(\lambda_0 \in [0,1]\). For any \(\lambda\) close to \(\lambda_0\), we can rewrite the parameterized BSPDIE in a form like \[\begin{cases} du = -\big[ L_{\lambda_0} u + B_{\lambda_0} q + M r + f_0 + (\lambda - \lambda_0)\left( (L_1 - L_0)u + (B_1 - B_0)q \right) \big] dt \\ \qquad\; + qdW_t + \displaystyle\int_Z r\,\tilde{\mu}(de,dt), \\ u|_{t=T} = \varphi. \end{cases}\] For a given triple \((\bar{u}, \bar{q}, \bar{r}) \in \mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\), define a mapping \(\Psi: (\bar{u}, \bar{q}, \bar{r}) \mapsto (u, q, r)\) by setting \((u, q, r)\) to be the strong solution to BSPDIE \[\begin{cases} du = -\left[ L_{\lambda_0} u + B_{\lambda_0} q + M r + f_0 + (\lambda - \lambda_0)\left( (L_1 - L_0)\bar{u} + (B_1 - B_0)\bar{q} \right) \right]dt \\ \qquad\;+ qdW_t + \displaystyle\int_Z r\,\tilde{\mu}(de,dt), \\ u|_{t=T} = \varphi. \end{cases}\] By the a priori estimate ?? again, together with the boundedness of operators in 22 , for any \((\bar{u}_1, \bar{q}_1, \bar{r}_1), (\bar{u}_2, \bar{q}_2, \bar{r}_2)\in\mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\), we have \[\begin{align} &||| \Psi(\bar{u}_1 - \bar{u}_2) |||_2^2 + ||| \Psi(\bar{q}_1 - \bar{q}_2) |||_1^2 + ||| \Psi(\bar{r}_1 - \bar{r}_2) |||_{\nu,1}^2 + \mathbb{E} \sup_{t \leq T} \|\Psi(\bar{u}_1 - \bar{u}_2)(t)\|_1^2 \\ \leq& C |\lambda - \lambda_0|^2 \left( ||| \bar{u}_1 - \bar{u}_2 |||_2^2 + ||| \bar{q}_1 - \bar{q}_2 |||_1^2 + ||| \bar{r}_1 - \bar{r}_2 |||_{\nu,1}^2 \right), \end{align}\] where \(C\) is a positive constant depending on given parameters. Hence, the mapping \(\Psi\) becomes a contraction in the space \(\mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\) if we take sufficiently small \(|\lambda - \lambda_0|\). In particular, this mapping is contractive if \(|\lambda - \lambda_0| \leq \frac{1}{2C}\).

By the fixed-point theorem, the equation is solvable in a neighborhood of \(\lambda_0\). Since the case \(\lambda = 0\) corresponds to a linear BSPDIE with frozen coefficients, whose existence of strong solution is already guaranteed by Proposition 6, we can initiate the continuation from \(\lambda = 0\). By dividing the interval \([0,1]\) into finitely many subintervals with a size less than \(\frac{1}{2C}\), we iteratively construct a sequence of strong solutions for finitely many \(\lambda \in [0,1]\) up to \(\lambda=1\). Actually, BSPDIE 8 appears as \(\lambda=1\). In this way, the existence of strong solution to BSPDIE 8 is proved. ◻

Proof. We prove the theorem by induction for the regularity index \(n\).

Base Case (\(n = 0\)). The conditions of Theorem 5 are satisfied. BSPDIE 8 admits a unique strong solution \((u, q, r) \in \mathbb{S}^2(H^1) \cap \mathbb{H}^2 \times \mathbb{H}^1 \times \mathbb{H}^{\nu,1}\) satisfying the estimate ?? which is actually ?? with \(n=0\).

Inductive Step. Assume Theorem 9 is true for some \(n = k \geq 0\), and hence BSPDIE 8 admits a unique strong solution \((u, q, r) \in \mathbb{S}^2(H^{k+1}) \cap \mathbb{H}^{k+2} \times \mathbb{H}^{k+1} \times \mathbb{H}^{\nu,k+1}\) satisfying the estimate ?? . We prove that it is still true for \(n = k+1\) if the conditions of Theorem 5 hold for \(|\alpha| \leq k+1\).

