THE EMM CONDITIONS IN A GENERAL MODEL FOR
INTEREST RATES.
KAIS HAMZA, SAUL JACKA, AND FIMA KLEBANER
Abstract. Assuming that the forward rates ftu are semimartingales, we give
conditions on their components for the discounted bond prices to be martingales.
To achieve this we give sufficient conditions for the integrated processes f¯tu =
Ru v
f dv to be semimartingales and identify their various components. We recover
0 t
the no-arbitrage conditions in well-known models in the literature, and finally,
we formulate a new random field model for interest rates and give its EMM
(no-arbitrage) condition.
1. Introduction
We consider a model for bond markets based on the forward rates, in which
forward rates are semimartingales. Heath, Jarrow and Morton (HJM) (1990,1992)
considered diffusion models for the forward rates, Shirakawa (1991) introduced
jumps into the model, and more general jump diffusion models were introduced by
Björk, Kabanov and Runggaldier (BKR) (1997) and Björk, Di Masi, Kabanov and
Runggaldier (BDMKR) (1997). Kennedy (1994) considered a Gaussian Markov
Field model for interest rates. A large part of these papers devoted attention
to finding no-arbitrage conditions in the models introduced. The semimartingale
model is the most general one if pricing is done by the principle of no-arbitrage, and
includes all of the above models as particular cases. The concept of “no-arbitrage”
or “no-free lunch” is appealing from the financial point of view, the models should
not allow for “free lunch”. The “no-arbitrage” concept is easy to define in simple
models of finitely many assets in discrete time, and the result known as the First
Fundamental Theorem of asset pricing asserts that “no-arbitrage” is equivalent to
1991 Mathematics Subject Classification. 60H15.
This work was supported by the Australian Research Council and EPSRC.
The research was done in part during the visit of Saul Jacka to the University of Melbourne.
1
2
Hamza K., Jacka S.D. and Klebaner F. C.
the existence of the Equivalent Martingale Measure (the EMM property), the measure under which the discounted security prices are martingales. It is harder to
define the concept of “no-arbitrage” in more general models, for example in BKR
(1997) and BDMKR (1997), this concept was defined by using measure-valued
portfolios and function-valued processes, to reflect the fact that in bond markets
there are infinitely many (continuum) traded securities (bonds parametrized by
their maturity). In continuous time the EMM is an essential assumption of the
models, see e.g. Harrison and Pliska (1981), Shiryaev (1999). Stricker (1990),
Delbaen and Schachermayer (1994) showed that in the models with finitely many
assets “no-free lunch with bounded risk” is equivalent to the EMM. They also
showed that the EMM property implies no-arbitrage. The semimartingale model
is the most general one with respect to no-arbitrage pricing, since if the security
price process is not a semimartingale then the market model admits arbitrage opportunities, Schachermayer (1993); on the other hand, if the security price process
is locally bounded and satisfies the no-free lunch with vanishing risk property for
simple integrands, then it must be a semimartingale, Delbaen and Schachermayer
(1994).
We do not go into definitions of “no-arbitrage” and its relation to the EMM
(the reader can find a list of results in Shiryaev (1999)), but rather assume the
EMM property and find its implications on the coefficients in the model. Often
in practice and in the literature the EMM conditions are called the “no-arbitrage”
conditions, we shall also use these terms interchangeably.
After giving the general result Theorem 2.3, we show how the EMM (no-arbitrage)
conditions in well-known models in the literature (HJM, Kennedy Gaussian field,
BDMKR) are recovered from our condition. Finally, we formulate a new random
field model for interest rates and give its EMM condition.
To obtain the EMM conditions one needs to work with integrated semimartinRu v
gales of forward rates f¯u =
f dv. To this end we give conditions for these
t
0
t
processes to be semimartingales and identify their various components. This contribution of the paper is of independent interest (in some of the literature the
semimartingale property of the integrated semimartingales was simply assumed).
3
A general model for interest rates
2. Semimartingale model for forward rates
We assume the existence of a continuum of bonds P (t, T ), 0 ≤ t ≤ T , where T
is a fixed horizon. We also assume the existence of a jointly measurable collection
of infinitesimal forward rates (implied by the bond at t with delivery at u), f˘u =
t
f˘(t, u), 0 ≤ t ≤ u ≤ T .
Forward rates and bonds are related by the identity
Z u
˘
(2.1)
P (t, u) = exp −
f (t, v)dv .
t
The bond market is said to have the EMM property if there exists a measure
Q ∼ P such that for each u ≤ T the discounted bond price process
Z t
u
St = S(t, u) = exp −
(2.2)
f˘(s, s)ds P (t, u)
0
is a Q-martingale. As usual, we have assumed the existence of a filtered probability
space (Ω, F, Ft , P ) satisfying the “usual conditions”.
We observe that the EMM condition can be restated as
Z u
˘
(2.3)
S(t, u) = EQ exp −
f (v, v)dv Ft .
0
As Musiela (1993) points out, the “HJM” parametrisation of the forward rate
curve is problematic in the sense that its domain of definition is not rectangular.
Musiela deals with this problem by introducing the time to maturity as an independent variable. We suggest extending the definition of f˘ by letting, for all t and
all u,
f (t, u) = f˘(t ∧ u, u).
(2.4)
Now (2.2) can be rewritten as
(2.5)
Stu
Z
= exp −
u
f (t, v)dv .
0
It is intuitively clear that by applying the logarithmic transformation to Stu and
then differentiating, we obtain that, under the EMM condition, ftu , as a process in
t, must be a semimartingale. Before we embrace this model, we show that, under
4
Hamza K., Jacka S.D. and Klebaner F. C.
the EMM condition, the “discounted forward rate” process is a Q-martingale; as a
consequence, we establish that ftu , must be a semimartingale. More precisely, let
φut = φ(t, u) = ftu Stu .
(2.6)
φut represents today’s “value” of the infinitesimal forward rate available at t for a
contract straddling u.
Theorem 2.1. Assume that φut ≥ 0, for all t and u, then the market has the
EMM property if and only if there exists Q ∼ P such that
(1) for Lebesgue almostZall u ∈ [0, T ], φut , as a process in t, is a Q-martingale;
u
(2) for all t, u ∈ [0, T ],
φ(t, v)dv is Q-integrable.
