Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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550 9 General Processes: Mathematical Facts
Mean Variance Hedging: f(x)=x
2
Any probability Q
∗
such that
(i) Q
∗
∈M
P
(
S)
(ii) E
P
((dQ
∗
/dP)
2
)=inf{E
P
((dQ/dP)
2
), Q ∈M
P
(
S) }
is called a minimal measure. The existence of Q
∗
is established in Mania and
Schweizer [618] for continuous processes. The existence of Q
∗
in the case of
discontinuous processes can fail. In the case r = 0, this probability is related
to the existence of a strategy π and an initial wealth v which minimize, for a
given H
E
P
⎛
⎝
v +
T
0
π
s
dS
s
− H
2
⎞
⎠
.
See also F¨ollmer and Schweizer [351].
9.7.3 Indifference Prices
Another method, studied by Davis [220], is to value contingent claims for an
agent endowed with a particular utility function. Related results have been
obtained by a number of authors in various contexts.
A different approach was initiated by Hodges and Neuberger [444]. We
briefly explain the framework of this approach. Let x be the initial endowment
of an agent and U a utility function. The reservation price of the contingent
claim H is defined as the infimum of h’s such that
sup
π
E[U(V
x+h,π
T
− H)] ≥ sup
π
E[U(V
x,π
T
)] ,
where the supremum is taken over the admissible strategies. The agent selling
the contingent claim starts with an initial endowment x+h. Using the strategy
π, he obtains a portfolio with terminal value V
x+h,π
T
and he has to deliver the
contingent claim H; hence, his terminal wealth is V
x+h,π
T
− H. He agrees to
sell the claim if his utility sup
π
E[U(V
x+h,π
T
− H)] is greater than his utility
when he does not sell the claim. The particular case where U is an exponential
function is studied in detail in Rouge and El Karoui [310], whereas Delbaen
et al. [231] make precise the link between this approach and the one based on
entropy.
We do not discuss here these interesting approaches which are based on
optimization portfolio theory (see Karatzas and Shreve [514]), but we refer the
reader to the papers of Hugonnier [452], Bouchard-Denize [111], Henderson
[430] and Musiela and Zariphopoulou [662], to the book of Pham [710]andto
the collective book [142].
10
Mixed Processes
In this chapter, we present stochastic calculus for mixed processes (also
often called jump-diffusions), i.e., loosely speaking they are processes whose
dynamics are driven by a pair of processes consisting of a Brownian motion
and a compound Poisson process. The jump-diffusion approach models the
large number of small movements from the diffusion process part, while the
jump process part captures rare large moves. It is worthwhile mentioning
that a jump-diffusion process is not a diffusion, in the usual acceptance of
this terminology, i.e., a Markov process with continuous paths; however, we
keep this commonly used terminology of jump-diffusion. We draw the reader’s
attention to the fact that, in general, mixed processes in our sense are not
Markov processes.
10.1 Definition
For the moment, up to Subsection 10.4.4, we only consider the restricted class
of mixed processes of the form
X
t
= X
0
+
t
0
h
s
ds +
t
0
f
s
dW
s
+
t
0
g
s
dM
s
.
Here, W is a Brownian motion and (M
t
= N
t
−
t
0
λ(s)ds, t ≥ 0) is the
compensated martingale associated with an inhomogeneous Poisson process
N with deterministic intensity λ(t). The processes M and W are independent
(a justification of this independence assumption is presented in Proposition
10.2.6.2)andf,g,h are predictable processes (with respect to the filtration
generated by the pair (W, M )) such that, for any t
t
0
|h
s
|ds < ∞,
t
0
f
2
s
ds < ∞,
t
0
|g
s
|λ(s)ds < ∞ a.s..
In what follows, we write X =(f, g,h) for such processes. In this general
setting, a mixed process is not a Markov process. The continuous martingale
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial
Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4
10,
c
Springer-Verlag London Limited 2009
551
552 10 Mixed Processes
part of the semi-martingale X is
t
0
f
s
dW
s
and the purely discontinuous
martingale part is
t
0
g
s
dM
s
.