Take the derivative of spatial variable on both sides of BSPDIE 8 . Since the coefficients have bounded derivatives up to order \(k+1\), the interchange of the derivative operator \(D^\alpha\) and the integration is permissible. Define new variables \[\bar{u} := D^\alpha u, \quad \bar{q} := D^\alpha q, \quad \bar{r} := D^\alpha r.\] Then \((\bar{u}, \bar{q}, \bar{r})\) satisfies BSPDIE \[\label{eq:b5} \begin{align} \bar{u}(t,x) =&D^\alpha \varphi(x) + \int_t^T \bigg[ a^{ij}(s,x) D_{ij} \bar{u}(s,x) + \sigma^i(s,x) D_i \bar{q}(s,x) + \tilde{f}_0(s,x) \bigg] ds \\ & - \int_t^T\bar{q}(s,x)dW_s - \int_t^T\int_Z \bar{r}(s,e,x)\tilde{\mu}(de,ds), \end{align}\tag{23}\] where \[\begin{align} \tilde{f}_0 ={}& D^\alpha f_0 + \sum_{\substack{|\beta|+|\gamma|=|\alpha| \\ |\beta|\geq 1}} C_{\alpha\beta} \left[(D^\beta a^{ij})(D^\gamma u_{x^i x^j}) + (D^\beta \sigma^i)(D^\gamma q_{x^i})\right] \\ &+ \sum_{|\beta|+|\gamma|=|\alpha|} C_{\alpha\beta} \left[(D^\beta b^i)(D^\gamma u_{x^i}) + (D^\beta c)(D^\gamma u) + (D^\beta \nu)(D^\gamma q)\right] \\ & - \int_Z D^\alpha u \, \nu(de) - \int_Z \sum_{|\beta|+|\gamma|=|\alpha|} C_{\alpha\beta} (D^\beta k^i(s,e,x))(D^\gamma u_{x^i}) \nu(de) \\ &+ \int_Z D^\alpha \left[ r(s,e, x + k(s,e,x)) - r(s, e, x) \right] \nu(de)+ \int_Z D^\alpha \left[u(s,x + k(s,e,x))\right] \nu(de). \end{align}\] Note that \(\tilde{f}_0 \in \mathbb{H}^0\) and satisfies the estimate \[\begin{align} ||| \tilde{f}_0 |||_0^2 &\leq C(K, \kappa, T) \left( ||| D^\alpha f_0 |||_0^2 + ||| u |||_{k+1}^2 + ||| q |||_{k+1}^2 + ||| r |||_{\nu,k+1}^2 \right) \\ &\leq C(K, \kappa, T) \left( ||| f_0 |||_{k+1}^2 + \mathbb{E} \| \varphi \|_{k+1}^2 \right), \end{align}\] where we used the regularity assumptions on \(f_0\) and \(\varphi\), and the induction hypothesis in the second inequality.

Now, applying Theorem 5 to the BSPDIE 23 , we have \[(\bar{u}, \bar{q}, \bar{r}) \in \mathbb{S}^2(H^1) \cap \mathbb{H}^2, \times \mathbb{H}^1\times \mathbb{H}^{\nu,1}.\] In particular, for the multi-index \(\alpha\) with \(|\alpha| = k+1\), it follows that \[({u}, {q}, {r}) \in \mathbb{S}^2(H^{k+2}) \cap \mathbb{H}^{k+3}, \times \mathbb{H}^{k+2}\times \mathbb{H}^{\nu,k+2},\] i.e., the solution has the desired regularity for \(n = k+1\). Moreover, the estimate ?? follows from the corresponding estimate in Theorem 5 with \(f_0=\tilde{f}_0\).

By mathematical induction, Theorem 9 holds for all nonnegative integers. ◻

Proof. All integrations by parts over \(\mathbb{R}^d\) in this proof are first justified for smooth and compactly supported functions, for which the boundary terms vanish. The general case follows by a standard density argument in Sobolev spaces.