0
Z
Proof of Theorem 2.1. First we observe that S(t, u) = exp −
u
f (t, v)dv , as
0
a function of u, is Lebesgue almost everywhere differentiable, with
S(t, u + ε) − S(t, u)
= −f (t, u)S(t, u) = −φ(t, u) (Leb a.e.)
ε→0
ε
lim
which, since S(t, 0) = 1, translates to
Z
S(t, u) = 1 −
u
φ(t, v)dv.
0
Assuming (1) and (2), and using the conditional Fubini Theorem, we obtain
Z u
EQ [S(t, u)|Fs ] = EQ 1 −
φ(t, v)dv Fs
0
Z u
Z u
= 1−
EQ [φ(t, v)|Fs ]dv = 1 −
φ(s, v)dv = S(s, u).
0
0
Conversely, if S u is a Q-martingale, then by the conditional Fatou Lemma,
S(s, u + ε) − S(s, u)
ε→0
ε
1
= lim {EQ [S(t, u)|Fs ] − EQ [S(t, u + ε)|Fs ]}
ε→0 ε
S(t, u) − S(t, u + ε) ≥ EQ lim
Fs = EQ [φ(t, u)|Fs ],
ε→0
ε
φ(s, u) = − lim
and φ(t, u) is a Q-supermartingale.
A general model for interest rates
5
On the other hand,
(2.7)
S(0, u) = EQ [S(u, u)]
Z u
= EQ 1 −
φ(u, v)dv
0
Z u
= 1−
EQ [φ(u, v)]dv
0
Z u
≥ 1−
φ(0, v)dv = S(0, u)
0
so we must have equality in (2.7), from which we conclude that
EQ [φ(u, v)] = φ(0, v),
(Leb a.e.)
which, together with the fact that φ(t, u) is a Q-supermartingale, implies that
φ(t, u) is a Q-martingale.
Remark The positivity requirement in Theorem 2.1 can clearly be relaxed to
the assumption that the forward rates are, locally in t and uniformly in u, bounded
below.
Corollary 2.1. If the market has the EMM property then
Z u
u
φt = EQ f (u, u) exp −
f (v, v)dv Ft .
0
Corollary 2.2. If the market has the EMM property then Stu , as a process in
t, is a semimartingale.
Proof of Corollary 2.2. It follows from Theorem 2.1 that, under the EMM conφu
dition, ftu = tu is a semimartingale (with respect to Q, and therefore with respect
St
to P ).
Next we write the canonical representation under Q of the semimartingale ftu ,
as a process in t, for each u ≤ T ,
(2.8)
f u = f0u + M u + B u + h(x) ∗ (µu − ν u ) + (x − h(x)) ∗ µu ,
where M u is a continuous local martingale, B u is a predictable process of finite
variation, h(x) is a bounded function with compact support satisfying h(x) = x in
6
Hamza K., Jacka S.D. and Klebaner F. C.
a neighbourhood of zero (e.g. h(x) = xI|x|≤1 ), µu is the jump measure of f u , i.e.
X
µu (dt, dx) =
ε(s,∆fsu ) (dt, dx),
{s:∆fsu 6=0}
ν u is the compensator of µu , and where we have used the notation
Z tZ
(J ∗ µ)t = (J(x) ∗ µ)t =
J(s, x)µ(ds, dx),
0
R
(see Liptser and Shiryaev p. 189, Jacod and Shiryaev p. 84).
As suggested by (2.1) and (2.2), the semimartingales f u will be integrated with
respect to u. For this to occur we shall first assume that all processes arising in
(2.8) are integrable in u. We shall not list the required assumptions here since in
order to carry out the calculations needed in this paper, more stringent conditions
will be imposed on these processes. These conditions will be stated in the results
below.
Thus assuming
that they
Z u
Z u exist, we introduce
Z u the following integrated
Z u processes:
f¯tu =
ftv dv, B̄tu =
Btv dv, M̄tu =
Mtv dv, µ̄u (dt, dx) =
µv (dt, dx)dv
0
0
0
Z u
Z u0
v
v
u
u
ν (dt, dx)dv. As a
µ ([0, t] × A)dv), ν̄ (dt, dx) =
(i.e. µ̄ ([0, t] × A) =
0
0
general rule, a “bar” will signify an integration in u.
Also let mu (dt, dx) be the jump measure of f¯tu , and nu (dt, dx) its compensator.
The following theorem gives sufficient conditions under which f¯u is a semimartingale, as well as identifies its various components.
Theorem
2.2. Assume that
Z T
hM u , M u iT du < +∞,
(M)
Z0 T
AuT du < +∞, where Au denotes the total variation process of B u ,
(FV)
Z 0T Z t Z
2 u
h(x) ν (ds, dx) du is locally integrable,
(J1)
Z0 T Z0 T ZR
u
|x − h(x)|µ (dt, dx) du < +∞.
(J2)
0
0
Then
(2.9)
¯u
f =
R
f¯0u
u
u
Z
+ M̄ + B̄ +
0
u
h ∗ (µv − ν v )dv + (x − h(x)) ∗ µ̄u ,
7
A general model for interest rates
where M̄ u is a continuous locally square integrable martingale with sharp bracket
Z uZ u
u
u
hM v , M w it dvdw,
M̄ , M̄ t =
0
0
u
Z
u
h ∗ (µv − ν v )dv is a purely dis-
B̄ is predictable and of finite variation, and
0
continuous local martingale. Furthermore, M̄ u is the continuous martingale part
of the semimartingale f¯u the jumps of which are given by
Z u
u
¯
∆ft =
∆ftv dv.
0
Proof of Theorem 2.2. Propositions 4.2 and 4.6 of the Appendix deal with the first
part of the theorem. Here we show the decomposition of f¯u . Since X v = h∗(µv −ν v )
is a purely discontinuous local martingale, its square bracket is
2
Z
X
v
v
v
v
[X , X ]t =
h (∆fs ) − h(x)ν ({s} × dx)
R
s≤t
"
≤ 2
X
h (∆fsv )2
2 #
h(x)ν v ({s} × dx)
Z
+
R
s≤t
Z
X
2 v
v 2
≤ 2
h (∆fs ) + h(x) ν ({s} × dx)
R
s≤t
Z t Z
≤ 2
Z tZ
2 v
h(x) µ (ds, dx) +
0
0
R
where we have used the fact that ν v ({s} Z× R)
Z≤ 1.
