In this section, we establish the convergence of the numerical approximations in Sect. 3 to the viscosity solution of the HJBVI (2.5). To simplify the notation, we will occasionally drop the terms \(\{K^\alpha u\}_{\alpha \in \mathbf{A }}\) and \(\{B^\alpha u\}_{\alpha \in \mathbf{A }}\) in (3.3), (3.5) and (3.6), and simply denote them by \(F(\mathbf{x},u,Du,D^2u)=0\) in the sequel.
Approximation by Non-singular Measures and Finite Control Sets
In this section, we shall study the approximations of HJBVI (2.5) with a non-singular measure and finite control set.
In fact, it is not difficult to see that (3.3) and (3.5) are consistent approximations of (2.5) in the viscosity sense (see e.g. [22, 29]), such that the comparison principle of (2.5) enables us to conclude that the solutions of (3.3) and (3.5) converge to that of (2.5) on compact sets as \(\varepsilon ,\delta \rightarrow 0\).
The remainder of this section thus focuses on obtaining an error estimate for these approximations in terms of the jump truncation size \(\varepsilon \) and control mesh size \(\delta \). Although it should be possible to extend the analytic arguments in Reference [22] to the present nonlinear setting, we follow a different path by first identifying the viscosity solution of (3.5) as the value function of a mixed control problem in terms of modified SDE and BSDE. This control-theoretical interpretation further enables us to develop a shorter proof of the convergence order of these approximations through a probabilistic argument.
We start with the truncation of singular measures. A possible way to work with a nonsingular jump measure is to introduce a Backward SDE with a modified driver and an approximative jump-diffusion dynamics where the small jumps part has been substituted by a rescaled diffusion coefficient of the Brownian motion W.
More precisely, we adopt the same probability space as introduced in Sect. 2, which supports the Brownian motion process W and the independent Poisson measure N(dt, de). For a given jump truncation size \(\varepsilon >0\), we define a modified diffusion coefficient \({\tilde{\sigma }}^\alpha \) as in (3.1), and also introduce a modified driver \(f^\varepsilon (\alpha ,t,x,y,z,k):={\hat{f}}(\alpha ,t, x,y,z,\int _{|e| \ge \varepsilon } k(e)\gamma (x,e)\nu (de)),\) where \({\hat{f}}\) is given in Assumption 2.1.
Next we shall modify the coefficients by including the control discretization. For any given control mesh size \(\delta >0\), we denote by \(\phi ^\delta \) the piecewise constant approximation (on the control variable \(\alpha \)) of a generic function \(\phi \) based on its value on \(\mathbf{A}_\delta \), where \(\phi =b, {\tilde{\sigma }}, \eta , f^\varepsilon \). Note that the Lipschitz continuity of the coefficients on the control parameter \(\alpha \) (see Assumption 2.1) and the condition (3.4) imply that there exists a constant \(C\ge 0\), such that it holds for any given \((\alpha ,t,x,u,p,k)\in \mathbf{A }\times [0,T]\times {{\mathbb {R}}}^d\times {{\mathbb {R}}}\times {{\mathbb {R}}}^d\times {{\mathbb {R}}}\) and discretization parameters \(\varepsilon ,\delta >0\) that
$$\begin{aligned} |b(\alpha ,x)-b^\delta (\alpha ,x)|+|\sigma (\alpha ,x)-{\tilde{\sigma }}^\delta ({\alpha },x)|&\le C\bigg (\delta +\bigg |\int _{|e|<\varepsilon } (1\wedge |e|^2)\, \nu (de)\bigg |^{\frac{1}{2}}\bigg ), \end{aligned}$$
(4.1)
$$\begin{aligned} |\eta (\alpha ,x,e)-\eta ^\delta (\alpha ,x,e)|&\le C\delta (1\wedge |e|), \end{aligned}$$
(4.2)
$$\begin{aligned} |{f}(\alpha ,t,x,u,p,k)-{f}^{\varepsilon ,\delta }({\alpha },t,x,u,p,k)|&\le C\bigg (\delta +\int _{|e|<\varepsilon } k(e)\gamma (x,e)\, \nu (de)\bigg ). \end{aligned}$$
(4.3)
For any given initial state \(x\in {{\mathbb {R}}}^d\), control \(\alpha \in \mathcal {A}_t^t\) and \(\tau \in \mathcal {T}_t^t\), we consider the modified controlled jump-diffusion process \((X^{\varepsilon ,\delta ,\alpha ,t, x}_s)_{ t \le s\le T}\) satisfying the following SDE: for each \(s\in [t,T]\),
$$\begin{aligned} X_s= & {} x+\int _t^s b^\delta (\alpha _v,X_v)\,dv+\int _t^s {\tilde{\sigma }}^\delta (\alpha _v,X_v)\,dW^t_v\nonumber \\&+\int _t^s\int _{|e|>\varepsilon } \eta ^\delta (\alpha _v,X_v,e)\,{\tilde{N}}^t(dv,de), \end{aligned}$$
(4.4)
and the BSDE with the modified controlled driver \(f^{\varepsilon ,\delta }(\alpha _s,s,X_s^{\varepsilon ,\delta ,\alpha ,t,x},y,z,k)\):
$$\begin{aligned} {\left\{ \begin{array}{ll} \!\begin{aligned} -dY^{\varepsilon ,\delta ,\alpha ,t,x}_{s,\tau }=&{}f^{\varepsilon ,\delta }(\alpha _s,s,X_s^{\varepsilon ,\delta ,\alpha ,t,x},Y^{\varepsilon ,\delta ,\alpha ,t,x}_{s,\tau },Z^{\varepsilon ,\delta ,\alpha ,t,x}_{s,\tau },K^{\varepsilon ,\delta ,\alpha ,t,x}_{s,\tau })ds\\ &{}\qquad -\,Z^{\varepsilon ,\delta ,\alpha ,t,x}_{s,\tau }dW^t_s-\int _{E} K^{\varepsilon ,\delta ,\alpha ,t,x}_{s,\tau }(e)\,{\tilde{N}}^t(ds,de),\qquad s\in [t,\tau );\\ Y^{\varepsilon ,\alpha ,t,x}_{\tau ,\tau }=&{}\xi (\tau ,X^{\varepsilon ,\alpha ,t,x}_\tau ). \end{aligned} \end{array}\right. }\nonumber \\ \end{aligned}$$
(4.5)
The coefficients of the above SDE and BSDE satisfy Assumption 2.1, and therefore the equations are well-posed.
Now for any each time \(t\in [0,T]\) and state \(x\in {{\mathbb {R}}}^d\), we consider the following value function:
$$\begin{aligned} u^{\varepsilon ,\delta }(t,x)= \sup _{\tau \in \mathcal {T}_t^t}\sup _{\alpha \in \mathcal {A}_t^t} \mathcal {E}^{f^{\varepsilon ,\delta ,\alpha }}_{t,\tau }[\xi (\tau , X^{\varepsilon ,\delta ,\alpha , t,x}_\tau )], \end{aligned}$$
(4.6)
subject to the controlled SDE (4.4), where the nonlinear expectation is induced by (4.5).
The following theorem shows that \(u^{\varepsilon ,\delta }\) is the unique viscosity solution of the HJBVI equation (3.5) introduced in Sect. 3.
Proposition 4.1
The function \((t,x)\mapsto u^{\varepsilon ,\delta }(T-t,x)\), with \(u^{\varepsilon ,\delta }\) defined by (4.6), is the unique viscosity solution of the HJBVI (3.5).
Proof
Due to the compactness of the set \(\mathbf A \), one can follow the same arguments in Reference [16, 17] and characterize \(u^{\varepsilon ,\delta }\) as the unique viscosity solution of an HJBVI with coefficients \(b^\delta ,{\tilde{\sigma }}^{\delta },\eta ^\delta , f^{\varepsilon ,\delta }\). Then it suffices to observe that such HJBVI is equivalent to (3.5) since its coefficients are piecewise linear in the control parameter. \(\square \)
Remark 4
Contrary to the case without control and optimal stopping studied in Reference [15], it is not clear that one can use a different approximation of the forward backward system by introducing an independent Brownian motion scaled with the standard deviation of small jumps. Indeed, the equations would be well-posed in an enlarged filtration \(\mathbb {G}\), but the control process is \(\mathbb {F}\)-predictable, with \(\mathbb {F} \subset \mathbb {G}\), which leads to difficulties in the derivation of the dynamic programming principle.
We now exploit the above control-theoretical characterization of the viscosity solution \(u^{\varepsilon ,\delta }\) and obtain a convergence order in terms of the jump truncation size \(\varepsilon \) and control mesh size \(\delta \). Let us first show the following uniform convergence result of the forward component \(X^{\varepsilon ,\delta , \alpha ,t,x}\) towards \(X^{\alpha ,t,x}\) when \(\varepsilon ,\delta \) tends to 0.
Lemma 4.2
For each \(\varepsilon ,\delta \in (0,1)\), \(p\ge 2\), \(t\in [0,T]\), \(x \in \mathbb {R}^d\) and \(\alpha \in \mathcal {A}_t^t\), it holds that
$$\begin{aligned}&\mathbb {E}\left[ \sup _{t \le s \le T} |X_s^{\varepsilon ,\delta , \alpha ,t,x}-X_s^{\alpha ,t,x}|^p\right] \nonumber \\&\quad \le C \bigg \{\delta ^p+ \bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg )^{p/2}+\bigg (\int _{|e|\le \varepsilon } |e|^p\nu (de)\bigg )\bigg \}, \end{aligned}$$
(4.7)
where C is a constant independent of \(t,x,\alpha ,\varepsilon \) and \(\delta \).