For \(\varepsilon>0\), let \(h_\varepsilon\in C^2(\mathbb{R})\) be a standard convex regularization of \((x^-)^2\) such that \(h_\varepsilon(x)=0\) for \(x\ge0\), and \(0\le h_\varepsilon(x)\le C(x^-)^2\), \(h_\varepsilon'(x)\le0\) and \(xh_\varepsilon'(x)\ge0\) for any \(x\in\mathbb{R}\). We choose the approximation such that \[h_\varepsilon(x)\to (x^-)^2,\qquad h_\varepsilon'(x)\to -2x^-,\qquad 0\le h_\varepsilon''(x)\le C,\] uniformly in \(\varepsilon\), and there exists a constant \(C_0>0\) independent of \(\varepsilon\) such that for any \(x\in\mathbb{R}\), \[|h_\varepsilon'(x)|^2\le C_0 h_\varepsilon(x),\;\;\;\text{and}\;\;\; |h_\varepsilon'(x)|^2\le C_0 h_\varepsilon(x)h_\varepsilon''(x).\] Such a family is obtained, for instance, by smoothing \((x^-)^2\) only on the interval \([-\varepsilon,0]\) and keeping it equal to \(x^2\) on \((-\infty,-\varepsilon]\) and to \(0\) on \([0,\infty)\). We apply Itô’s formula for a.e.\(x\in\mathbb{R}^d\) to the composition \(h_\varepsilon(u(t,x))\). In view of the regularity of the strong solution \(u\in\mathbb{S}^2(H^1)\cap\mathbb{H}^2\), this is justified by first mollifying \(u\) in the spatial variable, applying the finite-dimensional Itô formula with jumps to \(h_\varepsilon(u_n(t,x))\), and then passing to the limit \(u_n\to u\) in \(L^2(\Omega\times[0,T]\times\mathbb{R}^d)\). Then it turns out that \[\begin{align} d\,h_\varepsilon(u(t,x)) =&-h_\varepsilon'(u(t-,x))\Big[ a^{ij}(t,x)D_{ij}u(t,x)+b^i(t,x)D_i u(t,x)+c(t,x)u(t,x) \\ &\qquad\qquad\qquad\;\; +\int_Z\Big(u(t,x+k(t,e,x))-u(t,x)-k^i(t,e,x)D_i u(t,x)\Big)\nu(de) \\ &\qquad\qquad\qquad\;\; +\sigma^i(t,x)D_i q(t,x)+v(t,x)q(t,x)+f_0(t,x) \\ &\qquad\qquad\qquad\;\; +\int_Z\Big(r(t,e,x+k(t,e,x))-r(t,e,x)\Big)\nu(de) \Big]dt \\ &+h_\varepsilon'(u(t-,x))\,q(t,x)dW_t +\frac{1}{2} h_\varepsilon''(u(t-,x))|q(t,x)|^2dt \\ &+\int_Z\Big[ h_\varepsilon\big(u(t-,x)+r(t,e,x)\big)-h_\varepsilon(u(t-,x)) -h_\varepsilon'(u(t-,x))r(t,e,x) \Big]\nu(de)dt \\ &+\int_Z\Big[ h_\varepsilon\big(u(t-,x)+r(t,e,x)\big)-h_\varepsilon(u(t-,x)) \Big]\tilde{\mu}(de,dt). \end{align}\] Taking integration over \(\Omega\times[t,T]\times\mathbb{R}^d\), we have \[\label{eq:ito-neg} \begin{align} &\mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon(u)dx +\frac{1}{2}\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)|q|^2dxds \\ &+\mathbb{E}\int_t^T\int_{\mathbb{R}^d}\int_Z \Big[h_\varepsilon(u+r)-h_\varepsilon(u)-h_\varepsilon'(u)r\Big]\nu(de)dxds \\ =& \mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon(\varphi)dx +J_1+J_2+J_3+J_4+J_5, \end{align}\tag{24}\] where \[\begin{align} J_1&= \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)\,a^{ij}D_{ij}udxds,\\ J_2&= \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)\,b^iD_i udxds,\\ J_3&= \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)\,c\,udxds,\\ J_4&= \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)\,(\sigma^iD_i q+vq+f_0)dxds,\\ J_5&= \mathbb{E}\int_t^T\int_{\mathbb{R}^d} \int_Z h_\varepsilon'(u) \big[u(\cdot+k)-u-k^iD_i u+r(\cdot+k)-r\big]\nu(de)dxds. \end{align}\] Next we estimate \(J_1\)–\(J_5\) one by one.

