An Impulsive Differential Game Arising in Finance with Interesting Singularities

Abstract

We investigate a differential game motivated by a problem in mathematical finance. This game displays two interesting features.
On the one hand, one of the players, Pursuer say, may, and will, use infinitely large controls, i.e., impulses, producing “jumps” in the state variables. Standard
optimal trajectories are made of such a jump followed by a “coasting period” where Pexerts no control. This leads to barriers of a somewhat new type. But because the cost of jumps is only proportional to their
amplitude, some singular optimal trajectories arise where Puses an intermediary control, nonzero but finite. (In classical impulse control, there is a minimum positive cost to any use
of the control, forbidding such a mixed situation.)

On the other hand, the complete solution of the game exhibits a type of singularity, the existence of which had long been
conjectured (noticeably by Arik Melikyan in discussions with the first author) but, as far as we know, never shown in actual
examples: a two-dimensional focal manifold traversed by noncollinear optimal fields depending on the control used by Evader. It is on this manifold that intermediary controls for Parise.

Finally, we show that the Isaacs equation of a discrete-time version of the problem provides a discretization scheme that
converges to the value function of the differential game. This is done through the investigation of a (degenerate) quasi-variational
inequality and its viscosity solution, with the help of an equivalent, but nonimpulsive, differential game—a method of interest
per se that we credit to Joshua—to which we apply essentially the classical method of Capuzzo Dolcetta extended to differential
games by Pourtallier and Tidball, with some technical adaptations.