In the particular case where the coefficients f,g,h are constant, or
deterministic functions of time, the filtration generated by the process X is
equal to the filtration generated by the pair (W, M ) and the process X is an
inhomogeneous Markov process.
We shall also consider, in Subsection 10.2.4, the case of stochastic
differential equations where the coefficients f, g,h are defined in terms of a
given function of time and space, i.e., f
s
=
f(s, X
s
−
).
The jump times of the process X are those of N,thejumpsizeofX is
ΔX
t
= X
t
− X
t
−
= g
t
ΔN
t
.
Clearly, the model could be extended to the case where W and N are
multidimensional processes (see Shirakawa [789]) or to the case where M is
the compensated martingale of a compound Poisson process. To keep the
notation simple, we shall consider only one-dimensional processes and the
case where M is the compensated martingale of a Poisson process (except in
Subsection 10.4.4).
Some authors prefer to write the dynamics of a jump-diffusion process
using the Poisson process N instead of the compensated martingale. We have
chosen to write the dynamics in terms of martingales in order to recognize
easily the martingale part. However, writing the dynamics
X
t
= X
0
+
t
0
h
s
ds +
t
0
f
s
dW
s
+
t
0
g
s
dN
s
,
with
h
s
= h
s
− g
s
λ(s) is more convenient for recognizing the jump part.
10.2 Itˆo’s Formula
Let X =(f,g,h)andY =(
˜
f,˜g,
˜
h) be two mixed processes driven by the
same pair consisting of a Brownian motion W and an inhomogeneous Poisson
process N:
dX
t
= h
t
dt + f
t
dW
t
+ g
t
dM
t
=(h
t
− g
t
λ(t))dt + f
t
dW
t
+ g
t
dN
t
,
dY
t
=
˜
h
t
dt +
˜
f
t
dW
t
+˜g
t
dM
t
=(
˜
h
t
− ˜g
t
λ(t))dt +
˜
f
t
dW
t
+˜g
t
dN
t
.
We study the stability of this class of mixed processes under multiplication
and composition with a regular function.
10.2.1 Integration by Parts
The integration by parts formula (9.3.3)reads
d(XY )=X
−
dY + Y
−
dX + d[X, Y ]
10.2 Itˆo’s Formula 553
with
d[X, Y ]
t
= f
t
˜
f
t
dt + g
t
˜g
t
dN
t
=(f
t
˜
f
t
+ g
t
˜g
t
λ(t))dt + g
t
˜g
t
dM
t
.
Hence,
d(X
t
Y
t
)=(
˜
h
t
X
t
−
+ h
t
Y
t
−
+ f
t
˜
f
t
+ g
t
˜g
t
λ(t)) dt
+(
˜
f
t
X
t
−
+ f
t
Y
t
−
)dW
t
+(˜g
t
X
t
−
+ g
t
Y
t
−
+ g
t
˜g
t
)dM
t
,
from which we conclude that XY is a mixed process. Recall that, with respect
to the dt and dW
t
differentials, X
t
−
and Y
t
−
may be replaced by X
t
and Y
t
,
whereas the presence of left limits with respect to dM
t
is important.
10.2.2 Itˆo’s Formula: One-dimensional Case
The following result is obtained from the general Itˆo formula (9.4.1). We shall
write Itˆo’s formula in two forms, the “optional” one and the “predictable”
one.
Proposition 10.2.2.1 (Optional Itˆo Formula.) Let W be a Brownian
motion and M the compensated martingale associated with a Poisson process
N with deterministic intensity λ, independent of W .LetF be a C
1,2
function
defined on R
+
× R,andX a mixed process with dynamics
dX
t
= h
t
dt + f
t
dW
t
+ g
t
dM
t
.