For \(J_1\), the integration by parts gives \[\begin{align} J_1 &= -\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)a^{ij}D_i uD_j udxds -\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)(D_i a^{ij})D_j udxds. \end{align}\] For the second term on the right hand side of above equality, using the integration by parts again, we obtain \[\begin{align} -\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)(D_i a^{ij})D_j udxds =& -\mathbb{E}\int_t^T\int_{\mathbb{R}^d} D_j\big(h_\varepsilon(u)\big)(D_i a^{ij})dxds\\ =& \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u)D_{ij} a^{ij}dxds. \end{align}\] Therefore, \[J_1 = -\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)a^{ij}D_i uD_j udxds + \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u)D_{ij}a^{ij}dxds.\] By the uniform ellipticity of \(a^{ij}\) and the \(W^{2,\infty}\)-regularity in Assumption 7, we obtain \[J_1 \le -\alpha \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)|Du|^2dxds + \|D^2 a\|_{L^\infty}\,\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u)dxds.\]

For \(J_2\), since \(D_i(h_\varepsilon(u))=h_\varepsilon'(u)D_i u\), we have \[\begin{align} J_2 &= \mathbb{E}\int_t^T\int_{\mathbb{R}^d} b^i D_i(h_\varepsilon(u))dxds = -\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u) D_i b^idxds. \end{align}\] Hence, by the \(W^{1,\infty}\)-regularity of \(b\) in Assumption 7, we have \[J_2 \le C\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u)dxds.\]

For \(J_3\), since \(x h_\varepsilon'(x)\ge 0\) for any \(x\in\mathbb{R}\) and \(c\le 0\), we have \[J_3 = \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)\,c\,udxds \le 0.\]

For \(J_4\), the integration by parts yields \[\mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon'(u)\sigma^iD_i qdx = -\mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon''(u)\sigma^i D_i u qdx - \mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon'(u)D_i\sigma^i qdx.\] We now estimate each term related to \(J_4\) one by one. Let \(C_b:=\|D\sigma\|_{L^\infty}+\|v\|_{L^\infty}\). For the terms involving \(D_i\sigma^i\) and \(v\), due to \(|h_\varepsilon'(x)|^2\le C_0 h_\varepsilon(x)h_\varepsilon''(x)\) for any \(x\in\mathbb{R}\) and Young’s inequality, for any \(\eta>0\), \[\begin{align} \bigl|h_\varepsilon'(u)D_i\sigma^i q\bigr| \le C_b\sqrt{C_0}\,\sqrt{h_\varepsilon(u)}\,\sqrt{h_\varepsilon''(u)}\,|q| \le \frac{\eta}{2}\,h_\varepsilon''(u)|q|^2+\frac{C_b^2 C_0}{2\eta}\,h_\varepsilon(u), \end{align}\] and the same bound holds for \(h_\varepsilon'(u)v q\). Hence for any \(\eta>0\), \[\left|\mathbb{E}\int h_\varepsilon'(u)D_i\sigma^i qdx\right| + \left|\mathbb{E}\int h_\varepsilon'(u)v qdx\right| \le \eta\,\mathbb{E}\int h_\varepsilon''(u)|q|^2dx + \frac{C_b^2 C_0}{\eta}\,\mathbb{E}\int h_\varepsilon(u)dx .\] For the crossing term involving \(Du\) and \(q\), by the fact that \(|\sigma|^2\le\kappa^2\) and Young’s inequality again, it yields that \[\begin{align} -\mathbb{E}\int h_\varepsilon''(u)\sigma^i D_i u qdx \le \mathbb{E}\int h_\varepsilon''(u)|\sigma||Du||q|dx \le \frac{\alpha}{2}\mathbb{E}\int h_\varepsilon''(u)|Du|^2dx +\frac{\kappa^2}{2\alpha}\mathbb{E}\int h_\varepsilon''(u)|q|^2dx. \end{align}\] Summing these estimates, we obtain for any \(\eta>0\), \[\begin{align} J_4 \le& \frac{\alpha}{2}\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)|Du|^2dxds + \left(\frac{\kappa^2}{2\alpha}+\eta\right) \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)|q|^2dxds \\ & + \frac{C_b^2 C_0}{\eta}\,\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u)dxds +\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)f_0dxds . \end{align}\] Now choose \(\eta>0\) small enough so that \(\frac{\kappa^2}{2\alpha}+\eta<\frac{1}{2}\). This is possible since \(\kappa<\sqrt{\alpha}/2\) implies \(\frac{\kappa^2}{2\alpha}<\frac{1}{8}\), and we may take e.g.\(\eta=\frac{1}{8}\). Then after estimating \(J_5\) below, the terms involving \(|q|^2\) and \(|Du|^2\) on the right hand side of the \(J_4\) estimate will be absorbed into the corresponding nonnegative terms on the left hand side of 24 .