Proof
Fix \(\varepsilon ,\delta \in (0,1)\), \(\alpha \in \mathcal {A}_t^t\) and \(v \in [t,T]\). We have:
$$\begin{aligned}&{\mathbb {E}}\left[ \sup _{t \le u \le v} |X_u^{\varepsilon , \delta ,\alpha ,t,x}-X_u^{\alpha ,t,x}|^p\right] \\&\quad \le C \mathbb {E}\left[ \sup _{t \le u \le v} \left| \int _t^u (b^\delta (\alpha _s, X_s^{\varepsilon , \delta ,\alpha ,t,x})-b(\alpha _s, X_s^{\alpha ,t,x}))ds\right| ^p \right] \\&\qquad + C \mathbb {E}\left[ \sup _{t \le u \le v} \left| \int _t^u (\tilde{\sigma }^\delta (\alpha _s, X_s^{\varepsilon , \delta ,\alpha ,t,x})-\sigma (\alpha _s, X_s^{\alpha ,t,x}))dW_s\right| ^p \right] \\&\qquad +C \mathbb {E}\left[ \sup _{t \le u \le v} \left| \int _t^u \int _{|e| > \varepsilon }(\eta ^\delta (\alpha _s, X_s^{\varepsilon ,\delta , \alpha ,t,x},e)-\eta (\alpha _s, X_s^{\alpha ,t,x},e))\tilde{N}(ds,de)\right| ^p \right] \\&\qquad +C \mathbb {E}\left[ \sup _{t \le u \le v} \left| \int _t^u \int _{|e| \le \varepsilon }(\eta (\alpha _s, X_s^{\alpha ,t,x},e))\tilde{N}(ds,de)\right| ^p \right] , \end{aligned}$$
where C is a constant independent of \(\alpha \). The Burkholder–Davis–Gundy inequality, together with (4.1), (4.2), and the Lipschitz assumptions on the coefficients \(b, \sigma , \eta \) (see Assumption 2.1) lead to:
$$\begin{aligned}&\mathbb {E}\left[ \sup _{t \le u \le v} |X_u^{\varepsilon , \delta ,\alpha ,t,x}-X_u^{\alpha ,t,x}|^p\right] \\&\quad \le \, C\bigg \{\delta ^p+\bigg (\int _{|e|\le \varepsilon }(1 \wedge |e|^2)\nu (de)\bigg )^{\frac{p}{2}}\\&\qquad +\mathbb {E}\left[ \int _t^v \bigg (\sup _{t \le u \le s}\left| X_u^{\varepsilon ,\delta , \alpha ,t,x}-X_u^{\alpha ,t,x}\right| ^p\bigg )ds \right] \bigg \}\\&\qquad +C \mathbb {E}\left[ \left( \int _t^v \int _{|e|> \varepsilon }|\eta ^\delta (\alpha _s, X_s^{\varepsilon ,\delta , \alpha ,t,x},e)-\eta (\alpha _s, X_s^{\alpha ,t,x},e)|^2\nu (de)ds\right) ^{\frac{p}{2}} \right] \\&\qquad +C \mathbb {E}\left[ \int _t^v \int _{|e| > \varepsilon }|\eta ^\delta (\alpha _s, X_s^{\varepsilon ,\delta , \alpha ,t,x},e)-\eta (\alpha _s, X_s^{\alpha ,t,x},e)|^p\nu (de)ds \right] \\&\qquad + C \mathbb {E}\left[ \left( \int _t^v \int _{|e| \le \varepsilon }|\eta (\alpha _s, X_s^{\alpha ,t,x},e)|^2\nu (de)ds\right) ^{\frac{p}{2}}\right. \\&\qquad \left. + \left( \int _t^v \int _{|e| \le \varepsilon }|\eta (\alpha _s, X_s^{\alpha ,t,x},e)|^p\nu (de)ds\right) \right] \\&\quad \le \,C \bigg \{\mathbb {E}\left[ \int _t^v \left( \sup _{t \le u \le s}|X_u^{\varepsilon ,\delta , \alpha ,t,x}-X_u^{\alpha ,t,x}|^p\right) ds \right] \\&\qquad +\delta ^p+\bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg )^{\frac{p}{2}}+\bigg (\int _{|e|\le \varepsilon } |e|^p\nu (de)\bigg )\bigg \}, \end{aligned}$$
where the last inequality follows by the integrability assumption on the measure \(\nu \). Then we obtain the desired result (4.7) from the Gronwall’s inequality. \(\square \)
Based on the above estimate, we now show the convergence order of the viscosity function \(u^{\varepsilon ,\delta }\) of (3.5) towards u in terms of \(\varepsilon \) and \(\delta \).
Theorem 4.3
For any \(p>2\), there exists a constant \(C_p\) depending on p, such that for all \(\varepsilon ,\delta \in (0,1)\), we have
$$\begin{aligned}&\left| u^{\varepsilon ,\delta } (t,x)-u(t,x)\right| \le C_p \bigg \{ \delta +\bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg )^{\frac{1}{2}}\\&\quad +\bigg (\int _{|e|\le \varepsilon }|e|^p\nu (de)\bigg )^{\frac{1}{p}}\bigg \}, \quad (t,x) \in [0,T] \in \mathbb {R}^d. \end{aligned}$$
Proof
Fix \(\varepsilon ,\delta \in (0,1)\), \(t \in [0,T]\) and \(x \in \mathbb {R}^d\). The definitions of \(u^{\varepsilon ,\delta }\) and u imply that
$$\begin{aligned}&|u^{\varepsilon ,\delta }(t,x)-u(t,x)|^2 = \big |\sup _{\alpha \in \mathcal {A}_t^t } \sup _{\tau \in \mathcal {T}_t^t } \mathcal {E}^{f^{\varepsilon ,\delta ,\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\varepsilon ,\delta , \alpha ,t,x})\right] \nonumber \\&\quad -\sup _{\alpha \in \mathcal {A}_t^t } \sup _{\tau \in \mathcal {T}_t^t } \mathcal {E}^{f^{\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\alpha ,t,x})\right] \big |^2 \nonumber \\&\quad \le \sup _{\alpha \in \mathcal {A}_t^t } \sup _{\tau \in \mathcal {T}_t^t} \left| \mathcal {E}^{f^{\varepsilon ,\delta ,\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\varepsilon ,\delta , \alpha ,t,x})\right] - \mathcal {E}^{f^{\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\alpha ,t,x})\right] \right| ^2. \end{aligned}$$
(4.8)
Recall that, since \(\alpha \in \mathcal {A}_t^t\) and \(\tau \in \mathcal {T}_t^t\), \(\bigg |\mathcal {E}^{f^{\varepsilon ,\delta ,\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\varepsilon ,\delta , \alpha ,t,x})\right] - \mathcal {E}^{f^{\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\alpha ,t,x})\right] \bigg |\) is deterministic. By the a priori estimates on the spread between the first component of the solutions of two BSDEs with jumps (see Proposition A.4. in Reference [32]), we derive from (4.3) that there exist \(\beta >0\) and \(\eta >0\) independent on \(\tau \in \mathcal {T}_t^t\) and \(\alpha \in \mathcal {A}_t^t\), such that
$$\begin{aligned}&\left| \mathcal {E}^{f^{\varepsilon ,\delta ,\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\varepsilon ,\delta , \alpha ,t,x})\right] - \mathcal {E}^{f^{\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\alpha ,t,x})\right] \right| ^2 \\&\quad \le \mathbb {E}\left[ e^{\beta (\tau -t)} \left( \zeta (\tau , X_\tau ^{\varepsilon , \delta ,\alpha ,t,x})-\zeta (\tau , X_\tau ^{\alpha ,t,x})\right) ^2 \right] \\&\qquad +\eta \mathbb {E} \left[ \int _t^\tau e^{\beta (s-t)} \left( f(s,\alpha _s,X_s^{\alpha ,t,x},Y_{s,\tau }^{\alpha ,t,x},Z_{s,\tau }^{\alpha ,t,x},K_{s,\tau }^{\alpha ,t,x})\right. \right. \\&\qquad \left. \left. -f^{\varepsilon ,\delta }(s,\alpha _s,X_s^{\varepsilon ,\delta ,\alpha ,t,x},Y_{s,\tau }^{\alpha ,t,x},Z_{s,\tau }^{\alpha ,t,x},K_{s,\tau }^{\alpha ,t,x})\right) ^2 ds\right] \\&\quad \le C \bigg \{ \mathbb {E}\bigg [\sup _{t \le u \le T} |X_u^{\varepsilon ,\delta , \alpha ,t,x}-X_u^{\alpha ,t,x}|^2 \bigg ]+\delta ^2\\&\qquad +\mathbb {E}\bigg [ \int _t^\tau \bigg (\int _{|e| \le \varepsilon } K_{s,\tau }^{\alpha ,t,x}(e)\gamma (X_s^{\alpha ,t,x},e)\nu (de)\bigg )^2ds\bigg ] \\&\qquad + \mathbb {E}\bigg [ \int _t^\tau \bigg (\int _{|e|> \varepsilon } K_{s,\tau }^{\alpha ,t,x}(e)(\gamma (X_s^{\alpha ,t,x},e)-\gamma (X_s^{\varepsilon ,\delta ,\alpha ,t,x},e))\nu (de)\bigg )^2 ds\bigg ] \bigg \}, \end{aligned}$$
where C is a constant independent on \(t,x,\varepsilon ,\delta ,\alpha , \tau \), only depending on \(\beta \), \(\eta \), T and the Lipschitz constant of f.
