Suppose I have a single time series of losses $L$ that consists of two sub-parts $L_1$ and $L_2$.
Is there a relationship that relates the expected shortfall of $L$ to the expected shortfall of $L_1, L_2$
$$ {\rm{L = }}\left[ {\begin{array}{*{20}{c}} {{L_{1,T_1}}}\\ {{L_{2,T}}} \end{array}} \right] $$
We know that $T_1<T$.
Any reference is much appreciated.
Inequalities are fine.
Lets make it simpler. What if I assume that that $L_1, L_2\sim\mathcal{N}(\mu,\sigma^2)$ and $L\sim\mathcal{N}(\mu_1,\sigma_1,\mu_2,\sigma_2,w_1)$ (Bivariate Normal Mixture). Then based on Broda and Paolella (2011), I know that
$$ \begin{array}{c} {w_1} + {w_2} = 1\\ {F_L}\left( x \right) = {w_1}\Phi \left( {\frac{{x - {\mu _1}}}{{{\sigma _1}}}} \right) + {w_2}\Phi \left( {\frac{{x - {\mu _2}}}{{{\sigma _2}}}} \right)\\ {F_L}\left( q \right) = 1 - \alpha \\ E{S_\alpha }\left( L \right) = {w_1}\frac{{\Phi \left( {\frac{{q - {\mu _1}}}{{{\sigma _1}}}} \right)}}{{1 - \alpha }}\left( {{\mu _1} - {\sigma _1}\frac{{\phi \left( {\frac{{q - {\mu _1}}}{{{\sigma _1}}}} \right)}}{{\Phi \left( {\frac{{q - {\mu _1}}}{{{\sigma _1}}}} \right)}}} \right) + {w_2}\frac{{\Phi \left( {\frac{{q - {\mu _2}}}{{{\sigma _2}}}} \right)}}{{1 - \alpha }}\left( {{\mu _2} - {\sigma _2}\frac{{\phi \left( {\frac{{q - {\mu _2}}}{{{\sigma _2}}}} \right)}}{{\Phi \left( {\frac{{q - {\mu _2}}}{{{\sigma _2}}}} \right)}}} \right)\\ = {w^*}_1E{S_\alpha }\left( {{L_1}} \right) + {w^*}_2E{S_\alpha }\left( {{L_2}} \right) \end{array} $$
Even with simplistic assumptions, it depends on the weights in which I combine the 2 distributions in the mixture