Z
u
t
h(x) ν (ds, dx) ,
2 v
R
h(x) µ (dt, dx) dv is equiva2 v
Furthermore the local integrability of
0
0
R
Z u Z t Z
h(x)2 ν v (dt, dx) dv and (J1) implies that of the process
lent to that of
0
R
0
Z u
Z u
v
v
Xtv dv is a purely
[X , X ]t dv. Proposition 4.7 now concludes that
in t
0
0
Z u
v
∆Xt dv.
discontinuous local martingale with jumps given by
0
Z
Z
uZ
v
u
v
h(x)ν ({t} × dx), ∆f¯t =
h(x)ν v ({t} × dx)dv +
Finally since ∆Bt =
0
R
R
Z u
Z
Z u
Z u
v
v
v
v
h(∆ft ) − h(x)ν ({t} × dx) dv +
(∆ft − h(∆ft )) dv =
∆ftv dv. 0
R
0
0
Corollary 2.3. One may replace in the previous theorem (J1) by either
8
Hamza K., Jacka S.D. and Klebaner F. C.
Z
0
T
T
Z
(J1 )
(J100 )
Z0 T Z0 T
0
h(x) ν (dt, dx) du < +∞, or
ZR
u
|h(x)|ν (dt, dx) du is locally integrable.
Z
2 u
0
R
Proof of Corollary 2.3. Define the sequence of stopping times
Z T Z t Z
2 u
Tn = inf{t,
h(x) ν (dt, dx) du > n}.
0
0
R
Then by monotone convergence,
Z T Z Tn Z
2 u
h(x) ν (dt, dx) du
0
0
R
Z
T
Z
Tn −ε
Z
h(x) ν (dt, dx) du +
Z
= lim
ε↓0
0
0
T
2 u
R
0
Z
h(x)2 ν u ({Tn } × dx)du
R
2
≤ n + T sup h(x)
x
and (J1) follows. (J100 ) clearly implies (J1) since h(x)2 ≤ const × |h(x)|.
Theorem 2.3. Assume (M), (FV), (J1), (J2) and
(D) (e−x + x − 1) ∗ nT is locally integrable.
Then the discounted bond price process Stu given in (2.2) is a Q-martingale (i.e.
the bond market has the EMM property) if and only if
Z Z
1 u u
hM v , M w it dvdw
2 0 0
Z u
Z u
v
(2.10)
=
Bt dv +
((x − h(x)) ∗ ν v )t dv − (e−x + x − 1) ∗ nu t .
0
0
Before we prove the above theorem, we make the following remarks.
Remarks
(1) Condition (J100 ) gives a meaning to the, otherwise undefined, processes h∗µu
and h ∗ ν u . As a consequence, representation (2.8) simplifies to
(2.11)
f u = f0u + M u + V u + x ∗ µu ,
where M u is a continuous local martingale, V u is a continuous process of
finite variation, and µu is the jump measure of f u . The two representations
A general model for interest rates
9
then are related by
Vtu = Btu − (h ∗ ν u )t ,
(2.12)
and condition (2.10) becomes
Z Z
1 u u
hM v , M w it dvdw
2 0 0
Z u
Z u
v
(2.13)
=
Vt dv +
(x ∗ ν v )t dv − (e−x + x − 1) ∗ nu t
0
0
u
Furthermore x ∗ µ̄ = x ∗ m̄u and x ∗ ν̄ u = x ∗ n̄u which leads to the
following simplification of (2.13):
Z Z
Z u
1 u u
v
w
(2.14)
hM , M it dvdw =
Vtv dv − (e−x − 1) ∗ nu t .
2 0 0
0
(2) If f¯u is continuous (nu = 0), then differentiating (2.10) in u gives
Z u
(2.15)
hM u , M v it dv = Btu + ((x − h(x)) ∗ ν u )t .
0
u
(3) If the f are continuous, then Vtu is simply Btu , and condition (2.10) becomes
(by differentiating)
(2.16)
Btu
=
Vtu
Z
=
u
hM u , M v it dv.
0
(4) It is worth commenting on how and why the proposed model is different from
those of HJM or BDMKR. In fact to take full advantage of our model, the
filtration used has to be large enough to contain infinitely many Brownian
motions. Indeed, if the filtration in use reduces to being generated by
only finitely many Brownian motions, then our model would be, thanks to
the predictable representation property, no different to HJM or BDMKR.
For example, our model allows for processes driven by a Brownian sheet
rather than an n-dimensional Brownian motion. Likewise, the underlying
jump process allowed by our model could be as rich (information-wise) as
a Poisson sheet (or infinitely many independent Poisson processes).
(5) As Cont (1998) points out, the HJM model can be criticized for being
an infinite-dimensional diffusion driven by a finite number of independent
noises. The same criticism applies to the BDMKR model in that it is
an infinite-dimensional jump-diffusion process driven by a finite number of
10
Hamza K., Jacka S.D. and Klebaner F. C.
independent noises. Cont then suggests modelling the forward curves by an
infinite-dimensional diffusion driven by an infinite-dimensional Brownian
motion. The latter is defined by a family Wt of random linear functionals
on a Hilbert space H (say L2 ([0, +∞[)) satisfying
1
(a) ∀ψ ∈ H,
Wt (ψ) is a one-dimensional Brownian motion;
|ψ|
(b) ∀ψ, ψ 0 ∈ H, if ψ⊥ψ 0 , then the Brownian motions Wt (ψ) and Wt (ψ 0 )
are independent.
We emphasize here that the Brownian sheet W introduced in Example 3.4
encompasses such an infinite-dimensional noise. One has only to set
Z +∞
Wt (ψ) =
ψ(u)W (t, du).