Then,
F (t, X
t
)=F (0,X
0
)+
t
0
∂
s
F (s, X
s
) ds +
t
0
∂
x
F (s, X
s
−
)dX
s
+
1
2
t
0
∂
xx
F (s, X
s
)f
2
s
ds
+
s≤t
[F (s, X
s
) − F (s, X
s
−
) − ∂
x
F (s, X
s
−
)ΔX
s
] .
Here, the sum is taken over the almost surely finite number of jump times
which occur prior to t and is equal to
t
0
[F (s, X
s
−
+ g
s
) − F (s, X
s
−
) − ∂
x
F (s, X
s
−
)g
s
] dN
s
.
We may justify our terminology optional Itˆoformulafrom the fact that,
in general, the last term is only optional, in contrast with the last term in the
predictable Itˆoformula(10.2.1).
In the integral with respect to ds, X
s
−
or X
s
can be used. Hence, writing
Itˆo’s formula in a differential form, one obtains
554 10 Mixed Processes
dF (t, X
t
)=∂
t
F (t, X
t
) dt + ∂
x
F (t, X
t
−
)dX
t
+ ···
or
dF (t, X
t
)=∂
t
F (t, X
t
−
) dt + ∂
x
F (t, X
t
−
)dX
t
+ ··· .
We like to emphasize that these formulae are easy to memorize. Let us write
dX
t
= h
t
dt + f
t
dW
t
+ g
t
dM
t
.
In order to obtain Itˆo’s formula, we begin with “standard terms”
∂
t
Fdt+ ∂
x
FdX
t
+
1
2
f
2
t
∂
xx
Fdt. (S)
Next, we concentrate on the jumps terms both in (S) and in F (t, X
t
): the
process F (t, X
t
), has a jump of size F (t, X
t
) − F (t, X
t
−
) when the process
X jumps; the standard term (S) has a jump of size ∂
x
F (t, X
t
−
)ΔX
t
when X
jumps. It suffices to add to (S) the jumps of F (t, X
t
) and to subtract its own
jumps to obtain Itˆo’s formula.
We now present the predictable ItˆoFormula.
Proposition 10.2.2.2 (Predictable Itˆo Formula.) Let W be a Brownian
motion and M the compensated martingale associated with an inhomogeneous
Poisson process N with deterministic intensity λ, independent of W .LetF
be a C
1,2
function defined on R
+
× R,and
dX
t
= h
t
dt + f
t
dW
t
+ g
t
dM
t
.
Then,
F (t, X
t
)=F (0,X
0
)+
t
0
∂
x
F (s, X
s
)f
s
dW
s
+
t
0
[F (s, X
s
−
+ g
s
) − F (s, X
s
−
)] dM
s
+
t
0
∂
t
F (s, X
s
)+h
s
∂
x
F (s, X
s
)+
1
2
f
2
s
∂
xx
F (s, X
s
)
+ λ(s)[F (s, X
s
+ g
s
) − F (s, X
s
) − ∂
x
F (s, X
s
)g
s
]
ds .
(10.2.1)
It follows that the process (F (t, X
t
),t≥ 0) is a local martingale if and only if
∂
t
F (t, X
t
)+h
t
∂
x
F (t, X
t
)+
1
2
f
2
t
∂
xx
F (t, X
t
)
+ λ(t)[F (t, X
t
+ g
t
) − F (t, X
t
) − g
t
∂
x
F (t, X
t
)] = 0,dt× dP a.s..
10.2 Itˆo’s Formula 555
The use of the predictable Itˆo formula was important in obtaining this
last result. In the case where the coefficients f,g,h, λ are constant (or
deterministic, or even functions of time and state), we are led to solve the
PDEI (Partial Differential Equation with Integral term)
∂
t
F (t, x)+h∂
x
F (t, x)+
1
2
f
2
∂
xx
F (t, x)
+ λ[F (t, x + g) − F (t, x) − g∂
x
F (t, x)] = 0 , ∀x ∈ R, ∀t ≥ 0.