Full text

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An Impulsive Di?erential Game Arising in Finance
with Interesting Singularities∗
Pierre Bernhard†, Naïma El Farouq‡, Stéphane Thiery§
Abstract
We investigate a di?erential game motivated by a problem in mathematical ?nance.
This game displays two interesting features. On the one hand, one of the players,
Pursuer say, may, and will, use in?nitely large controls, i.e., impulses, producing
?jumps? in the state variables. Standard optimal trajectories are made of such a jump
followed by a ?coasting period? where P exerts no control. This leads to barriers of
a somewhat new type. But because the cost of jumps is only proportional to their
amplitude, some singular optimal trajectories arise where P uses an intermediary
control, nonzero but ?nite. (In classical impulse control, there is a minimum positive
cost to any use of the control, forbidding such a mixed situation.)
On the other hand, the complete solution of the game exhibits a type of singularity,
the existence of which had long been conjectured (noticeably by Arik Melikyan in
discussions with the ?rst author) but, as far as we know, never shown in actual
examples : a two-dimensional focal manifold traversed by noncollinear optimal ?elds
depending on the control used by Evader. It is on this manifold that intermediary
controls for P arise.
Finally, we show that the Isaacs equation of a discrete-time version of the problem
provides a discretization scheme that converges to the Value function of the di?eren-
tial game. This is done through the investigation of a (degenerate) quasi-variational
inequality and its viscosity solution, with the help of an equivalent, but nonimpulsive,
di?erential game ?a method of interest per se that we credit to Joshua ? to which
we apply essentially the classical method of Capuzzo-Dolcetta extended to di?erential
games by Pourtallier and Tidball, with some technical adaptations.
∗We acknowledge an important numerical work performed by intern students at the University of Nice
Sophia-Antipolis : Nicéphore Allaglo, Carole Bouvelot, Charlotte Pouderoux, and Laetitia Richter.
†I3S,UniversityofNiceSophia-Antipolis
Pierre.Bernhard@polytech.unice.fr
‡I3S, University of Nice Sophia-Antipolis and CNRS, Sophia-Antipolis, France, elfarouq@i3s.unice.fr.
§I3S, University of Nice Sophia-Antipolis and CNRS, Sophia-Antipolis, France, thiery@i3s.unice.fr
andCNRS,Sophia-Antipolis,France,
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An Impulsive Di?erential Game Arising in Finance with Interesting Singularities
1The Di?erential Game Considered
We consider a di?erential game arising in ?nance, speci?cally in the theory of option
pricing with an ?interval model?. (We refer to [4, 16] for the context in ?nance). This
is a game in two dimensions plus time with an integral payo?, or three dimensions plus
time with a terminal payo?, and two scalar controls (pursuer P and evader E), with the
peculiarity that the pursuer may, and will, use arbitrarily large control values, up to the
point of producing ?impulses?. Thus, this player may cause discontinuities in some state
variables, incurring a related cost.
1.1Dynamics
The (3-D) dynamics are as follows. We call (x,y,z) the state variables, and u and v
the controls of pursuer and evader respectively. The continuous (nonimpulsive) part of
the dynamics is given in terms of ε = sign(u) and two numbers C+1and C−1, also written
C+and C−respectively, with C+> 0, C−< 0, as follows :
˙ x = vx,
˙ y = vy + u,
˙ z = vy − Cεu,
(1)
(2)
(3)
with the control constraints on v speci?ed by two positive numbers α and β as
−α ≤ v ≤ β .
(4)
Since u is not bounded, we allow the pursuer to cause discontinuities in the state
variables at isolated time instants tkaccording to the rule
y(t+
z(t+
k) = y(t−
k) = z(t−
k) + uk,
k) − Cεkuk.
(5)
(6)
Of course, we have set y(t−
jump amplitude in y is uk∈ R, and εk= sign(uk).
To avoid an unessential discussion later on, we shall further assume that
k) = limt↑tky(t) and y(t+
k) = limt↓tky(t) and likewise for z. The
α ≤ β ,
and
0 < (1 + C+)(1 + C−) ≤ 1.
(7)
1.2 Payo?
The game is played over a ?xed time interval [0,T], and is a capture-evasion game of
kind, with capture de?ned in terms of a given positive number Z as
z(T) ≥ max{0, x(T) − Z} =: M(x(T)).
(8)
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Pierre Bernhard, Naïma El Farouq, Stéphane Thiery
Again this rather strange setup is motivated by its ?nance application in [4].
We may notice that, since z does not appear in the right-hand side of its dynamics, it
integrates so that (8) is equivalent to
?T
As a consequence, we may consider the game of degree in dimension 2 plus time with
state variables (x,y), the same dynamics (1) (2) and (5)(6), and payo? minumaxvG with
?T
z(0) ≥
0
(−vy + Cεu)dt +
?
k
Cεkuk+ M(x(T)).
G =
0
(−vy + Cεu)dt +
?
k
Cεkuk+ M(x(T)).
(9)
Let W(t,x,y) be its Value function, an initial state is capturable i? z ≥ W(0,x,y), so
that the graph of the Value function W is the barrier of the game of kind.
1.3Strategies
In this game, the pursuer chooses the function u(t), the jump instants tk, and the
jump amplitudes uk. It does so knowing past values of the state. It is a classical fact
that it will only use an (instantaneous) state feedback which we write symbolically u =
ϕ(t,X(t)), where X stands for the whole state. Admissible strategies are those such that
the dynamical equations have for any initial state a unique solution with y(·) uniformly
bounded over admissible v(·)'s.