Now we estimate the last two terms in the above inequality. For any given \(p\ge 2\), the uniform boundness of \(\zeta \), g and f with respect to \(t,x,\alpha \) and \(\tau \) (see Assumption 2.1), together with the a priori estimates for \(L^p\) solutions of BSDEs (see Proposition 2 in Reference [27]) gives us an uniform control on the \(\mathbb {H}^p_{t,\nu }\) norm of \(K_{\cdot ,\tau }^{\alpha ,t,x}\) (which only depends on p and the bounds of \(\zeta \), g, f and T). Using this result and the boundedness of the map \(\gamma \) (see Assumption 2.1), we derive from Hölder’s inequality that there exists a constant C independent on \(\tau \) and \(\alpha \) such that
$$\begin{aligned}&\mathbb {E}\bigg [ \int _t^\tau \bigg (\int _{|e| \le \varepsilon } K_{s,\tau }^{\alpha ,t,x}(e)\gamma (X_s^{\alpha ,t,x},e)\nu (de)\bigg )^2ds\bigg ]\\&\quad \le \mathbb {E}\bigg [ \int _t^\tau \bigg (\int _{|e| \le \varepsilon } (K_{s,\tau }^{\alpha ,t,x})^2(e) \nu (de)\bigg ) \bigg (\!\int _{|e| \le \varepsilon } \gamma ^2(X_s^{\alpha ,t,x},e)\nu (de)\bigg )ds\bigg ]\\&\quad \le C\bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg ). \end{aligned}$$
Furthermore, for any given \(p>1\), by using the Lipschitz continuity of the map \(\gamma \), we can obtain
$$\begin{aligned}&\mathbb {E}\bigg [ \int _t^\tau \bigg (\int _{|e|> \varepsilon } K_{s,\tau }^{\alpha ,t,x}(e)(\gamma (X_s^{\alpha ,t,x},e)-\gamma (X_s^{\varepsilon ,\delta ,\alpha ,t,x},e))\nu (de)\bigg )^2 ds\bigg ] \nonumber \\&\le C \mathbb {E}\bigg [ \int _t^\tau \bigg (\int _{|e|> \varepsilon } |K_{s,\tau }^{\alpha ,t,x}(e)||X_s^{\alpha ,t,x}-X_s^{\varepsilon ,\delta ,\alpha ,t,x}|(1\wedge |e|^2)\nu (de)\bigg )^2 ds\bigg ]\nonumber \\&\le C \mathbb {E}\bigg [ \sup _{t\le u\le T}|X_u^{\alpha ,t,x}-X_u^{\varepsilon ,\delta ,\alpha ,t,x}|^2 \int _t^\tau \bigg (\int _{E} |K_{s,\tau }^{\alpha ,t,x}(e)|(1\wedge |e|^2)\nu (de)\bigg )^2 ds\bigg ]\nonumber \\&\le C \mathbb {E}\bigg [ \sup _{t\le u\le T}|X_u^{\alpha ,t,x}-X_u^{\varepsilon ,\delta ,\alpha ,t,x}|^2 \bigg (\int _t^\tau \int _{E} |K_{s,\tau }^{\alpha ,t,x}(e)|^2\nu (de)ds\bigg ) \bigg (\int _{E} (1\wedge |e|^4)\nu (de)\bigg )\bigg ]\nonumber \\&\le C_p \bigg (\mathbb {E}\bigg [ \sup _{t\le u\le T}|X_u^{\alpha ,t,x}-X_u^{\varepsilon ,\delta ,\alpha ,t,x}|^{2p}\bigg ]\bigg )^{1/p}, \end{aligned}$$
(4.9)
where we have used Hölder’s inequality and the boundedness of the \(\mathbb {H}^{2p/(p-1)}_{t,\nu }\) norm of \(K_{\cdot ,\tau }^{\alpha ,t,x}\) for the last line. Consequently, by summarizing all the above estimates and using Lemma 4.2, we can obtain that
$$\begin{aligned}&\left| \mathcal {E}^{f^{\varepsilon ,\delta ,\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\varepsilon ,\delta , \alpha ,t,x})\right] - \mathcal {E}^{f^{\alpha }}_{t,\tau }\left[ \zeta (\tau , X_\tau ^{\alpha ,t,x})\right] \right| ^2 \\&\quad \le \, C \bigg \{ \mathbb {E}\bigg [\sup _{t \le u \le T} |X_u^{\varepsilon ,\delta , \alpha ,t,x}-X_u^{\alpha ,t,x}|^2 \bigg ]+\delta ^2\\&\qquad +\int _{|e|\le \varepsilon }|e|^2\nu (de)+\bigg (\mathbb {E}\bigg [ \sup _{t\le u\le T}|X_u^{\varepsilon ,\delta ,\alpha ,t,x}-X_u^{\alpha ,t,x}|^{2p}\bigg ]\bigg )^{\frac{1}{p}}\bigg \}\\&\quad \le \,C \bigg \{ \delta ^2+\int _{|e|\le \varepsilon }|e|^2\nu (de)+\bigg [\delta ^{2p}+\bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg )^p\\&\qquad +\bigg (\int _{|e|\le \varepsilon }|e|^{2p}\nu (de)\bigg )\bigg ]^{\frac{1}{p}}\bigg \}\\&\quad \le \,C \bigg \{ \delta ^2+\bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg )+\bigg (\int _{|e|\le \varepsilon }|e|^{2p}\nu (de)\bigg )^{\frac{1}{p}}\bigg \}, \end{aligned}$$
from which we can conclude the desired estimate by taking the supremum over \(\alpha \) and \(\tau \). \(\square \)
Theorem 4.3 extends the continuous dependence result for classical nonlocal HJBVIs in Reference [22, Theorem 4.4] to the HJBVIs with nonlinear drivers and state-dependent measures (the operator \(B^\alpha \) defined by (2.8) involves the measure \(\gamma (x,e)\nu (de)\), which depends on the spatial variable x).
Due to the presence of the state-dependent measure, in particular the term (4.9), our error estimate has an additional term \(\big (\int _{|e|\le \varepsilon }|e|^p\nu (de)\big )^{1/p}\), which in general cannot be compared to the term \(\big (\int _{|e|\le \varepsilon }|e|^2\nu (de)\big )^{1/2}\) without further information. For example, if \(\nu \) is finite around zero, then for small enough \(\varepsilon \), the fact that \(p>2\) and Jensen’s inequality lead to the following estimate:
$$\begin{aligned}&\bigg (\int _{|e|\le \varepsilon }|e|^2\nu (de)\bigg )^{\frac{p}{2}}=\nu (B_\varepsilon )^{\frac{p}{2}}\bigg (\int _{|e|\le \varepsilon }|e|^2\frac{\nu (de)}{\nu (B_\varepsilon )}\bigg )^{\frac{p}{2}}\\&\quad \le \nu (B_\varepsilon )^{\frac{p}{2}-1}\bigg (\int _{|e|\le \varepsilon }|e|^p{\nu (de)}\bigg )\le \int _{|e|\le \varepsilon }|e|^p{\nu (de)}, \end{aligned}$$
where we denote \(\nu (B_\varepsilon )=\nu (\{e\in E\mid 0<|e|\le \varepsilon \})\). On the other hand, if we assume that the singular measure \(\nu \) behaves similar to the Lévy measures of \(\alpha \)-stable processes around zero, in the sense that \(\nu \) admits a density \(\rho (e)\) such that it holds for some constants \(C> 0\) and \(\kappa \in [0, 2)\) that
$$\begin{aligned} 0\le \rho (e)\le C|e|^{-n-\kappa }, \quad |e|<1, \, e\in E={{\mathbb {R}}}^n\setminus \{0\}, \end{aligned}$$
then a direct computation shows \(\big (\int _{|e|\le \varepsilon }|e|^p\nu (de)\big )^{1/p}=\mathcal {O}(\varepsilon ^{(p-\kappa )/p})\), \(p\ge 2\), which implies the jump truncation error is dominated by the term \(\big (\int _{|e|\le \varepsilon }|e|^2\nu (de)\big )^{1/2}\), and consequently we recover the same convergence rate as that for the classical setting.
Approximation by Switching Systems
In this section, we study the approximation of (3.5) by switching systems. We adopt the following standard definition of a viscosity solution to switching systems of the form (3.6) (see [1, 6, 29] and references therein).
Definition 4.4
(Viscosity solution of switching system) A \({{\mathbb {R}}}^J\)-valued upper (resp. lower) semicontinuous function U is said to be a viscosity subsolution (resp. supersolution) of (3.6) if and only if for any point \(\mathbf{x}_0\) and for any \(\phi \in C^{1,2}({\bar{\mathcal {Q}}}_T)\) such that \(U_j-\phi \) attains its global maximum (resp. minimum) at \(\mathbf{x}_0\), one has
$$\begin{aligned}&F_{j*}^{\varepsilon ,\delta ,c}(\mathbf{x}_0,U_j(\mathbf{x}_0), D\phi (\mathbf{x}_0), D^2\phi (\mathbf{x}_0),\{K_\varepsilon ^\alpha \phi (\mathbf{x}_0)\}_{\alpha \in \mathbf{A }_\delta },\\&\qquad \{B^\alpha _\varepsilon \phi (\mathbf{x}_0)\}_{\alpha \in \mathbf{A }_\delta },\{U_k(\mathbf{x}_0)\}_{k\not =j})\le 0\\&\quad \big (resp. \quad F_j^{\varepsilon ,\delta ,c\,*}(\mathbf{x}_0,U_j(\mathbf{x}_0), D\phi (\mathbf{x}_0), D^2\phi (\mathbf{x}_0),\{K_\varepsilon ^\alpha \phi (\mathbf{x}_0)\}_{\alpha \in \mathbf{A }_\delta },\\&\qquad \{B^\alpha _\varepsilon \phi (\mathbf{x}_0)\}_{\alpha \in \mathbf{A }_\delta }, \{U_k(\mathbf{x}_0)\}_{k\not =j})\ge 0\big ). \end{aligned}$$
A continuous function is a viscosity solution of the HJBVI (3.6) if it is both a a viscosity sub- and supersolution.
Note that in the definition of the viscosity solution of \(F_j\), the test function only replaces \(U_j\) in the integrals and derivatives, while leaving the terms \(\{U_k\}_{k\not =j}\) unchanged.
