Large Maturity Behavior of Forward Entropic Risk Measures Sample Clauses

Large Maturity Behavior of Forward Entropic Risk Measures. ‌ This section provides another main result of this chapter, which is the large ma- turity behavior of the forward entropic risk measures on the European type risk positions with respect to the stochastic factors. Mathematically, define, for any t ≥ 0, L∞M (Ft) =o {ξt ∈ L∞(Ft) : ξt = −g(Vt), for some function g : Rd → R being uniformly bounded and Lipschitz continuous with a Lipschitz constant Cg}. Moreover, define LM =o ∪t≥0L∞M (Ft). In the remaining of this chapter, we shall study the forward entropic risk measures on the risk positions ξ ∈ LM . By Theorem 3.5.1, for any T ≥ 0 and ξT ∈ L∞M (FT ), t ρt(ξT ) = Y T,g, ∀t ∈ [0, T ], where (Y T,g = {Y T,g}t∈[0,T], ZT,g = {ZT,g}t∈[0,T]) is the solution of the finite hori- t t zon BSDE (3.5.14): t Y T,g = g(VT ) + T ∫ s − G(Vs, Zs, ZT,g)ds t T ∫ s (ZT,g)trdWs, ∀t ∈ [0, T ]. (3.6.17) t Thanks to the BSDE representation, the large maturity behavior of the forward entropic risk measure is then given by that of the solution Y T,g of the finite horizon BSDE (3.6.17). Hence, it is natural to relate the finite horizon BSDE (3.6.17) with an ergodic BSDE: ˆ ˆ ∫ T′ ˆ ˆ ∫ T′ ˆ Yt = YT′ + t G(Vs, Zs, Zs) − λ ds − t (Zs)trdWs, ∀Tr ≥ 0, t ∈ [0, T ], (3.6.18) where its solution triplet, if exist, is denoted by (Yˆ = {Yˆt}t≥0, Zˆ = {Zˆt}t≥0, λˆ).