0
Proof of Theorem 2.3. f¯tu can be written as
(2.17)
f¯u = f¯0u + M̄ u + B̄ u + h ∗ (µ̄u − ν̄ u ) + (x − h(x)) ∗ µ̄u ,
where M̄ u is a continuous locally square integrable martingale, B̄ u is predictable
and locally integrable, h ∗ (µ̄u − ν̄ u ) is a purely discontinuous local martingale,
and ν̄ u compensates µ̄u . Note that M̄ u is the continuous martingale part of the
semimartingale f¯u .
Combining (2.2) with (2.1) and (2.4), leads to
Z u
u
u
(2.18)
St = S0 exp −
f (t, v)dv = S0u exp − f¯tu .
0
By Itô’s formula
dStu
=
u
−St−
h
i
1 u
u
−∆f¯tu
u
u
¯
¯
+ ∆ft − 1 .
dft − d M̄ , M̄ t − e
2
Hence S u is the stochastic exponential of the process
1 dXtu = −df¯tu + d M̄ u , M̄ u t + d (e−x + x − 1) ∗ mu t ,
2
and as such is a local martingale if and only if X u itself is a local martingale.
Using (2.17), we can see that this happens if and only if
(2.19)
1 dB̄tu − d M̄ u , M̄ u t + d (x − h(x)) ∗ µ̄u t − d (e−x + x − 1) ∗ mu t
2
is a local martingale.
A general model for interest rates
11
Making use of (J2) and (D), we see that, for S u to be a local martingale, it is
necessary and sufficient that the process
(2.20)
1 dB̄tu − d M̄ u , M̄ u t + d (x − h(x)) ∗ ν̄ u t − d (e−x + x − 1) ∗ nu t
2
is a local martingale. But this process is predictable and of finite variation, therefore
it must be zero, and we obtain condition (2.10).
3. Applications
This section deals with existent models (Examples 4.1 to 3.3) by recovering the
results concerned from our general approach. It also presents a new family of
models (Example 3.6) which we call “Gaussian and Poisson random fields model”.
For clarity of the exposition, we shall introduce this model step by step. First,
we study a basic Gaussian random field model (Example 3.4). Next, we look at
the basic Poisson random field model (Example 3.5). Then combine the two for a
study of the general model.
Example 3.1. (HJM model) In the HJM model it is assumed that the forward
rates satisfy a diffusion-type stochastic differential equation in t for any t ≤ u ≤ T ,
(3.1)
df (t, u) = α(t, u)dt + σ(t, u)dW (t),
where W (t) is the standard Brownian motion. Using convention (2.4), we extend
this model to all t, u ∈ [0, T ] by setting α(t, u) = σ(t, u) = 0, for t > u.
It is clear that (3.1) is a particular case of (2.8) with the martingale and finitevariation parts as given below and nil jump part;
Z t
Z t
u
u
(3.2)
Mt =
σ(s, u)dW (s)
Bt =
α(s, u)ds.
0
0
Now
u
v
Z
t
hM , M it =
σ(s, u)σ(s, v)ds,
0
and our condition (M) becomes
Z
Z TZ u
2
σ(t, u) dtdu =
0
0
T
hM u , M u iT du < +∞.
0
While this is more restrictive than condition C1 of HJM (p. 80) which only requires
Ru
finiteness (not integrability) of 0 σ(t, u)2 dt, it makes HJM’s unpleasant condition
12
Hamza K., Jacka S.D. and Klebaner F. C.
C3 (p. 82) redundant. Theorem 2.10 now states that a necessary and sufficient
condition for the existence of an equivalent martingale measure is, for all t ≤ u,
Z t
Z uZ t
α(s, u)ds =
σ(s, u)σ(s, v)dsdv.
0
0
0
Differentiating in t, we recover the no-arbitrage condition for this model derived
by Heath, Jarrow and Morton (1992):
Z
(3.3)
α(t, u) = σ(t, u)
u
σ(t, v)dv.
t
Example 3.2. (Gaussian random field model, Kennedy 1994.) Consider the
Gaussian random field model for forward rates
f (t, u) = f (0, u) + Vtu + Mtu .
(3.4)
Here Mtu = M ([0, t]×[0, u]) where M is a Gaussian random measure with covariance
function C(s, t, u, v). Also, for each u, the Vtu ’s are deterministic, continuous and
of finite variation. Note that the Mtu ’s are continuous Gaussian martingales such
that E(Msu Mtv ) = Cov(Msu , Mtv ) = C(s, t, u, v) which is a function of (s ∧ t, u ∧ v).
In this case the f (t, u) form a family of continuous semi-martingales.
Now
hM u , M v it = E(Mtu Mtv ) = C(t, t, u, v).
u
Recall that ftu = ft∧u
, and we recover from our condition (2.15) Kennedy’s (1994)
no-arbitrage condition,
(3.5)
Vtu
Z
u
C(t ∧ v, t ∧ v, u, v)dv.
=
0
Example 3.3. (Jump-diffusion BKR, BDMKR model) In this model it is assumed that the forward rates satisfy a jump-diffusion stochastic differential equation (see BDMKR assumption 5.1) in t for any u ≤ T , (t ≤ u)
Z
(3.6) df (t, u) = α(t, u)dt + σ(t, u)dW (t) +
δ(t, x, u)(λ(dt, dx) − l(dt, dx)),
IR
where W (t) is the standard Brownian motion, λ(dt, dx) is a jump measure of a
semimartingale, l(dt, dx) is its continuous compensator, and the coefficients α(t, u),
A general model for interest rates
13
σ(t, u) and δ(t, x, u) are continuous in u and predictable in t, satisfying the following
conditions for all t ≤ u ≤ T :
Z
u
Z
u
|α(s, v)|dv ds < ∞
0
Z
t
u
Z
u
2
|σ(s, v)| dv ds < ∞
(3.7)
0
Z
t
u
Z Z
u
2
|δ(s, x, v)| dv l(ds, dx) < ∞.
