Convex order and increasing convex order for McKean-Vlasov processes with common noise

We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes $X = (X_t)_{t \in [0, T]}$ and $Y = (Y_t)_{t \in [0, T]}$, defined by the following McKean-Vlasov equations with common Brownian noise $B^0 = (B_t^0)_{t \in [0, T]}$: $$dX_t=b(t, X_t, \mathcal{L}^1(X_t))d t+\sigma(t, X_t, \mathcal{L}^1(X_t))d B_t+\sigma^0 (t, \mathcal{L}^1(X_t))d B^0_t$$ $$dY_t=\,\beta(t, Y_t, \mathcal{L}^1(Y_t\,))d t+\,\theta(t, Y_t\,, \mathcal{L}^1(Y_t\,))d B_t\,+\,\theta^0 (t, \mathcal{L}^1(Y_t\,))d B^0_t,$$ where $\mathcal{L}^1(X_t)$ (respectively $\mathcal{L}^1(Y_t)$) denotes a version of the conditional distribution of $X_t$ (resp. $Y_t$) given $B^0$. These results extend those established for standard McKean-Vlasov equations in [Liu-Pag\`es, 2023] and [Liu-Pag\`es, 2021]. Under suitable conditions, for a (non-decreasing) convex functional $F$ on the path space with polynomial growth, we show $\mathbb{E}[F(X) | B^0] \leq \mathbb{E}[F(Y) | B^0]$ almost surely. Moreover, for a (non-decreasing) convex functional $G$ defined on the product space of paths and their marginal distributions, we establish $$\mathbb{E} \Big[\,G\big(X, (\mathcal{L}^1(X_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big]\leq \mathbb{E} \Big[\,G\big(Y, (\mathcal{L}^1(Y_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big] \quad \text{almost surely}.$$ Similar convex order results are also established for the corresponding particle system. Finally, we explore applications of these results to stochastic control problems and to the interbank systemic risk model introduced in [Carmona-Fouque-Sun, 2015].