Now we present the comparison principle for bounded semicontinuous viscosity solutions of (3.6), which not only implies the uniqueness of bounded viscosity solutions of (3.6), but is also essential for our convergence analysis. The proof will be given in Appendix 1.
Theorem 4.5
Let \(U{=(U_1,U_2,...,U_J)}\) and \(V {=(V_1,V_2,...,V_J)}\) be bounded viscosity sub- and supersolutions, respectively, of (3.6) with \(U(0,\cdot )\le V(0,\cdot )\). Then it holds under Assumption 2.1 that \(U_j(\mathbf{x})\le V_j(\mathbf{x})\) for all \(j=1,\ldots , J\).
The following theorem demonstrates the convergence of the switching system to the finite control HJBVI (3.5) as the switching cost goes to 0. Convergence with order 1 / 3 is proved in Reference [6] by a different technique, for nonlocal Bellman equations without obstacles and nonlinear source terms.
We momentarily assume the switching system (3.6) to admit a viscosity solution bounded independently of the (small enough) switching cost c. We give a constructive proof of existence through our numerical schemes in Sect. 4.3.
Theorem 4.6
Under Assumption 2.1, let \(U^{\varepsilon ,\delta , c}=(U^{\varepsilon ,\delta , c}_1,\ldots , U^{\varepsilon ,\delta , c}_J)\) and \(u^{\varepsilon ,\delta }\) be the viscosity solution of (3.6) and (3.5), respectively. Then for fixed \(\varepsilon , \delta >0\), we have for each \(j=1,\ldots , J\) that \( U^{\varepsilon ,\delta , c}_j\rightarrow u^{\varepsilon ,\delta } \) uniformly on compact sets as \(c\rightarrow 0\).
Proof
Since \(\varepsilon \) and \(\delta \) are fixed for our analysis, we shall omit the dependence on \(\varepsilon \) and \(\delta \), and simply denote by \(U^{c}\) the solution of (3.6). Consider a sequence of switching costs \(c_m\rightarrow 0\) as \(m\rightarrow \infty \), and the corresponding viscosity solution \(U^{c_m}=(U^{c_m}_1,\ldots , U^{c_m}_J)\). We shall first prove by contradiction that
$$\begin{aligned} U^{c_m}_j(\mathbf{x})\ge \mathcal {M}_j U^{c_m},\quad \mathbf{x}\in \mathcal {Q}_T, \; j=1,\ldots , J. \end{aligned}$$
(4.10)
Suppose the statement is false, then there would exist \(k\not = j\) and \(\mathbf{x}_0\in \mathcal {Q}_T\) such that \(U^{c_m}_j(\mathbf{x}_0)< U^{c_m}_k(\mathbf{x}_0)-c_m\). We then obtain from the continuity of \(U_j^{c_m}\) and \(U^{c_m}_k\) that there exists a nonempty open ball B around \(\mathbf{x}_0\) such that
$$\begin{aligned} U^{c_m}_j(\mathbf{x})< U^{c_m}_k(\mathbf{x})-c_m, \quad \mathbf{x}\in B. \end{aligned}$$
On the other hand, due to the fact that semi-jets are nonempty on a dense set (see e.g. [25, Lemma 8 on pp. 23]), there exists a \(C^2\) function \(\phi \) such that \(U^{c_m}_j-\phi \) attains its minimum at some point in B, say \(\mathbf{x}_1\). Hence we deduce from the fact that \(U^{c_m}_j\) is a supersolution that
$$\begin{aligned} U^{c_m}_j(\mathbf{x}_1)\ge \mathcal {M}_j U^{c_m}(\mathbf{x}_1)\ge U^{c_m}_k(\mathbf{x}_1)-c_m, \end{aligned}$$
which leads to a contradiction.
We now introduce the following functions through a relaxed limit: for \(j=1,\ldots , J\),
$$\begin{aligned} {\overline{U}}_j(x)=\lim _{r\rightarrow \infty }\sup _{m>r}\sup _{|\mathbf{y}-\mathbf{x}|<1/r}U^{c_m}_j(\mathbf{y}),\quad \underline{U}_j(x)=\lim _{r\rightarrow \infty }\inf _{m>r}\inf _{|\mathbf{y}-\mathbf{x}|<1/r}U^{c_m}_j(\mathbf{y}). \end{aligned}$$
(4.11)
It is not hard to check \({\overline{U}}_1=\ldots ={\overline{U}}_J\equiv {\overline{U}}\) and \(\underline{U}_1=\ldots =\underline{U}_J\equiv \underline{U}\). In fact, for any given \(j,k\in \{1,\ldots , J\}\), \(j\not =k\), \(\mathbf{x}\in \mathcal {Q}_T\), and \(m,r\in {\mathbb {N}}\), we obtain from (4.10) that \(U^{c_m}_j(\mathbf{y})\ge U^{c_m}_k(\mathbf{y})-c_m\) for \(\mathbf{y}\in \mathcal {Q}_T\), and hence
$$\begin{aligned} \sup _{m>r}\sup _{|\mathbf{y}-\mathbf{x}|<1/r}U^{c_m}_j(\mathbf{y})\ge \sup _{m>r}\sup _{|\mathbf{y}-\mathbf{x}|<1/r}U^{c_m}_k(\mathbf{y})-\sup _{m>r}c_m. \end{aligned}$$
Letting \(r\rightarrow \infty \) leads to the fact that \({\overline{U}}_j\ge {\overline{U}}_k\) for all \(j\not = k\). The statement for \(\{\underline{U}_j\}\) can be shown similarly.
Since it is clear that \({\overline{U}}\) and \(\underline{U}\) is bounded upper and lower semicontinuous, respectively, we now aim to show \({\overline{U}}\) and \(\underline{U}\) is respectively a sub- and supersolution of (3.5). Then the strong comparision principle gives us \({\overline{U}}\le \underline{U}\), which implies \(U={\overline{U}}= \underline{U}\) is the unique viscosity solution of (3.5). Uniform convergence on compact sets follows from a variation of Dini’s theorem (See Remark 6.4 in Reference [12]).
We start by showing \({\overline{U}}\) is a subsolution of (3.5). Let \(\phi \in C^{1,2}\) and \({\overline{U}}-\phi \) have a strict global maximum at \({\hat{\mathbf{x}}}_0\in {\bar{\mathcal {Q}}}_T\), then there will be a sequence \(c_m\rightarrow 0\) such that for each \(j\in \{1,\ldots , J\}\), we have \({\hat{\mathbf{x}}}^j_m\rightarrow {\hat{\mathbf{x}}}_0\), \(U_j^{c_m}({\hat{\mathbf{x}}}^j_m)\rightarrow {\overline{U}}({\hat{\mathbf{x}}}_0)\), and \(U_j^{c_m}-\phi \) attains a global maximum at \({\hat{\mathbf{x}}}^j_m\). Since \(U_j^{c_m}\) is a subsolution of (3.6) with \(c_m\), if we have \({\hat{\mathbf{x}}}_0\in \{0\}\times {{\mathbb {R}}}^d\), \(U_j^{c_m}({\hat{\mathbf{x}}}^j_m)\le g({\hat{\mathbf{x}}}^j_m)\) for infinitly many m and a fixed j, then it is clear that \({\overline{U}}({\hat{\mathbf{x}}}_0)\le g({\hat{\mathbf{x}}}_0)\). Therefore, without loss of generality, we assume for all m and j that
$$\begin{aligned}&\min \bigg [U_j^{c_m}({\hat{\mathbf{x}}}^j_m)-\zeta ({\hat{\mathbf{x}}}^j_m),\; \min \big (\phi _{t}({\hat{\mathbf{x}}}^j_m)-L_\varepsilon ^{\alpha _j} \phi ({\hat{\mathbf{x}}}^j_m)\nonumber \\&\quad -f^{\alpha _j}({\hat{\mathbf{x}}}^j_m,U^{c_m}_{j}({\hat{\mathbf{x}}}^j_m),{\tilde{\sigma }}^{\alpha _j}\cdot D\phi ({\hat{\mathbf{x}}}^j_m),B_\varepsilon ^{\alpha _j} \phi ({\hat{\mathbf{x}}}^j_m)); \nonumber \\&\quad U^{c_m}_{j}({\hat{\mathbf{x}}}^j_m)-\mathcal {M}_jU^{c_m}({\hat{\mathbf{x}}}^j_m)\big )\bigg ]\le 0. \end{aligned}$$
(4.12)