Finally, we turn to the jump term \(J_5\). For any \((\omega,s,e,x)\in \Omega \times [0,T] \times Z\times\mathbb{R}^d\), set \(V(x)=V(\omega,s,e,x):=u(s,x)+r(s,e,x)\). Then \[\begin{align} u(s,x+k(s,e,x))-u(s,x)+r(s,e,x+k(s,e,x))-r(s,e,x) =V(x+k(s,e,x))-V(x). \end{align}\] Define \[\mathcal{Z}_\varepsilon:= \mathbb{E}\int_t^T\int_{\mathbb{R}^d}\int_Z \Big[h_\varepsilon(u+r)-h_\varepsilon(u)-h_\varepsilon'(u)r\Big]\nu(de)dxds.\] A direct calculation gives \[\begin{align} J_5-\mathcal{Z}_\varepsilon =& \mathbb{E}\int_t^T\int_{\mathbb{R}^d}\int_Z \Big[ h_\varepsilon'(u)\big(V(x+k(s,e,x))-V(x)-k^i(s,e,x)D_i u\big) \Big]\nu(de)dxds \\ &-\mathbb{E}\int_t^T\int_{\mathbb{R}^d}\int_Z \Big[ h_\varepsilon(V(x))-h_\varepsilon(u)-h_\varepsilon'(u)(V(x)-u) \Big]\nu(de)dxds. \end{align}\] Since \(h_\varepsilon\) is convex, for any \(x\), \[h_\varepsilon(V(x+k(s,e,x))) \ge h_\varepsilon(u(x)) + h_\varepsilon'(u(x))\big(V(x+k(s,e,x))-u(x)\big).\] Consequently, \[\begin{align} &\big[h_\varepsilon'(u)\big(V(x+k(s,e,x))-V(x)-k^i(s,e,x)D_i u\big)\big] -\big[h_\varepsilon(V(x))-h_\varepsilon(u)-h_\varepsilon'(u)(V(x)-u)\big]\\ \le &h_\varepsilon(V(x+k(s,e,x)))-h_\varepsilon(V(x))-k^i(s,e,x)D_i h_\varepsilon(u). \end{align}\] Integrating over \(\mathbb{R}^d\times Z\), we obtain \[J_5-\mathcal{Z}_\varepsilon \le \mathbb{E}\int_t^T\int_{\mathbb{R}^d}\int_Z\Big[ h_\varepsilon(V(x+k(s,e,x)))-h_\varepsilon(V(x))-k^i(s,e,x) D_i\big(h_\varepsilon(u)\big)\Big]\nu(de)dxds.\] By Assumption 8, for any \((\omega,s,e)\in \Omega \times [0,T] \times Z\), \[\int_{\mathbb{R}^d} h_\varepsilon(V(x+k(s,e,x)))dx = \int_{\mathbb{R}^d} h_\varepsilon(V(x))dx.\] Hence \[\int_{\mathbb{R}^d}\big[h_\varepsilon(V(x+k(s,e,x)))-h_\varepsilon(V(x))\big]dx = 0.\] Thus \[J_5-\mathcal{Z}_\varepsilon \le -\mathbb{E}\int_t^T\int_Z\int_{\mathbb{R}^d} k^i(s,e,x)D_i h_\varepsilon(u(s,x))dx\,\nu(de)ds.\] By integration by parts, \[-\int_{\mathbb{R}^d} k^i(s,e,x)D_i h_\varepsilon(u(s,x))dx = \int_{\mathbb{R}^d} D_i k^i(s,e,x)\,h_\varepsilon(u(s,x))dx.\] Noticing from Assumptions 8 that \(\partial_xk\) is bounded, we obtain \[J_5 \le C\,\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u(s,x))dxds+\mathcal{Z}_\varepsilon.\] The term \(\mathcal{Z}_\varepsilon\) also appears on the left hand side of 24 , and hence it cancels in the estimate below.