Exercise 10.2.2.3 Let dX
t
= h
t
dt + σ
t
dW
t
+ ϕ
t
dM
t
be a mixed process and
S
t
= e
X
t
. Prove that the dynamics of S are
dS
t
= S
t
−
h
t
+
1
2
σ
2
t
+(e
ϕ
t
− 1 − ϕ
t
)λ(t)
dt + σ
t
dW
t
+(e
ϕ
t
− 1)dM
t
.
Conversely, if
dS
t
= S
t
−
(μ
t
dt + σ
t
dW
t
+ ψ
t
dM
t
)
with ψ
t
> −1, prove that S
t
= e
Y
t
,whereY is a mixed process.
Exercise 10.2.2.4 Let dS
t
= S
t
−
(bdt + σdW
t
+ φdM
t
)whereb, σ and φ are
constant coefficients and φ>−1. Let Y
t
=(S
t
)
−1
.Provethat
dY
t
= −Y
t
−
b − σ
2
+ λ
φ
1+φ
− φ
dt + σdW
t
+
φ
1+φ
dM
t
.
Hint: The jumps of S occur when the Poisson process N jumps, and the sizes
of the jumps are ΔS
t
= φS
t
−
ΔN
t
, hence S
t
= S
t
−
(1 + φΔN
t
). The coefficient
of dM (or of dN) may also be obtained by looking at the size of the jumps of
the process S
−1
.
10.2.3 Multidimensional Case
Let F be a C
1,2
function defined on R
+
× R
d
and let X
t
=(X
i
(t),i ≤ d)be
a d-dimensional mixed process with components having dynamics
X
i
(t)=X
i
(0) +
t
0
h
i
(s) ds +
t
0
f
i
(s)dW
s
+
t
0
g
i
(s)dM
s
,
where for simplicity, we have taken W and M uni-dimensional and where
g
i
are predictable processes and f
i
,h
i
are optional. Then, the optional Itˆo
formula is
F (t, X
t
)=F (0,X
0
)+
t
0
∂
s
F (s, X
s
) ds +
d
i=1
t
0
∂
x
i
F (s, X
s−
)dX
i
(s)
+
1
2
d
i,j=1
t
0
∂
x
i
x
j
F (s, X
s
)f
i
(s)f
j
(s) ds
+
s≤t
[F (s, X
s
) − F (s, X
s
−
) −
d
i=1
∂
x
i
F (s, X
s
−
)g
i
(s)ΔN
s
] ,
556 10 Mixed Processes
A further multidimensional generalization (with a d-dimensional Brownian
motion and a k-dimensional Poisson process) could be developed.
Exercise 10.2.3.1 Prove that the predictable Itˆo’s formula is
F (t, X
t
)=F (0,X
0
)+
t
0
∂
s
F (s, X
s
) ds
+
d
i=1
t
0
∂
i
F (s, X
s
)f
i
(s)dW
s
+
d
i=1
t
0
∂
i
F (s, X
s
)h
i
(s)ds
+
1
2
d
i,j=1
t
0
∂
ij
F (s, X
s
)f
i
(s)f
j
(s) ds
+
t
0
[F (s, X
s
−
+ g
s
) − F (s, X
s
−
)] dM
s
+
t
0
[F (s, X
s
+ g
s
) − F (s, X
s
) −
d
i=1
∂
i
F (s, X(s))g
i
(s)]λ(s) ds ,
where X
s
=(X
i
s
; i =1,...,d).
10.2.4 Stochastic Differential Equations
Proposition 10.2.4.1 Assume that λ is bounded and μ, σ, φ are functions:
R
+
× R → R which satisfy
|μ(t, x) − μ(t, y)| + |σ(t, x) − σ(t, y)| + |φ(t, x) − φ(t, y)|≤C|x − y|, ∀t, x, y
|μ(t, 0)| + |σ(t, 0)| + |φ(t, 0)|≤C, ∀t.