We have two cases. If there exists \(j\in \{1,\ldots , J\}\) and a subsequence of \(c_m\) such that \(U_j^{c_m}({\hat{\mathbf{x}}}^j_m)-\zeta ({\hat{\mathbf{x}}}^j_m)\le 0\), then by passing to the limit \(m\rightarrow \infty \), we have \({\overline{U}}({\hat{\mathbf{x}}}_0)-\zeta ({\hat{\mathbf{x}}}_0)\le 0\). Otherwise, by passing to subsequence, without loss of generality we can assume \(U_j^{c_m}({\hat{\mathbf{x}}}^j_m)-\zeta ({\hat{\mathbf{x}}}^j_m)>0\) holds for all j and m. Then for each \(m\in {\mathbb {N}}\), we can choose \(j_m\in \{1,\ldots , J\}\) and \({\hat{\mathbf{x}}}^{j_m}_m\) such that
$$\begin{aligned} (U_{j_m}^{c_m}-\phi )({\hat{\mathbf{x}}}^{j_m}_m)=\max _{j=1,\ldots , J}(U_{j}^{c_m}-\phi )({\hat{\mathbf{x}}}^{j}_m)=\max _{j=1,\ldots , J}\max _{\mathbf{x}}(U_{j}^{c_m}-\phi )(\mathbf{x}), \end{aligned}$$
and deduce from (4.12) that
$$\begin{aligned} \min \big (\phi _{t}({\hat{\mathbf{x}}}^{j_m}_m)-L_\varepsilon ^{\alpha _{j_m}} \phi ({\hat{\mathbf{x}}}_m)-&f^{\alpha _{j_m}}({\hat{\mathbf{x}}}^{j_m}_m,U^{c_m}_{{j_m}}({\hat{\mathbf{x}}}^{j_m}_m),{\tilde{\sigma }}^{\alpha _{j_m}}\cdot D\phi ({\hat{\mathbf{x}}}^{j_m}_m),B_\varepsilon ^{\alpha _{j_m}}\phi ({\hat{\mathbf{x}}}^{j_m}_m)); \nonumber \\&U^{c_m}_{{j_m}}({\hat{\mathbf{x}}}^{j_m}_m)-\mathcal {M}_{j_m}U^{c_m}({\hat{\mathbf{x}}}^{j_m}_m)\big )\le 0. \end{aligned}$$
(4.13)
Our choice of \(j_m\) implies \((U_{j_m}^{c_m}-\phi )({\hat{\mathbf{x}}}^{j_m}_m)\ge (U_{k}^{c_m}-\phi )({\hat{\mathbf{x}}}^{j_m}_m)\) for all \(k\not = j_m\), and thus \(U^{c_m}_{{j_m}}({\hat{\mathbf{x}}}^{j_m}_m)>\mathcal {M}_{j_m}U^{c_m}({\hat{\mathbf{x}}}^{j_m}_m)\). Consequently we obtain from (4.13) that
$$\begin{aligned} \phi _{t}({\hat{\mathbf{x}}}^{j_m}_m)-L_\varepsilon ^{\alpha _{j_m}} \phi ({\hat{\mathbf{x}}}_m)-f^{\alpha _{j_m}}({\hat{\mathbf{x}}}^{j_m}_m,U^{c_m}_{{j_m}}({\hat{\mathbf{x}}}^{j_m}_m),{\tilde{\sigma }}^{\alpha _{j_m}}\cdot D\phi ({\hat{\mathbf{x}}}^{j_m}_m),B_\varepsilon ^{\alpha _{j_m}} \phi ({\hat{\mathbf{x}}}^{j_m}_m))\le 0. \end{aligned}$$
Since we only have finite many choices of \(j_m\), by passing to a further subsequence if necessary, we can assume that \(j_m\rightarrow j_0\), then letting \(m\rightarrow \infty \) and using the continuity of the equation, we have
$$\begin{aligned} \phi _{t}({\hat{\mathbf{x}}}_0)-L_\varepsilon ^{\alpha _{j_0}} \phi ({\hat{\mathbf{x}}}_0)-f^{\alpha _{j_0}}({\hat{\mathbf{x}}}_0,{\overline{U}}({\hat{\mathbf{x}}}_0),{\tilde{\sigma }}^{\alpha _{j_0}}\cdot D\phi ({\hat{\mathbf{x}}}_0),B_\varepsilon ^{\alpha _{j_0}} \phi ({\hat{\mathbf{x}}}_0))\le 0. \end{aligned}$$
Since \(\alpha _{j_0}\in \mathbf{A }_\delta \) is an admissible control, we obtain
$$\begin{aligned} \min _{\alpha \in \mathbf{A }_\delta }\big \{\phi _{t}({\hat{\mathbf{x}}}_0)-L_\varepsilon ^{\alpha } \phi ({\hat{\mathbf{x}}}_0)-f^{\alpha _{j}}({\hat{\mathbf{x}}}_0,{\overline{U}}({\hat{\mathbf{x}}}_0),{\tilde{\sigma }}^{\alpha }\cdot D\phi ({\hat{\mathbf{x}}}_0),B_\varepsilon ^\alpha \phi ({\hat{\mathbf{x}}}_0))\big \}\le 0, \end{aligned}$$
and conclude that \({\overline{U}}\) is a subsolution of (3.6).
We now proceed to show \(\underline{U}\) is a supersolution. If \(\phi \in C^{1,2}\) and \({\overline{U}}-\phi \) has a strict global mimimum at \({\hat{\mathbf{x}}}_0\in {\bar{\mathcal {Q}}}_T\), then for any given \(j\in \{1,\ldots , J\}\), there will be sequences \(c_m\rightarrow 0\), \({\hat{\mathbf{x}}}_m\rightarrow {\hat{\mathbf{x}}}_0\), \(U_j^{c_m}({\hat{\mathbf{x}}}_m)\rightarrow \underline{U}({\hat{\mathbf{x}}}_0)\), and \(U_j^{c_m}-\phi \) attains a global mimimum at \({\hat{\mathbf{x}}}_m\). Using the fact that \(U_j^{c_m}\) is asupersolution to (3.6), we have (by ignoring the term \(U^{c_m}_{j}({\hat{\mathbf{x}}}^j_m)-\mathcal {M}_jU^{c_m}({\hat{\mathbf{x}}}^j_m)\)):
$$\begin{aligned}&\min \bigg [U_j^{c_m}({\hat{\mathbf{x}}}_m)-\zeta ({\hat{\mathbf{x}}}_m),\; \phi _{t}({\hat{\mathbf{x}}}_m)-L_\varepsilon ^{\alpha _j} \phi ({\hat{\mathbf{x}}}_m)\\&\quad -f^{\alpha _j}({\hat{\mathbf{x}}}_m,U^{c_m}_{j}({\hat{\mathbf{x}}}_m),{\tilde{\sigma }}^{\alpha _j}\cdot D\phi ({\hat{\mathbf{x}}}_m),B_\varepsilon ^{\alpha _j} \phi ({\hat{\mathbf{x}}}_m))\bigg ]\ge 0, \end{aligned}$$
then passing \(m\rightarrow \infty \) enables us to conclude for any \(j\in \{1,\ldots , J\}\),
$$\begin{aligned}&\min \bigg [\underline{U}({\hat{\mathbf{x}}}_0)-\zeta ({\hat{\mathbf{x}}}_0),\; \phi _{t}({\hat{\mathbf{x}}}_0)-L_\varepsilon ^{\alpha _j} \phi ({\hat{\mathbf{x}}}_0)\\&\quad -f^{\alpha _j}({\hat{\mathbf{x}}}_0,\underline{U}({\hat{\mathbf{x}}}_0),{\tilde{\sigma }}^{\alpha _j}\cdot D\phi ({\hat{\mathbf{x}}}_0),B_\varepsilon ^{\alpha _j} \phi ({\hat{\mathbf{x}}}_0))\bigg ]\ge 0, \end{aligned}$$
which completes our proof. \(\square \)
General Discrete Approximation to the Switching System
In this section, we establish the convergence of the piecewise constant policy approximation of (3.10) to the solution of the switching system (3.6). We will first summarize all the required conditions to guarantee the convergence, and perform the analysis under these assumptions. Then we will demonstrate in Sect. 4.4 that these conditions are in fact satisfied by the numerical scheme (3.18) proposed in Sect. 3 .
We assume the scheme (3.10) satisfies the following conditions introduced in Reference [29]:
Condition 1
-
(1)
(Positive interpolation.) Let \({\tilde{U}}^n_{k,i(j)}\) be the interpolant of the k-th grid onto the i-th point \(\mathbf{x}_{j,i}^n\) of the j-th grid, and \(N^k(j,i,n)\) be the neighboursFootnote 2 to the point \(\mathbf{x}_{j,i}^n\) on the k-th grid \(\Omega _{k,h}\). Then there exist weights \(\{\omega ^n_{k,i(j),a}\}_{a\in N^k(j,i,n)}\) satisfying \(\omega ^n_{k,i(j),a}\ge 0\) and \(\sum _{a\in N^k(j,i,n)}\omega ^n_{k,i(j),a}=1\), such that we can write
$$\begin{aligned} {\tilde{U}}^n_{k,i(j)}=\sum _{a\in N^k(j,i,n)}\omega ^n_{k,i(j),a} U^n_{k,a}. \end{aligned}$$
(4.14)
-
(2)
(Weak monotonicity.) The scheme (3.10) is monotone with respect to \(U^n_{j,i}\) and \({\tilde{U}}^{n}_{k,i(j)}\), i.e., if
$$\begin{aligned} V^n_{j,i}\ge U^n_{j,i}, \quad \forall (i,j,n); \quad {\tilde{V}}^{n}_{k,i(j)}\ge {\tilde{U}}^{n}_{k,i(j)}, \quad \forall (i,k,n), \end{aligned}$$
then we have
$$\begin{aligned}&G_j(\mathbf{x}^n_{j,i},h,U^{n+1}_{j,i}, \{V^{b+1}_{j,a}\}_{{(a,b)\!\not =\!(i,n)}}, \{ {\tilde{V}}^n_k\}_{k\not =j})\nonumber \\&\quad \le G_j(\mathbf{x}^n_{j,i},h,U^{n+1}_{j,i}, \{U^{b+1}_{j,a}\}_{(a,b)\!\not =\!(i,n)}, \{ {\tilde{U}}^n_k\}_{k\not =j}). \end{aligned}$$
(4.15)
-
(3)
(\(\ell ^\infty \) stability.) The solution \(U^{n+1}_{j,i}\) of the scheme (3.10) exists and is bounded uniformly in h and c.