Combining the estimates from \(J_1\) to \(J_5\), we have \[\label{mz3} \begin{align} &\mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon(u)dx +(\frac{1}{2}-\frac{\kappa^2}{2\alpha}-\eta)\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)|q|^2dxds +\frac{\alpha}{2}\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon''(u)|Du|^2dxds \\ \le& \mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon(\varphi(x))dx + C\mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon(u)dxds + \mathbb{E}\int_t^T\int_{\mathbb{R}^d} h_\varepsilon'(u)f_0dxds. \end{align}\tag{25}\] Moreover, since \(h_\varepsilon'(x)\le 0\) for any \(x\in\mathbb{R}\), we have \[h_\varepsilon'(u)f_0 = h_\varepsilon'(u)f_0^+ - h_\varepsilon'(u)f_0^- \le -h_\varepsilon'(u)f_0^-.\] By Young’s inequality, together with \(|h_\varepsilon'(x)|^2\le C_0 h_\varepsilon(x)\), \[h_\varepsilon'(u)f_0\leq -h_\varepsilon'(u)f_0^- \le \frac{1}{2}|h_\varepsilon'(u)|^2+\frac{1}{2}|f_0^-|^2 \le \frac{C_0}{2} h_\varepsilon(u)+\frac{1}{2}|f_0^-|^2.\] Therefore, it yields from 25 that \[\mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon(u(t,x))dx \le \mathbb{E}\int_{\mathbb{R}^d} h_\varepsilon(\varphi(x))dx + C\mathbb{E}\int_t^T \int_{\mathbb{R}^d} h_\varepsilon(u(s,x))dxds + C\mathbb{E}\int_t^T\int_{\mathbb{R}^d}|f_0^-(s,x)|^2dxds.\] Taking \(\varepsilon\rightarrow0\), by the dominated convergence theorem we have \[\mathbb{E}\int_{\mathbb{R}^d} |u^-(t,x)|^2dx \le \mathbb{E}\int_{\mathbb{R}^d} |\varphi^-(x)|^2dx + C\int_t^T \mathbb{E}\int_{\mathbb{R}^d}|u^-(s,x)|^2dxds + C\mathbb{E}\int_t^T\int_{\mathbb{R}^d}|f_0^-(s,x)|^2dxds,\] which leads to the negative part estimate in Lemma 11 by an application of Gronwall’s inequality. ◻

Proof of Theorem 10. Set \[\bar{u} = u_1 - u_2,\quad \bar{q} = q_1 - q_2,\quad \bar{r} = r_1 - r_2,\quad \bar{\varphi} = \varphi_1 - \varphi_2,\quad \bar{f} = f_{01} - f_{02}.\] By the linearity of BSPDIE 8 , the triple \((\bar{u},\bar{q},\bar{r})\) satisfies the same equation with the generator \((\bar{\varphi},\bar{f})\). The conditions of Theorem 10 indicate \[\bar{\varphi}(x) \ge 0\;\;\;\text{and} \;\;\;\bar{f}(t,x) \ge 0.\] and hence \[\bar{\varphi}^- = 0\;\;\;\text{and} \;\;\;\bar{f}^- = 0.\] By Lemma 11 we have \[\mathbb{E} \int_{\mathbb{R}^d} [\bar{u}(t,x)^-]^2 dx \le C e^{C(T-t)} \left( \mathbb{E} \int_{\mathbb{R}^d} [\bar{\varphi}(x)^-]^2 dx + \mathbb{E} \int_t^T \int_{\mathbb{R}^d} [\bar{f}(s,x)^-]^2 dx ds \right) = 0.\] Consequently, for any \(t\in[0,T]\), \[\mathbb{E} \int_{\mathbb{R}^d} [\bar{u}(t,x)^-]^2 dx = 0,\] which implies \[\bar{u}(t,x)^- = 0 \quad \text{a.e. a.s. }\] Therefore, for any \(t\in[0,T]\), \[u_1(t,x) - u_2(t,x)=\bar{u}(t,x) \ge 0 \quad \text{a.e. a.s. }\] ◻