[proofplan] We verify the three defining properties of a submartingale. Adaptedness and integrability follow directly from the corresponding properties of $X_t$ and from the hypotheses on $\varphi(X_t)$. The only substantive point is the conditional expectation inequality: conditional Jensen gives $\mathbb E[\varphi(X_t)\mid \mathcal F_s] \geq \varphi(\mathbb E[X_t\mid \mathcal F_s])$, and the martingale property identifies the inner conditional expectation with $X_s$. [/proofplan] [step:Verify adaptedness and integrability of the transformed process] For each $t \in T$, the random variable $X_t:\Omega \to I$ is $\mathcal F_t$-measurable because $(X_t)_{t \in T}$ is adapted. Since $\varphi:I\to\mathbb R$ is Borel measurable, the composition $\varphi(X_t):\Omega\to\mathbb R$ is $\mathcal F_t$-measurable. Thus $(\varphi(X_t))_{t \in T}$ is adapted to $(\mathcal F_t)_{t \in T}$. The hypothesis $\mathbb E[|\varphi(X_t)|] < \infty$ gives integrability of $\varphi(X_t)$ for every $t \in T$. [/step] [step:Apply conditional Jensen and the martingale identity] Fix $s,t \in T$ with $s \leq t$. The martingale property gives \begin{align*} \mathbb E[X_t \mid \mathcal F_s] &= X_s \end{align*} almost surely. The hypotheses needed for the [Conditional Jensen Inequality](/theorems/1149) are satisfied: $X_t$ is integrable, $X_t \in I$ almost surely, $\varphi$ is convex, and $\varphi(X_t)$ is integrable. Therefore \begin{align*} \mathbb E[\varphi(X_t) \mid \mathcal F_s] &\geq \varphi\left(\mathbb E[X_t \mid \mathcal F_s]\right) \\ &= \varphi(X_s) \end{align*} almost surely. [/step] [step:Conclude the submartingale property] The preceding steps show that $(\varphi(X_t))_{t \in T}$ is adapted, integrable at every time, and satisfies \begin{align*} \mathbb E[\varphi(X_t) \mid \mathcal F_s] &\geq \varphi(X_s) \end{align*} almost surely whenever $s \leq t$. These are precisely the defining properties of a submartingale with respect to $(\mathcal F_t)_{t \in T}$. [/step]