Then the SDE
dX
t
= μ(t, X
t
)dt + σ(t, X
t
)dW
t
+ φ(t, X
t
−
)dM
t
,X
0
= x
0
given,
admits a unique (pathwise) solution. The solution is an inhomogeneous
Markov process.
Proof: See Bass [58, 57], Dol´eans-Dade and Meyer [256], Ikeda and Watanabe
[456] or Jacod and Shiryaev [471].
Proposition 10.2.4.2 Let W be a Brownian motion, N the random measure
associated with a ν-compound Poisson process, and β
i
,γ
i
,δ
i
,i=0, 1 continu-
ous functions, with δ
1
> −1. The solution of the equation
dX
t
=(β
0
(t)+β
1
(t)X
t
) dt +(γ
0
(t)+γ
1
(t)X
t
) dW
t
+
(δ
0
(t, x)+δ
1
(t, x)X
t
−
) N(dt, dx)
10.2 Itˆo’s Formula 557
is
X
t
= Z
−1
t
β(t)
1+
t
0
Z
s
β(s)
(β
0
(s) − γ
0
(s)γ
1
(s))ds
+
t
0
Z
s
β(s)
γ
0
(s)dW
s
+
t
0
Z
s
−
β(s)
δ
0
(s, x) N(ds, dx)
where
Z
t
= E(−γ
1
W )
t
exp
−
t
0
ln(1 + δ
1
(s, x)) N(ds, dx)
and β(t) = exp
t
0
β
1
(u)du.
Proof: The proof is obtained from Itˆo’s calculus using the method of
variation of constants. This is a particular case of affine equations of the
form dX
t
= X
t
−
dY
t
+ dH
t
. See Gapeev [372] for a study of the solutions in
this particular case.
Comment 10.2.4.3 Despite the affine property of the coefficients, this
model is not what is called in mathematical finance an affine model. We shall
present the later in Subsection 10.4.4.
10.2.5 Feynman-Kac Formula
As in the Brownian motion case (see Subsection 1.5.6), we can interpret the
expectations of certain functionals (see below) as the values of solutions of
integro-differential equations. Consider the case where
dX
t
= μ(t, X
t
)dt + σ(t, X
t
)dW
t
+ φ(t, X
t
−
)dM
t
.
The Markov property implies that, if h is a real-valued Borel function and T
a fixed time, the conditional expectation E(h(X
T
)|F
t
)isoftheformH(t, X
t
).
Let us assume that H is a C
1,2
function. Since H(t, X
t
) is a martingale, its
predictable finite variation part is equal to zero. Therefore, from (10.2.1), we
are led to consider the PDEI
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∂H
∂t
(t, x)+μ(t, x)
∂H
∂x
(t, x)+
σ
2
(t, x)
2
∂
2
H
∂x
2
(t, x)
+λ(t)[H(t, x + φ(t, x))−H(t, x) − φ(t, x)
∂H
∂x
(t, x)] = 0 ,
H(T,x)=h(x) .
(10.2.2)
If H solves (10.2.2), then H(t, X
t
) is a local martingale. If h is bounded, we
obtain
E(h(X
T
)|F
t
)=H(t, X
t
) .
558 10 Mixed Processes
10.2.6 Predictable Representation Theorem
Let M be the compensated martingale associated with an inhomogeneous
Poisson process N with deterministic intensity and W a Brownian motion
defined on the same space. We assume that W and M are independent and
we denote by F the canonical filtration generated by the pair (W, N ) i.e.,
F
t
= σ(W
s
,N
s
,s≤ t)=σ(W
s
,M
s
,s≤ t).
Theorem 10.2.6.1 Let Z be a square integrable F-martingale. Then, there
exist two predictable processes (ϕ, ψ) such that Z = z + ϕW + ψM, with
E
t
0
ϕ
2
s
ds
< ∞, E
t
0
ψ
2
s
λ(s)ds
< ∞, ∀t.