0
IR
t
Although this model may seem similar to equation (2.8), there are fundamental
differences which we underline next. First note that δ(t, x, u)λ(dt, dx) is not the
jump measure of ftu ; it may not even be integer-valued and therefore not a jump
measure. This model (3.6) generalizes the HJM model by adding a jump component, but in both cases the processes ftu are driven by the same Brownian motion
W (t) and the dependence on u is modelled by taking the diffusion coefficient σ(t, u)
to be a function of u. Similarly, in BKR and in BDMKR the jumps of all processes
ftu are modelled by the same underlying jump process with jump measure λ(dt, dx)
and the dependence on u is modelled by taking the jump coefficient δ(t, x, u) to be
a function of u as well. Our model, on the other hand, allows the driving processes
(the continuous part as well as the jump part) to depend on u.
Using convention (2.4), we can extend this model to all t, u ∈ [0, T ] by setting
α(t, u) = σ(t, u) = δ(t, x, u) = 0, for t > u.
First we identify the various parameters appearing in our model in terms of those
appearing in this model. Let ξt be the underlying jump process with jump measure
λ(dt, dx), then ∆ftu = 1(∆ξt 6=0) δ(t, ∆ξt , u), and
X
µu ([0, t] × A) =
1A (∆fsu )1(∆fsu 6=0)
s≤t
(3.8)
X
=
s≤t
Also
X
s≤t
|∆fsu |
Z tZ
1A (δ(s, ∆ξs , u))1(∆ξs 6=0) =
1A (δ(s, x, u))λ(ds, dx).
0
IR
Z tZ
|δ(s, x, u)|λ(ds, dx), so that the integrability of δ(s, x, u)
=
0
IR
with respect to λ(ds, dx), assumed by (3.6), implies that the jumps of f u are
14
Hamza K., Jacka S.D. and Klebaner F. C.
summable and therefore that the simplified model of (2.11) applies. Now the semiX
martingale ftu −
∆fsu is continuous and therefore special. Identifying the cons≤t
tinuous martingale parts and the continuous finite-variation parts, we find that
Z t
Z tZ
Z t
u
u
α(s, u)ds −
δ(s, x, u)l(ds, dx).
σ(s, u)dW (s) and Vt =
Mt =
0
0
0
IR
The final step in our identification procedure is the specification of the compensator ν u . For any predictable J(t, x) ≥ 0,
Z t Z
Z t Z
u
u
E
J(s, x)ν (ds, dx) = E
J(s, x)µ (ds, dx)
0
IR
0
IR
Z t Z
= E
0
IR
Z t Z
= E
0
J(s, δ(s, x, u))λ(ds, dx)
J(s, δ(s, x, u))l(ds, dx) ,
IR
so that
(3.9)
Z tZ
u
ν ([0, t] × A) =
1A (δ(s, x, u))l(ds, dx).
0
IR
We will also need the jump measure of the integrated process. The fact that the
processes f u are driven by the same Brownian motion (i.e. W (t)), for the martingale part, the same jump process (i.e. ξt ), for the jump part, and the same process
of finite variation (i.e. t and l(dt, dx)), and taking into account the integrability
conditions given in (3.7), we can write with the convention that a “bar” signifies
an integration of that quantity with respect to u, that
Z
¯
(3.10) df (t, u) = ᾱ(t, u)dt + σ̄(t, u)dW (t) +
δ̄(t, x, u)(λ(dt, dx) − l(dt, dx)).
IR
This automatically provides us with the jumps of f¯u ,
∆f¯tu = 1(∆ξt 6=0) δ̄(t, ∆ξt , u),
and ultimately with the jump measure mu ,
Z tZ
u
m ([0, t] × A) =
1A (δ̄(s, x, u))λ(ds, dx),
0
IR
and its compensator nu ,
u
Z tZ
n ([0, t] × A) =
1A (δ̄(s, x, u))l(ds, dx).
0
IR
A general model for interest rates
15
Before we are able to apply Theorem 2.3 we need to check that conditions (M),
(FV), (J1) and (J2) are satisfied. Conditions (FV) and (M) are nothing but respectively the
Z uand
Z u first
Z second partsof 3.7. Using (3.9), we rewrite the third part
of (3.7) as
x2 ν v (ds, dx) dv < +∞. (J10 ) immediately follows.
t
0
IR
Thus condition (2.10) becomes
Z
Z t
Z tZ 1 t 2
−δ̄(s,x,u)
σ̄ (s, u)ds =
ᾱ(s, u)ds −
e
+ δ̄(s, x, u) − 1 l(ds, dx),
2 0
0
0
IR
which is the condition obtained by BDMKR (see BDMKR equation (5.15) and note
1
that D corresponds to −δ̄, and a to σ̄ 2 − ᾱ). Furthermore, in the particular case
2
where l(dt, dx) = kt (dx)dt, then differentiation yields the condition
Z 1 2
σ̄ (t, u) = ᾱ(t, u) −
e−δ̄(t,x,u) + δ̄(t, x, u) − 1 kt (dx),
2
IR
which is condition (5.16) in BMDKR. Further, differentiating with respect to u we
obtain condition ( see also (24) of BKR)
Z
−δ̄(t,x,u)
kt (dx).
α(t, u) = σ(t, u)σ̄(t, u) +
δ(t, x, u) 1 − e
IR
Example 3.4. (Basic Gaussian Random field model.)
Assume that
(3.11)
ftu
=
f0u
+
Vtu
Z tZ
+
u
σ(s, v)W (ds, dv),
0
0
where σ(t, u) is deterministic, W (t, u) is a Brownian sheet, and for any fixed u, Vtu
is finite-variation and continuous.
W is a Gaussian random measure on R2+ and thus, for any pair (A, B) of disjoint
Borel sets in R2+ , W (A) and W (B) are independent zero-mean Gaussian random
variables with variances Leb(A) and Leb(B).
We shall impose the following assumption on σ(t, u):
Z TZ TZ u
σ(s, v)2 dvdsdu < +∞.
0
0
0
As we will see later, this is nothing but condition (M). We shall also assume that
condition (FV) is satisfied. That is, if Au is the total-variation process of V u , then
Z T
AuT du < +∞.
0
16
Hamza K., Jacka S.D. and Klebaner F. C.