-
(4)
(Consistency.) Let \(\varepsilon ,\delta , c\) be fixed. For any test functions \(\phi _j\in C^{1,2}({\bar{\mathcal {Q}}}_T)\) and continuous \(\varphi _k\), there exist function \(\omega _1(h)\) and \(\omega _2(\xi )\), possibly depending on \(\varepsilon \), such that \(\omega _1(h)\rightarrow 0\) as \(h\rightarrow 0\), \(\omega _2(\xi )\rightarrow 0\) as \(\xi \rightarrow 0\), and
$$\begin{aligned} \begin{aligned}&|G_j(\mathbf{x}^{n+1}_{j,i},h,\phi ^{n+1}_{j,i}+\xi , \{\phi ^{b+1}_{j,a}\}_{(a,b)\!\not =\!(i,n)}+\xi , \{{\tilde{\varphi }}^n_k\}_{k\not =j})\\&\quad -F^{\varepsilon ,\delta , c}_j(\mathbf{x}^{n+1}_{j,i},\phi _j(\mathbf{x}^{n+1}_{j,i}), D\phi _j(\mathbf{x}^{n+1}_{j,i}), D^2\phi _j(\mathbf{x}^{n+1}_{j,i}),\\&\quad \{{\tilde{\varphi }}_k(\mathbf{x}^{n}_{j,i})\}_{k\not =j})|\le \omega _1(h)+\omega _2(\xi ). \end{aligned} \end{aligned}$$
(4.16)
Remark 5
As pointed out in Reference [29], Condition 1 (1)–(2) are weaker than the standard condition that the scheme is monotone in \(U^n_{k,\alpha }\) (see e.g. [3]). By only requiring that the interpolation has positive coefficients and that the numerical scheme is monotone in the interpolant \({\tilde{U}}^n_{k,\alpha }\), we are allowing the usage of high order nonlinear interpolations among different grids (e.g., the monotonicity preserving interpolations in Reference [19]).
Also note the contrast to the linear interpolant (3.11) used in (3.12) and (3.13) for the construction of a monotone approximation to the integral operators.
We now present the convergence of the discrete approximation to the switching system.
Theorem 4.7
Under Assumptions 2.1, the solution to any scheme of the form (3.10) satisfying Condition 1 converges to the viscosity solution of (3.6) uniformly on bounded domains.
The proof is essentially the same as that in Reference [29] and is omitted. We remark that in the proof, we construct the solution of the switching system directly from the numerical solutions. Since the solution of the scheme (3.10) is uniformly bounded, Theorems 4.5 and 4.7 immediately give the existence and uniqueness of a bounded viscosity solution to the switching system (3.6).
Corollary 4.8
Under Assumption 2.1 and the existence of a scheme satisfying Condition 1, the switching system (3.6) admits a unique viscosity solution bounded uniformly in c.
A Specific Implicit Scheme for the Switching System
In this section, we analyze the implicit scheme (3.18) and demonstrate that it satisfies Condition 1, which subsequently implies its convergence to the switching system.
The following estimates are essential for our consistency and stability analysis.
Lemma 4.9
Under Assumption 2.1, there exists C independent of \(h,k,\varepsilon , \delta \) such that for any test functions \(\phi _j\in C^{1,2}({\bar{\mathcal {Q}}}_T)\) and \(\varepsilon <1\) that
$$\begin{aligned}&|A^{\alpha }_{\varepsilon ,h,k}\phi ^{n+1}_{j,i}+K^{\alpha ,1}_{\varepsilon ,h}\phi ^{n+1}_{j,i} -A_\varepsilon ^{\alpha } \phi _j(\mathbf{x}_{j,i}^{n+1})\\&\quad -K_\varepsilon ^{\alpha } \phi _j(\mathbf{x}_{j,i}^{n+1})|\le C \left( \frac{h^2}{k^2}+\frac{h^2}{\varepsilon ^2}+\omega (\mathbf{x}^{n+1}_{j,i},k)\right) ,\\&|B^{\alpha }_{\varepsilon ,h}\phi ^{n+1}_{j,i}-B_\varepsilon ^{\alpha }\phi (\mathbf{x}^{n+1}_{j,i})|\le C\frac{h^2}{\varepsilon }. \end{aligned}$$
for some \(\omega (\mathbf{x}^{n+1}_{j,i},k)\) such that \(\omega (\cdot , k)\rightarrow 0\) as \(k\rightarrow 0\) uniformly on compact neighbourhoods of \(\mathbf{x}^{n+1}_{j,i}\).
Proof
We first derive the estimate for \(B^{\alpha }_{\varepsilon ,h}\phi ^{n+1}_{j,i}\). It follows from \(|\eta ^\alpha |\le C\) and the definitions of \(B^{\alpha }_{\varepsilon ,h}\phi \) and \(B_\varepsilon ^{\alpha }\phi \) that
$$\begin{aligned}&|B^{\alpha }_{\varepsilon ,h}\phi ^{n+1}_{j,i}-B_\varepsilon ^{\alpha }\phi (\mathbf{x}^{n+1}_{j,i})|\\&\quad \le \int _{|e|\ge \varepsilon }|\mathcal {I}_h[\phi (t^{n+1},x_{j,i}+\cdot )](\eta ^\alpha (x_{j,i},e))\\&\quad -\phi (t^{n+1},x_{j,i}+\eta ^\alpha (x_{j,i},e))| \gamma (x_{j,i},e)\,\nu (de)\\&\quad \le Ch^2|D^2\phi |_{B(\mathbf{x}^{n+1}_{j,i},C)}\int _{|e|\ge \varepsilon }(1\wedge |e|)\,\nu (de)\le C\frac{h^2}{\varepsilon }, \end{aligned}$$
where we have used the fact that \(|\mathcal {I}_h[\phi ]-\phi |_{B(\mathbf{x}^{n+1}_{j,i},C)}\le C|D^2\phi |_{B(\mathbf{x}^{n+1}_{j,i},C)}h^2\). Similar arguments give us that \(|K^{\alpha ,1}_{\varepsilon ,h}\phi ^{n+1}_{j,i}-K^{\alpha ,1}_{\varepsilon }\phi (\mathbf{x}^{n+1}_{j,i})|\le Ch^2|D^2\phi |_{B(\mathbf{x}^{n+1}_{j,i},C)}\int _{|e|\ge \varepsilon }\,\nu (de)\le C\frac{h^2}{\varepsilon ^2}\).
We then infer from Taylor’s theorem with an integral remainder that the truncation errors of the local terms can be bounded by
$$\begin{aligned}&|A^{\alpha }_{\varepsilon ,h,k}\phi ^{n+1}_{j,i}-A^{\alpha }_{\varepsilon }\phi (\mathbf{x}^{n+1}_{j,i})-b^\alpha _{\varepsilon }(x_{j,i})\cdot D\phi (\mathbf{x}^{n+1}_{j,i})|\\&\quad \le C|D^2\phi |_{B(\mathbf{x}^{n+1}_{j,i},C)}\frac{h^2}{k^2}+\omega (\mathbf{x}^{n+1}_{j,i},k) \end{aligned}$$
for some function \(\omega (\mathbf{x}^{n+1}_{j,i},k)\) such that \(\omega (\cdot , k)\rightarrow 0\) as \(k\rightarrow 0\) uniformly on compact neighbourhoods of \(\mathbf{x}^{n+1}_{j,i}\), which enables us to deduce that
$$\begin{aligned}&|A^{\alpha }_{\varepsilon ,h,k}\phi ^{n+1}_{j,i}+K^{\alpha ,1}_{\varepsilon ,h}\phi ^{n+1}_{j,i} -A^{\alpha }_\varepsilon \phi _j(\mathbf{x}_{j,i}^{n+1})-K^{\alpha } _\varepsilon \phi _j(\mathbf{x}_{j,i}^{n+1})|\\&\quad \le |A^{\alpha }_{\varepsilon ,h,k}\phi ^{n+1}_{j,i}-A^{\alpha }_{\varepsilon }\phi (\mathbf{x}^{n+1}_{j,i})-b^\alpha _{\varepsilon }(x_{j,i})\cdot D\phi (\mathbf{x}^{n+1}_{j,i})|+|K^{\alpha ,1}_{\varepsilon ,h}\phi ^{n+1}_{j,i}-K^{\alpha ,1}_{\varepsilon }\phi (\mathbf{x}^{n+1}_{j,i})|\\&\quad \le C\left( \frac{h^2}{k^2}+\frac{h^2}{\varepsilon ^2}+\omega (\mathbf{x}^{n+1}_{j,i},k)\right) . \end{aligned}$$
\(\square \)
Lemma 4.10
Under Assumption 2.1 there exists C independent of \(h,k,\varepsilon , \delta \) such that for all \(\varepsilon <1\)
$$\begin{aligned} \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}\le \frac{C}{h\varepsilon }\wedge \frac{1}{\varepsilon ^2},\quad \sum _{m\not =0}\beta ^{\alpha ,n}_{h,m,i}\le \frac{C}{h}\wedge \frac{1}{\varepsilon }, \quad \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i} \le \frac{C}{k^2}, \end{aligned}$$
where \(\kappa ^{\alpha ,n}_{h,m,i}\), \(\beta ^{\alpha ,n}_{h,m,i}\), and \(d^{\alpha ,n}_{h,k,m,i}\) are defined in (3.14) and (3.16), respectively.
Proof
We shall only prove the estimate for \(\kappa ^{\alpha ,n}_{h,m,i}\), since the estimate for \(\beta ^{\alpha ,n}_{h,m,i}\) follows from a similar argument, and the estimate for \(d^{\alpha ,n}_{h,k,m,i}\) follows directly from the fact that \(\sum _m \omega _m=1\).
