Dynamic Hedging with Stochastic Differential Utility

ility space 2 . We also assume throughout the paper that probabilistic
statements are in the context of this filtered probability space.
2. V denotes the space of predictable square-integrable processes 3 ,such
that
½
·Z t
V ≡ predictable υ :[0,T] × Ω → R ¯
¯E
0
¸
¾
υ 2 sds < ∞,t∈ [0,T] ,
where Ω is the state space, and predictable means measurable with respect
to the σ-algebra generated by left-continuous processes adapted
to the agent’s filtration, that is, υ t depends only on information available
up to time t.
3. There exist M assets to be hedged, whose value is described by an
M-dimensional Markov process S, with the stochastic differential representation
dS t = µ t (S t )dt + σ t (S t )dB t , (1)
where µ is M-dimensional, σ is (M × N)-dimensional and µ m ∈ V and
σ mn ∈ V for all m and n (hence, the Markov process S is well defined) 4 .
4. There are K futures contracts available for trade at each instant of
2 ” 0 ” indicates transpose, or differentiation when there is only one argument in the
function.
3 If T = ∞, then the square integrability condition changes to E £R ∞
e βt υ 2 0 sds ¤ < ∞,
where β is a constant characterized in Appendix C of Duffie and Epstein (1992).
4 Henceforth, we omit the dependence of the parameters on S t for simplicity.
4