Z tZ
Mtv
v
σ being deterministic,
=
σ(s, w)W (ds, dw) is a Gaussian martingale
0
0
Z tZ v
0
with variance
σ(s, w)2 dsdw. Furthermore, for v < v 0 , Mtv can be written as
0
0
Z t Z v0
v
the sum of the two independent random variables Mt and
σ(s, w)W (ds, dw).
0
v
It follows that
D
v
M ,M
v0
D
E
= M
v∧v 0
,M
v∧v 0
Z tZ
E
t
v∧v 0
=
t
0
σ(s, w)2 dwds,
0
and the EMM condition for this model is given by
Z Z Z Z
Z u
0
1 u u t v∧v
2
0
σ(s, w) dwdsdvdv =
Vtv dv.
2 0 0 0 0
0
Differentiating with respect to u, we obtain the following EMM condition:
Z uZ tZ v
(3.12)
σ(s, w)2 dwdsdv = Vtu .
0
0
0
Note that Kennedy’s condition (see Example 4.2) canZbe recovered
from the
s∧t Z u∧v
above by noting that the covariance function C(s, t, u, v) =
σ(s0 , v 0 )2 ds0 dv 0 .
0
0
The no-arbitrage condition in the SPDE model (Hamza and Klebaner (1995))
can also be easily recovered from (3.12).
Example 3.5. (Basic Poisson Random field model.)
Assume that
(3.13)
ftu
=
f0u
+
Vtu
Z tZ
+
u
λ(s, v)N (ds, dv),
0
0
where N (t, u) is a Poisson sheet, λ(s, v) is deterministic, and for any fixed u, Vtu is
of finite variation, continuous and satisfies condition (FV).
As for the Brownian sheet, N is a Poisson random measure on R2+ such that,
for any pair (A, B) of disjoint Borel sets in R2+ , N (A) and N (B) are independent
Poisson random variables with means Leb(A) and Leb(B).
We shall impose the following assumption on λ(s, v):
Z TZ TZ u
|λ(s, v)|dvdsdu < +∞.
0
0
0
As we will see later, this is nothing but conditions (J100 ) and (J2) combined.
17
A general model for interest rates
In the sequel, u is fixed. Let (T1 , U1 ), (T2 , U2 ), . . . be the random points defining
the restriction of the Poisson sheet N to [0, u] × [0, u]. The numbering of this
sequence will be such that Tn < Tn+1 . Let, for v ≤ u,
Z tZ v
X
v
v
λ(s, w)N (ds, dw) =
λ(Tn , Un )1Tn ≤t 1Un ≤u .
Nt = N ([0, t] × [0, v]), Zt =
0
0
n
Then, with
U (t) = Un , for Tn ≤ t < Tn+1 ,
we can write that
∆ftv = ∆Ztv = λ(t, U (t))∆Ntv = λ(t, U (t))1U (t)≤v ∆Ntu .
Note that if ∆Ntv = 1 then ∆Ntu = 1 for each u ≥ v. Since Ztv (as a process in t)
has independent increments, the compensator of its jump measure is deterministic.
This statement is obtained by a monotone-class argument identical to that of Jacod
and Shiryaev p. 71. In the computation below, we use the fact that Nsv , as a process
in v, is Poisson, and therefore that the conditional law of U (s) given the entire path
of Nθu (as a process in θ), is uniform on the interval [0, u].
ν v ([0, t] × A) = E[µv ([0, t] × A)] =
X
P[∆Zsv ∈ A∗ ]
(A∗ = A \ {0})
s≤t
=
X
P[λ(s, U (s))1U (s)≤v ∈ A∗ , ∆Nsu 6= 0]
s≤t
X =
E P[λ(s, U (s))1U (s)≤v ∈ A∗ |Nθu , θ ≥ 0]1∆Nsu 6=0 ]
s≤t
=
=
=
=
=
X 1 Z u
E
1w≤v 1A∗ (λ(s, w))dw1∆Nsu 6=0
u 0
s≤t
#
Z v "X
1
E
1A∗ (λ(s, w))∆Nsu dw
u 0
s≤t
Z v Z t
1
u
E
1A∗ (λ(s, w))dNs dw
u 0
0
Z Z
1 v t
1A∗ (λ(s, w))udsdw
u 0 0
Z tZ v
1A∗ (λ(s, w))dwds.
0
0
18
Hamza K., Jacka S.D. and Klebaner F. C.
It follows that for any φ,
Z tZ
v
v
φ∗ (s, λ(s, w))dwds,
(φ ∗ ν )t =
(3.14)
0
0
where φ∗ (s, x) = φ(s, x)1x6=0 . Note that φ∗ (s, x) = φ(s, x) if φ(s, 0) = 0.
We now turn to the jump measure of the integrated process and notice that, V̄ u
being continuous, the jumps of f¯u are those of Z̄ u , which we now evaluate.
Z̄tu
Z
u
Z t Z
v
=
0
0
Z tZ u
λ(s, w)N (ds, dw) dv =
(u − w)λ(s, w)N (ds, dw).
0
0
0
Its jump measure can thus be obtained by simply replacing λ(s, w) by (u−w)λ(s, w)
in (3.14). We find that for any deterministic φ,
Z tZ u
u
(3.15)
(φ ∗ n )t =
φ∗ (s, (u − w)λ(s, w))dwds.
0
0
Applying (2.13), we find the following EMM condition:
Z u
Z u
v
(x ∗ ν v )t dv = (e−x + x − 1) ∗ nu t
Vt dv +
0
0
Z tZ
u
e−(u−w)λ(s,w) + (u − w)λ(s, w) − 1 dwds
=
0
0
Z tZ
u
−(u−w)λ(s,w)
e
=
0
u
Z
(x ∗ ν v )t dv − tu,
dwds +
0
0
that is
Z
u
Vtv dv
Z tZ
=
0
0
u
e−(u−w)λ(s,w) dwds − tu.
0
Differentiating with respect to u gives
Z tZ u
v
(3.16)
Vt = −
λ(s, v)e−(u−v)λ(s,v) dvds.
0
0
Example 3.6. (Gaussian and Poisson Random field model.)