The definition of \(\kappa ^{\alpha ,n}_{h,m,i}\) and the integrability property (2.1) of \(\nu \) imply that
$$\begin{aligned} \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}&=\sum _{m\not =0}\int _{|e|>\varepsilon } \omega _m(\eta ^\alpha (x_i,e);h)\,\nu (de)\\&=\sum _{m\not =0}\int _{|e|>\varepsilon } \omega _m(\eta ^\alpha (x_i,e);h)1_{\{\eta ^\alpha (x_i,e)\in \text {supp}\,\omega _m\}}\,\nu (de)\\&=\sum _{m\not =0}\int _{|e|>\varepsilon } \big (\omega _m(\eta ^\alpha (x_i,e);h)-\omega _m(0;h)\big )1_{\{\eta ^\alpha (x_i,e)\in \text {supp}\,\omega _m\}}\,\nu (de)\\&\le \int _{|e|>\varepsilon }\sum _{m\not =0} |D\omega _m|_0|\eta ^\alpha (x_i,e)|1_{\{\eta ^\alpha (x_i,e)\in \text {supp}\,\omega _m\}}\,\nu (de)\\&\le \frac{C}{h}\int _{|e|>\varepsilon }(1\wedge |e|)\,\nu (de)\\&\le \frac{C}{h}\int _{|e|>\varepsilon }\frac{1\wedge |e|}{\varepsilon }(1\wedge |e|)\,\nu (de)=\frac{C}{h\varepsilon }\int _{|e|>\varepsilon }(1\wedge |e|^2)\,\nu (de)\le \frac{C}{h\varepsilon }. \end{aligned}$$
Alternatively, it follows directly from the identity \(\sum _{m\in {\mathbb {Z}}^d}\omega _m(\cdot ;h)\equiv 1\) that
$$\begin{aligned}&\sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}=\sum _{m\not =0}\int _{|e|>\varepsilon } \omega _m(\eta ^\alpha (x_i,e);h)\,\nu (de)\\&\quad \le \int _{|e|>\varepsilon }\,\nu (de)\le \frac{1}{\varepsilon ^2}\int _{|e|>\varepsilon }(1\wedge |e|^2)\,\nu (de), \end{aligned}$$
which leads us to the desired estimates. \(\square \)
Remark 6
Since we have not used any information on the exact behavior of the nonsingular measure \(\nu \) around zero, the estimates for the nonlocal terms in Lemmas 4.9 and 4.10 are not optimal for many specific cases. If one can estimate upper bounds of the density of the Lévy measure, or equivalently estimate the (pseudo-differential) orders of the nonlocal operators \(K^\alpha \) and \(B^\alpha \), more precise results for the truncation error of the singular measure can be deduced (Reference [5]).
The next lemma presents some important properties of the Lax–Friedrichs numerical flux for Lipschitz continuous Hamiltonian, which are crucial for our subsequent analysis. We refer readers to Reference [11] for a proof of these statements. Then the following hold:
Lemma 4.11
Let \({\tilde{f}}\) as in (3.17) and \((\mathbf{x}^{n}_{j,i}, u, k)\in \Omega _{j,h}\times {{\mathbb {R}}}\times {{\mathbb {R}}}\), and suppose Assumption 2.1 and the condition \(\theta >C\lambda \) hold, where C is the Lipschitz constant of the Hamiltonian \({\bar{f}}\).
-
(1)
(Consistency.) For any test functions \(\phi \in C^{1,2}([0,T]\times {{\mathbb {R}}}^d)\), we have
$$\begin{aligned} |{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},u, \Delta \phi ^{n}_{j,i},k)-{\bar{f}}^{\alpha }(\mathbf{x}_{{j,i}}^{{n}},u, D\phi (\mathbf{x}^{n}_{j,i}),k)|\le Ch^2/\Delta t. \end{aligned}$$
-
(2)
(Monotonicity.) If \(V^n_{j,i}\ge U^n_{j,i}\), for all i, j, n, then we have
$$\begin{aligned} \Delta t{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},u, \Delta V^{n}_{j,i},k)+2d\theta V_{j,i}^n\ge \Delta t{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},u, \Delta U^{n}_{j,i},k)+2d\theta U_{j,i}^n. \end{aligned}$$
-
(3)
(Stability.) For any bounded functions U and V, we have
$$\begin{aligned}&|(\Delta t{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},u, \Delta V^{n}_{j,i},k)+2d\theta V_{j,i}^n)\\&\quad -\,( \Delta t{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},u, \Delta U^{n}_{j,i},k)+2d\theta U_{j,i}^n)| \le 2d\theta |U-V|_0. \end{aligned}$$
Proposition 4.12
Suppose Assumption 2.1, the positive interpolation property in Condition 1 and the condition \(\theta >C\lambda \) hold. Then we have the following:
-
(1)
There exists a unique bounded solution \(U^{n}\) of the scheme (3.18).
-
(2)
The scheme is \(\ell ^\infty \) stable and weakly monotone. It is consistent with the switching system (3.6) provided \(h^2/\Delta t\rightarrow 0\) and \(h/k\rightarrow 0\) as \(h,k,\Delta t \rightarrow 0\) (\(\varepsilon \) is fixed here).
Proof
We start to establish the existence and uniqueness of a bounded solution of (3.18) in (1) by an induction argument. It is clear the statement holds for \(t^0=0\) since \(U^0=g\) is bounded. Now we assume that \(\{U_j^{n-1}\}_{j=1}^J\) are bounded functions on \(h{\mathbb {Z}}^d\) and consider the time point \(t^n\). The positive interpolation property implies the interpolation step among different grids does not increase the \(\ell ^\infty \) norm of the solution, and hence \(U_j^{n-\frac{1}{2}}\) is bounded for each \(j=1,\ldots , J\).
For each \(\rho >0\) and \(j=1,\ldots , J\), we define the operator \(\mathcal {P}:U_j^n\rightarrow U_j^n\) by
$$\begin{aligned} \mathcal {P}U_{j,i}^n=U_{j,i}^n-\rho \cdot (\text {left-hand side of } (3.18)), \quad i\in {\mathbb {Z}}^d, \end{aligned}$$
with a given function \(U^{n-\frac{1}{2}}_j\). By virtue of the fact that fixed points to the equation \(\mathcal {P}U_{j}^n=U_j^n\) are precisely the solutions to (3.18), it suffices to establish that for small enough \(\rho \), the operator \(\mathcal {P}\) is a contraction on \(\ell ^\infty ({\mathbb {Z}}^d)\), i.e., the Banach space of bounded functions on \(h{\mathbb {Z}}^d\) employed with the sup-norm, which along with the contraction mapping theorem leads to the desired results. (Similar contraction operators have been introduced in References [7, 13] to demonstrate the well-posedness of their numerical schemes).
For any bounded functions \(U_j^n\) and \(V_j^n\), the definitions of \(\mathcal {P}\), \(A^{\alpha }_{\varepsilon ,h,k}\) and \(K^{\alpha ,1}_{\varepsilon ,h}\) give that
$$\begin{aligned}&\mathcal {P}U_{j,i}^n-\mathcal {P}V_{j,i}^n\nonumber \\ \le&(1-\rho )(U_{j,i}^n-V_{j,i}^n)+\rho \Delta t \bigg [\sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}[(U_{j,m}^n-V_{j,m}^n)-(U_{j,i}^n-V_{j,i}^n)]\nonumber \\&+ \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}[(U_{j,i+m}^n-V_{j,i+m}^n)-(U_{j,i}^n-V_{j,i}^n)]\nonumber \\&+{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},U^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}) -{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},V^{n}_{j,i},\Delta V^{n}_{j,i},B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}) \bigg ]\nonumber \\ \le&(1-\rho -\rho \Delta t \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}-\rho \Delta t \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i})(U_{j,i}^n-V_{j,i}^n)\nonumber \\&\quad +\rho \Delta t( \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+\sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i})|U^{n}_{j}-V^{n}_{j}|_0\nonumber \\&+\rho \Delta t \big ( {\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},U^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}) -{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},V^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}) \big ) \end{aligned}$$
(4.17)
$$\begin{aligned}&+\rho \Delta t \big ( {\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},V^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}) -{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},V^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}) \big ) \end{aligned}$$
(4.18)
$$\begin{aligned}&+\rho \Delta t \big ( {\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},V^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}) -{\tilde{f}}^{\alpha }(\mathbf{x}^{n}_{j,i},V^{n}_{j,i},\Delta V^{n}_{j,i},B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i})\big ). \end{aligned}$$
(4.19)
It remains to estimate (4.17), (4.18) and (4.19). Lemma 4.11 (3) enables us to bound (4.19) by \( -\rho 2d\theta (U^{n}_{j,i}-V^{n}_{j,i})+\rho 2d\theta |U^{n}_{j}-V^{n}_{j}|_0. \) We then derive upper bounds for (4.17) and (4.18) depending on whether \(U^n_{j,i}-V^n_{j,i}\) or \(B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}-B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}\) is positive. If \(U^n_{j,i}-V^n_{j,i}>0\), the monotonicity of f in y implies that (4.17) is bounded above by \(-\rho \Delta tC(U^n_{j,i}-V^n_{j,i})\), while if \(U^n_{j,i}-V^n_{j,i}<0\), the Lipschitz continuity of f in y enables us to bound (4.17) by \(\rho \Delta tC|U^n_{j,i}-V^n_{j,i}|=-\rho \Delta tC(U^n_{j,i}-V^n_{j,i})\).
We then discuss the sign of \(B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}-B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}\). Suppose \(B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}-B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}<0\), then we obtain from the monotonicity of f in k that (4.18)\(\le 0\). Consequently we obtain that
$$\begin{aligned} \mathcal {P}U_{j,i}^n-\mathcal {P}V_{j,i}^n \le&(1-\rho -\rho \Delta t \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}\nonumber \\&-\rho \Delta t \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}-\rho \Delta t C-\rho 2d\theta )(U_{j,i}^n-V_{j,i}^n)\nonumber \\&+\rho (\Delta t \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+\Delta t\sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}+ 2d\theta )|U^{n}_{j}-V^{n}_{j}|_0\nonumber \\ \le&(1-\rho -\rho \Delta t C)|U_{j}^n-V_{j}^n|_0, \end{aligned}$$
(4.20)
provided that \(1-\rho (1+2d\theta )-\rho \Delta t\big ( \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+\sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}+ C)>0\), which is satisfied for small enough \(\rho \).