If Z is a local martingale, there exist two predictable processes (ϕ, ψ) such
that Z = z + ϕW + ψM.
Proof: Let F ∈ L
2
(F
W
∞
)andG ∈ L
2
(F
N
∞
). Then, from the predictable
representation theorem for a Brownian filtration (resp. for a Poisson filtration)
there exist two predictable processes ϕ and ψ such that
F = E(F )+
∞
0
ϕ
s
dW
s
,G= E(G)+
∞
0
ψ
s
dM
s
,
with
E
∞
0
ϕ
2
s
ds
< ∞, E
∞
0
ψ
2
s
λ(s)ds
< ∞.
Define
F
t
= E(F |F
W
t
)=E(F )+
t
0
ϕ
s
dW
s
,G
t
= E(G|F
N
t
)=E(G)+
t
0
ψ
s
dM
s
.
These processes are F-martingales and, since W and M are independent (see
Lemma 9.5.4.1), the integration by parts formula yields
F
t
G
t
= F
0
G
0
+
t
0
F
s
−
dG
s
+
t
0
G
s
−
dF
s
= F
0
G
0
+
t
0
F
s
−
ψ
s
dM
s
+
t
0
G
s
−
ϕ
s
dW
s
.
The result for square integrable martingales now follows from the totality in
L
2
(F
∞
) of these products FG. The result for local martingales can be easily
deduced.
We now discuss the independence assumption:
Proposition 10.2.6.2 Let G be a given filtration. If W is a G-Brownian
motion and N a G-Poisson process, then W and N are independent.
10.3 Change of Probability 559
Proof: Let F ∈ L
2
(F
W
∞
)andG ∈ L
2
(F
N
∞
). With the same notation as in
the proof of the previous theorem, we obtain
FG = E(F )E(G)+
∞
0
F
s
dG
s
+
∞
0
G
s−
dF
s
and E(FG)=E(F )E(G). Note that we have used the fact that (F
s
,s ≥ 0)
and (G
s
,s≥ 0) are G-martingales.
Comment 10.2.6.3 See Chou and Meyer [180], Davis [219, 222], Jacod [468]
and Watanabe [837] for complements. Note again that the hypothesis that W
is a Brownian motion and N a Poisson process in the same filtration is very
important.
10.3 Change of Probability
Throughout this section, we always assume that W is a Brownian motion
and M is the compensated martingale of an inhomogeneous Poisson process
with deterministic intensity λ(t), with W and M independent. The filtration
generated by W and M is denoted by F.
10.3.1 Exponential Local Martingales
Let γ and ψ be two F-predictable processes such that γ
t
> −1. The solution
of
dL
t
= L
t
−
(ψ
t
dW
t
+ γ
t
dM
t
),L
0
> 0 (10.3.1)
is the strictly positive exponential local martingale
L
t
= L
0
s≤t
(1 + γ
s
ΔN
s
) e
−
R
t
0
γ
s
λ(s)ds
exp
t
0
ψ
s
dW
s
−
1
2
t
0
ψ
2
s
ds
= L
0
exp
t
0
ln(1 + γ
s
)dN
s
−
t
0
λ(s)γ
s
ds +
t
0
ψ
s
dW
s
−
1
2
t
0
ψ
2
s
ds
= L
0
exp
t
0
ln(1 + γ
s
)dM
s
+
t
0
[ln(1 + γ
s
) − γ
s
]λ(s)ds
× exp
t
0
ψ
s
dW
s
−
1
2
t
0
ψ
2
s
ds
.
In the last expression, the exponential term is the Dol´eans-Dade exponential
martingale of ψW, whereas the first one is the Dol´eans-Dade exponential
martingale of γM, therefore one can write
L
t
= L
0
E(γM)
t
E(ψW )
t
.