Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
the marginal distribution of ξ is the unique solution of an integro-differential
equation : this is the Kolmogorov equation satisfied by the Markov process
X mimicking ξ, extending thus the Kolmogorov forward equation to a
non-Markovian setting and to discontinuous semimartingales (Theorem 2.3).
Section 2.3 shows how this result may be applied to processes whose
jumps are represented as the integral of a predictable jump amplitude with
respect to a Poisson random measure
t t t
∀t ∈ [0,T] ζt = ζ0+ βsds+ δsdWs+ ψs(y) Ñ(dsdy), (1.15)
0
0
a representation often used in stochastic differential equations with jumps.
In Section 2.4, we show that our construction may be applied to a large class
of semimartingales, including smooth functions of a Markov process (Section
2.4.1), such as
f : R d → R ξt = f(Zt)
for f regular enough and Zt taken as the unique solution of a certain ddimensional
stochastic integro-differential equation, and time-changed Lévy
processes (Section 2.4.2), such as
t
ξt = LΘt Θt =
0
0
θsds, θt > 0,
with L a scalar Lévy process with triplet (b,σ 2 ,ν).
1.2.2 Chapter 3: forward PIDEs for option pricing
The standard option pricing model of Black-Scholes and Merton [20, 75],
widely used in option pricing, is known to be inconsistent with empirical
observations on option prices and has led to many extensions, which include
state-dependent volatility, multiple factors, stochastic volatility and jumps
[30]. Whilemorerealisticfromstatistical point ofview, thesemodelsincrease
the difficulty of calibration or pricing of options.
Since the seminal work of Black, Scholes and Merton [20, 75] partial
differential equations (PDE) have been used as a way of characterizing and
efficiently computing option prices. In the Black-Scholes-Merton model and
various extensions of this model which retain the Markov property of the
risk factors, option prices can be characterized in terms of solutions to a
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