On the other hand, if \(B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}-B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i}>0\), the Lipschitz continuity of f in k enables us to bound (4.18) by \(C (B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}-B^{\alpha }_{\varepsilon ,h}V^{n}_{j,i})\), which along with (3.13) implies again (4.20) provided that the the following condition is satisfied:
$$\begin{aligned} 1-\rho (1+2d\theta )-\rho \Delta t\bigg ( \sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+\sum _{m\not =0}(\kappa ^{\alpha ,n}_{h,m,i}+\beta ^{\alpha ,n}_{h,m,i})+ C\bigg )>0,\qquad \end{aligned}$$
(4.21)
which holds for small enough \(\rho \). This completes the proof that \(\mathcal {P}\) is a contraction operator.
We now proceed to establish the \(\ell ^\infty \) stability of the scheme. Let \(\{U_j^{n-1}\}_{j=1}^J\) be the solutions to (3.18). By expressing the discrete operators \(A^{\alpha }_{\varepsilon ,h,k}\) and \(K^{\alpha ,1}_{\varepsilon ,h}\) in the monotone form (3.15) and (3.12), and substituting them into (3.18), we obtain that
$$\begin{aligned}&[1+2d\theta +\Delta t\big (\sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+ \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}\big )]U^n_{j,i}\\&\qquad -\Delta t\big (\sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}U^n_{j,m}+\sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}U^{n}_{j,i+m}\big )\\&\quad =U^{n-\frac{1}{2}}_{j,i}+\Delta t {\tilde{f}}^{\alpha }(\mathbf{x}^n_{j,i},U^{n}_{j,i},\Delta U^{n}_{j,i}, B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i})+2d\theta U^{n}_{j,i}, \end{aligned}$$
from which we can deduce
$$\begin{aligned}&[1+2d\theta +\Delta t\big (\sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+ \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}\big )]U^n_{j,i}\nonumber \\&\qquad -\Delta t\big (\sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}+\sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}\big )|U^{n}_{j,i}|_0\nonumber \\&\quad \le \Delta t\big [ f^{\alpha }(\mathbf{x}^n_{j,i},U^{n}_{j,i},\Delta U^{n}_{j,i},B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i})- f^{\alpha }(\mathbf{x}^n_{j,i},0,\Delta U^{n}_{j,i},B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i})\big ] \end{aligned}$$
(4.22)
$$\begin{aligned}&\qquad +\Delta t\big [f^{\alpha }(\mathbf{x}^n_{j,i},0,\Delta U^{n}_{j,i},B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i})- f^{\alpha }(\mathbf{x}^n_{j,i},0,\Delta U^{n}_{j,i},0) \big ]\nonumber \\&\qquad +|U^{n-\frac{1}{2}}_j|_0+( \Delta t [{\tilde{f}}^{\alpha }(\mathbf{x}^n_{j,i},0,\Delta U^{n}_{j,i},0)-{\tilde{f}}^{\alpha }(\mathbf{x}^n_{j,i},0,0,0)]\nonumber \\&\quad \quad +2d\theta U^{n}_{j,i})+\Delta t {\tilde{f}}^{\alpha }(\mathbf{x}^n_{j,i},0,0,0) . \end{aligned}$$
(4.23)
Using similar arguments as those for the upper bound of (4.17), we deduce that (4.22) is bounded above by \(-\Delta tCU^n_{j,i}\) independent of the sign of \(U^n_{j,i}\).
Suppose now \(B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}<0\), then we obtain from the monotonicity of f in k that (4.23) is nonpositive. Then the \(\ell ^\infty \) stability of the numerical flux and the boundedness of \(f^{\alpha }(\mathbf{x},0,0,0)\) yield that
$$\begin{aligned} (1+\Delta tC)|U^n_{j}|_0\le |U^{n-\frac{1}{2}}_{j}|_0+ \Delta tC_1. \end{aligned}$$
(4.24)
Here C is the constant from Assumption 2.1 and \(C_1>0\) is a large enough constant that we will choose later. On the other hand, if \(B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}>0\), the Lipschitz continuity of f in k enables us to bound (4.23) by \(C B^{\alpha }_{\varepsilon ,h}U^{n}_{j,i}\), which along with (3.12) implies again (4.24).
With the estimate (4.24) in hand, we are ready to derive a uniform bound for the solutions \(\{U_j^n\}\), which is independent of h and c. The proof follows from an inductive argument. Let us introduce the notation \(|U^n|_0=\max _{1\le j\le J}|U_j^n|_0\) for each n and define the term \(a_0=\max (|g|_0,|\zeta |_0)\), then it is clear that \(a_0\ge \max (|U^0|_0,|\zeta |_0)\). Suppose we have \(a_{n-1}\) such that \(a_{n-1}\ge \max (|U^{n-1}|_0,|\zeta |_0)\). Then the definition of \(U^{n-\frac{1}{2}}_{j,i}\) implies that \(|U^{n-\frac{1}{2}}_j|_0\le \max (|\zeta |_0,|U^{n-1}|_0)\le a_{n-1} \). Define the term
$$\begin{aligned} a_{n}:=\frac{1}{1+\Delta t C}a_{n-1}+\Delta t C_1, \end{aligned}$$
with the same constants as those in (4.24), then we have \(|U^{n}|_0\le a_n\). To proceed by induction, we further require \(a_n\ge |\zeta |_0\). Since \(a_{n-1}\ge |\zeta |_0\) and C is fixed, it suffices to require \(C_1\ge C|\zeta |_0\). In this way, we can construct a sequence \(\{a_n\}\), such that \(|U^{n}|_0\le a_n\), but \(a_n\) is uniformly bounded independent of c, h and \(\Delta t\), and hence this completes the proof of \(\ell ^\infty \) stability.
We now study the weak monotonicity of the scheme. Let \(V^n_{j,i}\ge U^n_{j,i}\) and \({\tilde{V}}^{n}_{k,i(j)}\ge {\tilde{U}}^{n}_{k,i(j)}\) for all i, j, k, n, then we have \(V^{n+\frac{1}{2}}_{j,i}\ge U^{n+\frac{1}{2}}_{j,i}\). Moreover the monotonicity of f in k and the weak monotonicity of \({\tilde{f}}\) imply that
$$\begin{aligned}&\sum _{m\in {\mathbb {Z}}^d}d^{\alpha ,n}_{h,k,m,i}U^{n+1}_{j,m}+ \sum _{m\not =0}\kappa ^{\alpha ,n}_{h,m,i}U^{n+1}_{j,i+m}\\&\quad + {\tilde{f}}^{\alpha }(\mathbf{x}^{n+1}_{j,i},U^{n+1}_{j,i}, \Delta U^{n+1}_{j,i},\sum _{m\not =0}\beta ^{\alpha ,n}_{h,m,i}[U^{n+1}_{j,i+m}-U^{n+1}_{j,i}]) \end{aligned}$$
is nondecreasing with \(\{U^{b+1}_{j,a}\}_{(a,b)\!\not =\!(i,n)}\), which gives the weak monotonicity of the scheme (3.18).
Finally we study the consistency of the scheme. By using the Lipschitz continuity of \(x\rightarrow \min (x,a)\), it is clear that it suffices to bound
$$\begin{aligned} (I_1):=&\Delta t\big (A^{\alpha }_{\varepsilon ,h,k}\phi ^{n+1}_{j,i}+K^{\alpha ,1}_{\varepsilon ,h}\phi ^{n+1}_{j,i}+ {\tilde{f}}^{\alpha }(\mathbf{x}^{n+1}_{j,i},\phi ^{n+1}_{j,i}+\xi ,\Delta \phi ^{n+1}_{j,i}, B^{\alpha }_{\varepsilon ,h}\phi ^{n+1}_{j,i})\big )\\ (I_2):=&\bigg |\frac{\phi ^{n+1}_{j,i}-\phi _{j,i}^{n}}{\Delta t}-\big (A^{\alpha }_{\varepsilon ,h,k}\phi ^{n+1}_{j,i}+K^{\alpha ,1}_{\varepsilon ,h}\phi ^{n+1}_{j,i}\\&\quad + {\tilde{f}}^{\alpha }(\mathbf{x}^{n+1}_{j,i},\phi ^{n+1}_{j,i}+\xi ,\Delta \phi ^{n+1}_{j,i},B^{\alpha }_{\varepsilon ,h}\phi ^{n+1}_{j,i})\big )\\&\quad -\phi _{j,t}(\mathbf{x}_{j,i}^{n+1})-A_\varepsilon ^{\alpha } \phi _j(\mathbf{x}_{j,i}^{n+1})-K_\varepsilon ^{\alpha } \phi _j(\mathbf{x}_{j,i}^{n+1})\\&\quad - f^{\alpha }(\mathbf{x}^{n+1}_{j,i},\phi (\mathbf{x}^{n+1}_{j,i}),D\phi (\mathbf{x}^{n+1}_{j,i}),B_\varepsilon ^{\alpha }\phi (\mathbf{x}^{n+1}_{j,i}))\bigg |, \end{aligned}$$
which can be estimated by using Lemmas 4.9, 4.11, and the Lipschitz continuity of f. \(\square \)
Remark 7
The contraction operator \(\mathcal {P}\) is introduced to demonstrate our scheme admits a unique solution for any given discretization parameters \(\Delta t\), h, k and \(\varepsilon \). However, due to its low convergence rate, it is not advisable to implement this contraction mapping directly to solve the nonlinear equation (3.18). In fact, Lemma 4.10 and the stability condition (4.20) restrict the contraction constant of \(\mathcal {P}\) to admit a lower bound depending on the spatial discretization of the diffusion operator. This undesirable dependence of \(\Delta t\) on k can be avoided by considering the mapping T defined by (3.19), which is implicit in the local terms. It has been shown that for small enough h, the contraction constant of T is proportional to \(\theta \), which can be chosen to achieve a rapid convergence.