## 3 Trust-Optimized 6G-IoT Framework
### Trust Management System
## Appendix A Mathematical Derivation of the Governing SPDE

We begin by expanding the terms in the governing stochastic partial differential equation (SPDE):

\[\mathbf{I}_{i}(t)=\int_{0}^{t}\rho(\mathbf{y}_{i}(\tau),\mathbf{y}_{i}(\tau- \Delta t))e^{-\beta(t-\tau)}d\tau, \tag{15}\]9 \[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]0 \[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]1

The linear operator \(\mathcal{L}_{i}\mathbf{M}_{i}(x,t)\) is expanded as

\[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]2 \[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]3 \[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]4

The nonlinear term \(\mathbf{B}_{i}(\mathbf{M}_{i}(x,t),\mathbf{u}_{i}(t))\) is expressed as

\[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]5

For the integral term involving the kernel function \(\mathbf{K}_{i}(x,\xi,t)\),

\[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]6

Finally, the stochastic term is represented as

\[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]7

By combining all terms, we arrive at the final governing equation:

\[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]8 \[\mathbf{P}_{i}(t)=\mathbb{E}\Big{[}\big{(}\hat{\mathbf{X}}_{i}(t+\Delta t)- \mathbf{X}_{i}(t+\Delta t)\big{)}^{2}\Big{]}, \tag{16}\]9 \[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]0 \[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]1

Thus, we have derived the complete expression of the governing SPDE. ## Appendix B Mathematical Derivation of the Central Server SDEs

Starting with the matrix-valued SDE governing the central server's resource allocation:

\[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]2The expansion of the terms leads to

\[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]3 \[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]4 (101) \[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]5

Next, for the optimization process, the Hamilton-Jacobi-Bellman (HJB) equation is

\[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]6 \[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]7

Expanding the terms,

\[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]8

thereby leading to the final form of the HJB equation:

\[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\]9 \[\quad+(\mathbf{C}_{r}(t)+\mathbf{D}_{r}(t)\mathbf{W}_{r}(t))d \mathbf{W}_{r}(t), \tag{17}\]0

This completes the derivation of the SDE and its optimization through the HJB equation.