Partial Differential Equations - Modelling and ... - ResearchGate

We have proved
Calibration of Lévy Processes with American Options 267
Proposition 5. Under the assumptions of Proposition 4, there exist a positive
constant C and a constant λ ≥ 0 such that, for all u ∈ V α if α>1/2 or for
all u ∈ V 1 if α ≤ 1/2,
〈Bu,u + 〉≥C|u + | 2 V α − λ‖u +‖ 2 L 2 (R +) . (28)
A weak maximum principle for parabolic problems stems from Proposition 5.
The Integro-Differential Operator when the Volatility σ is Positive
When σ > 0, the space V 1 plays a special role. Thus, we use the shorter
notation V = V 1 .
With B defined in (13), we introduce the integro-differential operator A:
Av = − σ2 x 2 ∂ 2 v
2 ∂x 2 + rx∂v + Bv. (29)
∂x
If σ>0, and if (α, ψ) satisfy the assumptions of Proposition 4, then
• A is a continuous operator from V to V −1 ,
• we have the Gårding inequality: there exist c > 0andλ ≥ 0suchthat
〈Av, v〉 ≥c|v| 2 V − λ‖v‖ 2 L 2 (R +) , ∀v ∈ V, (30)
• for any v ∈ V ,
〈Av, v + 〉≥c|v + | 2 V − λ‖v + ‖ 2 L 2 (R +) , (31)
• the operator A + λI is one to one and continuous from V 2 onto L 2 (R + ),
with a continuous inverse.
Remark 3. Note that the assumption that ψ>0nearz = 0 is not necessary
for A to have the above properties: indeed, since σ>0, Gårding’s inequality
holds even if ψ = 0 near 0. The main advantage of this assumption is rather
that it permits a clear identification of the kernel’s singularity at z =0.
3 The Variational Inequality when the Volatility
σ is Positive
We are ready to write the variational inequalities corresponding to the linear
complementarity problem (15)–(18).
We introduce the closed subspace of V :
K = {v ∈ V, v(x) ≥ u ◦ (x) in R + }. (32)