Dynamic Hedging with Stochastic Differential Utility

space L isthe(possiblyunbounded)linearoperatorA defined by:
T t φ − φ
Aφ = lim .
t↓0 t
The domain D(A) isthesubspaceofL for which this limit exists.
If T t is a strongly continuous contraction semigroup, we can reconstruct T t
using its infinitesimal generator A (see Ethier and Kurtz (1986), Proposition
2.7 of Chapter 1). Thus the Markov process can be parameterized using A.
The space C of continuous functions on a compact state space endowed
with the sup-norm is a common domain for a semigroup. For instance, the
generator A d of a multivariate diffusion process is an extension of the secondorder
differential operator:
A d φ (x) ≡ d dt E x [φ (X t )] |t=0+
= µ · φ x + 1 2 tr [Σφ xx] ,
where
dx t = µ (x t ) dt + Λ (x t ) dB t ,
tr is the trace operator, and
Λ (x t ) Λ 0 (x t )=Σ (x t ) .
For a formal proof see, for instance, Oksendal (1995). For more details, see
Ait-Sahalia, Hansen and Scheinkman (2002), and Hansen and Scheinkman
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