Assume that
(3.17)
ftu
=
f0u
+
Vtu
Z
+
0
t
Hsu
Z
u
Z
Z
Ks
σ(s, v)W (ds, dv) +
0
t
0
u
λ(s, v)N (ds, dv),
0
where σ(s, v) and λ(s, v) are deterministic, W (t, u) is a Brownian sheet, N (t, u) is
a Poisson sheet, and for any fixed u, Vtu is finite variation and continuous, and Htu
and Kt are predictable.
A general model for interest rates
19
Theorem 3.1. The EMM condition for this model is given by
(3.18)
Z uZ
0
t
Hsu Hsv
0
Z
v
2
Vtu
σ(s, w) dwdsdv =
t
Z
+
0
Z
Ks
0
u
λ(s, v)e−(u−v)Ks λ(s,v) dvds.
0
When there is no continuous martingale part (σ = 0 or H = 0), (3.18) simplifies
to
t
Z
Vtu
=−
Z
u
λ(s, v)e−(u−v)Ks λ(s,v) dvds.
Ks
0
0
In the absence of jumps (λ = 0), (3.18) simplifies to
Z uZ t
Z v
u v
σ(s, w)2 dwdsdv = Vtu .
Hs Hs
0
0
0
Proof of Theorem 3.1. Let
Lut
Z tZ
u
=
σ(s, v)W (ds, dv), so that
0
Mtu
0
t
Z
Hsu dLus .
=
0
The properties of the Gaussian martingale Lut were discussed in Example 3.4. They
imply that
(3.19)
D
u
M ,M
u0
Z
E
=
t
The jump part of
ftv
t
0
Hsu Hsu
0
Z
u∧u0
σ(s, v)2 dvds.
0
can be expressed in terms of the process
Z tZ v
v
Zt =
λ(s, w)N (ds, dw)
0
0
introduced in Example 3.5 as
Ytv
t
Z
Ks dZsv .
=
0
It immediately follows, that with the notations of Example 3.5, the jumps of ftv
are given by
∆ftv = ∆Ytv = Kt ∆Ztv = Kt λ(t, U (t))1U (t)≤v ∆Ntu .
The compensators ν u and nu are then found to be
Z tZ u
u
1A∗ (Ks λ(s, v)) dvds,
ν ([0, t] × A) =
0
nu ([0, t] × A) =
0
Z tZ
u
1A∗ ((u − v)Ks λ(s, v)) dvds.
0
0
20
Hamza K., Jacka S.D. and Klebaner F. C.
Now the result is obtained by differentiating (2.14).
4. Appendix
In the sequel, we assume given a stochastic basis (Ω, (Ft )t≥0 , P) on which Mtv
and Btv are families, indexed by v ≤ u, of local martingales and processes of finite
variation respectively. Let Avt be the total variation process of Btv and suppose
that for almost every v ≤ u, B0v = 0. We also assume given a family of random
measures µv with compensators ν v . Results on a finite horizon [0, T ] immediately
follow by stopping at T .
v
Z uProposition 4.1. Suppose thatZ Bu t is a family of increasing processes for which
Btv dv < +∞. Then B̄tu =
Btv dv is an increasing process and ∆B̄tu =
0
Z0 u
v
∆Bt dv.
0
Proof of Proposition 4.1. By monotone convergence we see that
u
B̄t−
Z
=
u
v
Bt−
dv.
0
The result immediately follows.
Z
u
Z
u
Avt dv < +∞. Then B̄tu =
Btv dv is a
0Z
0
u
u
v
process of finite variation and ∆B̄t =
∆Bt dv. If for almost every v ≤ u, B v
Proposition 4.2. Suppose that
0
is predictable then Zso is B̄ u .
u
Furthermore, if
Avt dv is locally integrable and Ctv denotes the compensator (or
0
Z u
u
v
u
Ctv dv.
dual predictable projection) of Bt , then the compensator of B̄t is C̄t =
0
Proof of Proposition 4.2.
Btv
=
(Avt
+
Btv )
−
Avt
is a measurable decomposition
v
v
of Btv as a difference of two increasingZprocesses. Since
Z u |Bt | ≤ At , theZintegrability
u
u
v
v
v
u
v
Avt dv is a
(At + Bt )dv −
Bt dv =
condition on Vt implies that B̄t =
0
0
0
process of finite variation as a difference of two increasing processes. The jumps of
B̄tu are obtained by applying proposition 4.1 to Avt + Btv and Avt .
Since the Ctv ’s are predictable, then the same goes for the approximating sums
of C̄tu . Consequently C̄tu , as a limit of predictable processes, is predictable itself.
Because the total variation
of C v is smaller than the compensator of Av , the
Z u
local integrability of
Avt dv and the above imply that C̄tu is of finite variation.
0
The rest follows by Fubini’s theorem.
A general model for interest rates
Proposition 4.3. ν̄ u (dt, dx) =
Ru
0
21
ν v (dt, dx)dv is the compensator of µ̄u (dt, dx).
Proof of Proposition 4.3 Since a point-wise limit of predictable processes is preRu
dictable, ν̄ u (dt, dx) = 0 ν v (dt, dx)dv is predictable. Let H(t, x) ≥ 0 be predictable. Then, by Fubini’s theorem and the definition of compensators,
Z ∞Z
Z uZ ∞Z
u
v
E
H(t, x)µ̄ (dt, dx) = E
H(t, x)µ (dt, dx) dv
0
IR
0
0
IR
Z u Z ∞Z
=
E
H(t, x)µv (dt, dx) dv
0
0
IR
Z u Z ∞Z
=
E
H(t, x)ν v (dt, dx) dv
0
0
IR
Z uZ ∞Z
= E
H(t, x)ν v (dt, dx) dv
0
0
IR
Z ∞Z
= E
H(t, x)ν̄ u (dt, dx) ,
0
IR
and the result is proved.
PropositionZ
4.4. Suppose
u, Mtv is a martingale and
that for almost every vZ ≤
u
u
|Mtv |dv < +∞. Then M̄tu =
that for all t, E
Mtv dv is a martingale and
0
0
Z u
u
v
∆M̄t =
∆Mt dv.
0
Proof of Proposition
shows that for s < t,
Z u 4.4. Asimple
Z uuse of Fubini’s theorem
Z u
Mtv dv|Fs =
E[M̄tu |Fs ] = E
E[Mtv |Fs ]dv =
Msv dv = M̄su . In other
0
words, M̄tu is a martingale.