We are looking for capturable states of the game of kind. It is known that this is
equivalent to looking for the upper value of the game of degree, and that then, whether
the evader plays open loop or closed loop is irrelevant. Thus we may always assume that v
is chosen open loop, as a measurable time function from [0,T] into [−α,β]. (This remark
will play an important role in the investigation of the convergence.)
2 A Geometric Analysis : The Isaacs-Breakwell Theory
2.1 Jumps as Ordinary Trajectories
In [4], we introduced a quasi-variational inequality (QVI) naturally related to the
game of degree with impulse controls. However, due to its very degenerate nature, it is
not accounted for by the literature on viscosity solutions of ?rst order QVI such as [3, 2].
We prefer to use the 3-D plus time representation (1), (2), (3), (5), (6), and the formulation
as a game of kind, and apply to it the geometrical tools of the semipermeability.
In that representation, jumps are just trajectories orthogonal to the t axis. As a matter
of fact, Equations (5) and (6) show that these trajectories are also orthogonal to the x
axis and have a slope either −C+or −C−in the (y,z) plane. We stress the following fact.
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An Impulsive Di?erential Game Arising in Finance with Interesting Singularities
Proposition 2.1 Given a smooth two-dimensional manifold M transverse to the jump
trajectories, the hypersurface made of jump trajectories of the same slope through each
point of M is a ?safe hypersurface? for P. (i.e., E cannot force the state to cross it
against P's will.)
Proof Indeed by choosing a jump, P causes the state to traverse these trajectories in no
time, so that E's control v has no time to act. (P has chosen to be in the dynamics (5),
(6) where v does not enter.)
We shall in e?ect construct manifolds y = ˇ y(t,x), z = ˇ z(t,x) for some functions ˇ y and
ˇ z, construct barriers made of jump trajectories reaching that manifold, and show that
upon reaching it, P still has a means of preventing a crossing of the composite surface.
2.2The Natural Barrier
We proceed with the classical construction of the natural barrier through the boundary
of the capture set which here is t = T, z = M(x), y arbitrary. This has been published in
[4]. We summarize it up here.
The natural barrier is made up of two sheets, one towards x ≤ Z and one towards
x ≥ Z. They are given below, together with a corresponding inward semipermeable
normal as the vector (n,p,q,1) (corresponding to the state variables (t,x,y,z)) leading
to Isaacs'?main equation?
?
and the adjoint equations
0 = max
u
inf
v∈[−α,β]
n + v
?
px + (q + 1)y
?
+ u(q − Cε)
?
,
˙ p = −vp,
˙ q = −v(q + 1).
(10)
(11)
The analysis depends on the fact that the maximum in u of (q − Cε)u is reached at
u = 0 provided that C−≤ q ≤ C+. (Remember that ε = sign(u)). When q leaves that
range, there is no maximum anymore. (Or u should be in?nite : we shall have a jump.)
Sheet α towards x ≤ Z.
v∗= −α and
We set the parameters x(T) = s ≤ Z, y(T) = r. It yields
sheet (α) semipermeable normal να
n(t) = αr,
p(t) = 0,
q(t) = e−α(T−t)− 1,
1 = 1.
t = t
x(t) = seα(T−t),
y(t) = reα(T−t),
z(t) = r(eα(T−t)− 1),
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Pierre Bernhard, Naïma El Farouq, Stéphane Thiery
This is a valid solution as long as q ≥ C−, i.e., for t ≥ tαwith
e−α(T−tα)= 1 + C−,i.e.,T − tα=1
αln
?
1
1 + C−
?
.
(12)
Sheet β towards x ≥ Z.
and
On this sheet, x(T) = s ≥ Z, y(T) = r. We ?nd that v∗= β,
sheet (β) semipermeable normal νβ
n(t) = β(s − r),
x(t) = se−β(T−t),
y(t) = re−β(T−t),
z(t) = r(e−β(T−t)− 1) + s − Z ,
This is a valid solution as long as q ≤ C+, i.e., for t ≥ tβwith
eβ(T−tβ)= 1 + C+,i.e.,
t = t
p(t) = −eβ(T−t),
q(t) = eβ(T−t)− 1,
1 = 1.
T − tβ=1
βln(1 + C+).
(13)
From the hypothesis (7), we have tα< tβ.
Moreover, from ?nal states on the boundary z = x − Z ≥ 0 of the admissible set, a
2-D singular sheet can be constructed with r = s, v arbitrary, leading to
?
x = y = z − Z = sexp
−
?T
t
v(τ)dτ
?
,
−p = q + 1 = exp
??T
t
v(τ)dτ
?
.
Intersection and Composite Barrier.
along a two-dimensional edge D that spans the domain t ≥ tβ, Ze−β(T−t)≤ x ≤ Zeα(T−t),
and that can be parametrized by (t,x) as y = ˇ y(t,x), z = ˇ z(t,x) given by
The two main sheets (α) and (β) intersect
ˇ y(t,x) =
(xeβ(T−t)− Z)
eβ(T−t)− e−α(T−t),ˇ z(t,x) = (1 − e−α(T−t))ˇ y(t,x).
(14)
Notice that for x = Z exp(−β(T −t)), we have ˇ y = ˇ z = 0, which corresponds to the sheet
(α) with r = 0. For smaller x's, only the sheet (α) plays a role. We ?nd it convenient to
extend the de?nition of ˇ y and ˇ z by 0 for both. For x = Z exp(α(T − t)), we have ˇ y = x,
ˇ z = x − Z, which corresponds to the sheet (β) with r = s = Z exp((α + β)(T − tβ)). For
larger x's, only the sheet (β) plays a role. Again, we extend the de?nitions of ˇ y = x and
ˇ z = x − Z to larger x's.
We easily check that D is an E-dispersal line. States ?above? (with larger z's) are
indeed capturable, and the edge does not ?leak? since P's control on both sheets is the
same : u = 0. Therefore, this same control prevents crossing of both barrier sheets.
The singular sheet x = y = z + Z is imbedded in the sheet (β). But it can be used
against v = −α until time tα. In the region x ≤ Z exp(α(T −t)) it plays no role. However,
it will be seen to play a prominent role in the region x ≥ Z exp(α(T −t)) for t ≤ tβ, when
the sheet (β) does not exist. There it behaves as a manifold drawn on an extension of the
sheet (α) for s ≥ Z.
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