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Calibration of Lévy Processes with American Options 263
Calibration of σ and k is an inverse problem for finding the coefficients
of a variational inequality involving a partial integro-differential operator.
The main goal of the paper is to study the last least square optimization
problem theoretically, for a special parameterization of k, see (25) below,
with σ bounded away from 0, and to give necessary optimality conditions. The
results presented here have their discrete counterparts when the variational
inequalities are discretized with finite elements of finite differences. Numerical
results will be presented in a forthcoming paper.
2 Preliminary Results
2.1 Change of Unknown Function in the Forward Problem
It is helpful to change the unknown function: we set
u ◦ (x) =(S − x) + , u(t, x) =P (t, x) − x + S. (14)
The function u satisfies: for t ∈ (0,T]andx>0,
∂u
∂t − σ2 x 2 ∂ 2 u
2 ∂x 2 + rx∂u + Bu ≥−rx, (15)
∂x
u(t, x) ≥ u ◦ (x), (16)
( ∂u
∂t − σ2 x 2 ∂ 2 )
u
2 ∂x 2 + rx∂u ∂x + Bu + rx (u − u ◦ )=0. (17)
The initial condition for u is
u(t =0,x)=u ◦ (x), x > 0. (18)
For writing the variational inequalities stemming from (15)–(18), we need
to introduce suitable weighted Sobolev spaces. In particular, fractional order
weighted Sobolev spaces will be useful for studying the non-local part of the
operator.
2.2 Functional Setting
Sobolev Spaces on R
For a real number s, let the Sobolev space H s (R) be defined as follows: the distribution
w defined on R belongs to H s (R) if and only if its Fourier transform
ŵ satisfies
∫
(1 + ξ 2 ) s |ŵ(ξ)| 2 dξ < +∞.
R