0
0
u
u
Now
Z u martingale Mt , M
Z t−u = E[Mt |Ft− ]. Thus M̄t− = E[M̄t |Ft− ] =
Z u recall thatfor any
v
Mtv dv|Ft− =
E[Mtv |Ft− ]dv =
Mt−
dv and the result follows.
E
0
0
0
The next proposition requires the following lemma. It is known to hold for
continuous local martingales (see Revuz and Yor, p. 120).
Lemma 4.1. Let M and N be two local martingales, denote by ≺ M, N the
total variation process of [M, N ]. Then
q
≺ M, N t ≤ 2 [M, M ]t [N, N ]t .
22
Hamza K., Jacka S.D. and Klebaner F. C.
X
Proof of Lemma 4.1. It follows from [M, N ]t = hM c , N c it +
∆Ms ∆Ns , that
s≤t
≺ M, N t ≤ ≺ M c , N c t +
X
|∆Ms ||∆Ns |
s≤t
sX
q
X
c
c
c
c
≤
hM , M it hN , N it +
(∆Ms )2
(∆Ns )2
s≤t
s≤t
q
≤ 2 [M, M ]t [N, N ]t .
v
Proposition 4.5. Suppose that
v ≤ u, Mt is a (square
Z u inteZ u for almost every
grable) martingale such that E
hM v , M v i∞ dv < +∞. Then M̄tu =
Mtv dv
0
0
is a square integrable martingale and
(4.1)
u
M̄ , M̄
u
u
Z
t
Z
u
hM v , M w it dvdw.
=
0
0
Proof of Proposition 4.5. First note that for any t, both E (M̄tu )2 and E
Z
u
|Mtv |dv
2
0
are less than or equal to
" Z
2 #
Z
u
v
E
|Mt |dv
=E
0
u
Z
0
u
|Mtv Mtw |dvdw
Z
u
Z
=
0
0
u
E[|Mtv Mtw |]dvdw
0
Z u
2
v 2 1/2 w 2 1/2
v 2 1/2
≤
E (Mt )
E (Mt )
dvdw =
E (Mt )
dv
0
0
0
Z u
Z u
v 2
E (Mt ) dv = u
≤ u
E [hM v , M v it ] dv
0
0
Z u
hM v , M v i∞ dv ,
≤ uE
Z
u
Z
u
0
which is finite. This and Proposition 4.4 establish that M̄tu is a square-integrable
martingale. Its sharp bracket is obtained as follows. First, Lemma 4.1 shows that
q
the total variation of the process [M v , M w ]t is less than or equal to 2 [M v , M v ]t [M w , M w ]t .
Since
Z
u
v
v
Z
0
v
v
hM , M i∞ dv < +∞
[M , M ]∞ dv = E
E
u
0
23
A general model for interest rates
and
Z
u
Z
u
Z
q
u
[M v , M v ]t [M w , M w ]t dvdw =
0
0
q
2
[M v , M v ]t dv
0
Z
u
[M v , M v ]t dv,
≤ u
0
Z
uZ
u
[M v , M w ]t dvdw is an integrable process
0
0
Z uZ u
of finite variation, the compensator of which is
hM v , M w i. Finally, let τ be
it follows from Proposition 4.2 that
0
0
a finite stopping time. Then
Z uZ u
Z u Z u
u 2
v
w
= E
Mτ Mτ dvdw =
E[Mτv Mτw ]dvdw
E (M̄τ )
0
0
0
0
Z uZ u
=
E [[M v , M w ]τ ] dvdw
0
0
Z u Z u
v
w
= E
[M , M ]τ dvdw
0
Z
0
u
Z
u
v
w
hM , M iτ dvdw ,
= E
0
0
which shows (4.1) and completes the proof.
Proposition
u, Mtv is continuous and
Z u 4.6. Suppose that for almost every vZ ≤
u
hM v , M v it dv < +∞. Then M̄tu =
Mtv dv is a locally squarethat for all t,
0
0
integrable martingale and (4.1) holds.
Proof of Proposition 4.6. Let Tn = inf{t :
monotone convergence
Z u
Z
v
v
hM , M iTn dv = lim
0
ε↓0
0
Ru
0
hM v , M v it dv > n}. Then, by
u
hM v , M v iTn −ε dv ≤ n,
and the local martingales M v stopped at Tn are square-integrable martingales
(E hM v , M v iTn < +∞ (a.e.)) which satisfy the integrability condition of Proposition 4.5. The result immediately follows from the application of proposition 4.5.
Proposition 4.7. Suppose that for almost every v ≤ Zu, Mtv is a purely disconu
[M v , M v ]t dv is locally
tinuous local martingale such that the increasing process
0
24
Hamza K., Jacka S.D. and Klebaner F. C.
Z
u
Mtv dv is a locally square-integrable purely discontinuous
0
Z u
u
martingale such that ∆M̄t =
∆Mtv dv, and (4.1) holds.
integrable. Then
M̄tu
=
0
Z
u
Proof of Proposition 4.6. Let Tn be such that E
0
v
v
[M , M ]Tn dv
< +∞.
v
Then
Z u(for almost every
v ≤ u) M is a locally square-integrable martingale and
E
[M v , M v iTn dv < +∞. Therefore we can apply Proposition 4.5 and Propo0
sition 4.4 to the local martingales M v stopped at Tn .
We now establish the
purely discontinuous feature of M̄ u . Let Lt be a continuous square-integrable (or
u even
bounded)
martingale
and
τ
be
a
finite
stopping
time.
Then
E
M̄τ Lτ =
Z u
Z u
E
Mτv dvLτ =
E [Mτv Lτ ] dv = 0 since M v is purely discontinuous and L is
0
0
continuous.
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Department of Mathematics, Monash University, Clayton, Victoria, Australia
E-mail address: Kais.Hamza@sci.monash.edu.au
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
E-mail address: s.d.jacka@warwick.ac.uk
Department of Mathematics, Monash University, Clayton, Victoria, Australia
E-mail address: Fima.Klebaner@sci.monash.edu.au