Jump-Diffusion Risk-Sensitive Asset Management

Jump-Diffusion Risk-Sensitive Asset Management

Jump-Diffusion Risk-Sensitive Asset
Management

Mark Davis and S´bastien Lleo
e

Department of Mathematics
Imperial College London

Full paper available at http://arxiv.org/abs/0905.4740v1

Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management

Outline

Outline

1 Introduction

2 The Risk-Sensitive Investment Problem

3 Solving the Stochastic Control Problem
Change of Measure
The HJB PDE
Identifying a (Unique) Candidate Optimal Control
Veriﬁcation Theorem
Existence of a C 1,2 Solution to the HJB PDE

4 Concluding Remarks


Introduction

Introduction

Risk-sensitive control is a generalization of classical stochastic
control in which the degree of risk aversion or risk tolerance of the
optimizing agent is explicitly parameterized in the objective
criterion and inﬂuences directly the outcome of the optimization.
In risk-sensitive control, the decision maker’s objective is to select
a control policy h(t) to maximize the criterion

1
J(x, t, h; θ) := − ln E e −θF (t,x,h) (1)
θ
where t is the time, x is the state variable, F is a given reward
function, and the risk sensitivity θ ∈ (0, ∞) is an exogenous
parameter representing the decision maker’s degree of risk aversion.


Introduction

Jacobson [?], Whittle [?], Bensoussan [?] led the theoretical
development of risk sensitive control.

Risk-sensitive control was first applied to solve financial problems
by Lefebvre and Montulet [?] in a corporate finance context and by
Fleming [?] in a portfolio selection context. However, Bielecki and
Pliska [?] were the first to apply the continuous time risk-sensitive
control as a practical tool that could be used to solve ‘real world’
portfolio selection problems. A major contribution was made by
Kuroda and Nagai [?] who introduced an elegant solution method
based on a change of measure argument which transforms the risk
sensitive control problem in a linear exponential of quadratic
regulator.


The Risk-Sensitive Investment Problem

Let (Ω, {Ft } , F, P) be the underlying probability space.

Take a market with a money market asset S0 with dynamics
dS0 (t)
= a0 + A0 X (t) dt, S0 (0) = s0 (2)
S0 (t)
and m risky assets following jump-diﬀusion SDEs
N
dSi (t) ¯
= (a + AX (t))i dt + σik dWk (t) + γi (z)Np (dt, dz),
Si (t − ) Z
k=1
Si (0) = si , i = 1, . . . , m (3)
X (t) is a n-dimensional vector of economic factors following
dX (t) = (b + BX (t))dt + ΛdW (t), X (0) = x (4)


Note:
W (t) is a Rm+n -valued (Ft )-Brownian motion with
components Wk (t), k = 1, . . . , (m + n).
¯
Np (dt, dz) is a Poisson random measure (see e.g. Ikeda and
Watanabe [?]) deﬁned as
¯
Np (dt, dz)
˜
Np (dt, dz) − ν(dz)dt =: Np (dt, dz) if z ∈ Z0
=
Np (dt, dz) if z ∈ ZZ0

the jump intensity γ(z) satisﬁes appropriate well-posedness
conditions.
assume that

ΣΣ > 0 (5)



The wealth, V (t) of the investor in response to an investment
strategy h(t) ∈ H, follows the dynamics

dV (t)
= a ˆ
a0 + A0 X (t) dt + h (t) ˆ + AX (t) dt + h (t)ΣdWt
V (t − )
+ ¯
h (t)γ(z)Np (dt, dz) (6)
Z

with initial endowment V (0) = v , where ˆ := a − a0 1,
a
ˆ
A := A − 1A0 and 1 ∈ Rm denotes the m-element unit column
vector.

The objective is to maximize a function of the log-return of wealth
1 1
J(x, t, h; θ) := − ln E e −θ ln V (t,x,h) = − ln E V −θ (t, x, h)
θ θ
(7)


By Itˆ,
o
t
e −θ ln V (t) = v −θ exp θ g (Xs , h(s); θ)ds χh
t (8)
0

where
1
g (x, h; θ) = a ˆ
(θ + 1) h ΣΣ h − a0 − A0 x − h (ˆ + Ax)
2
1 −θ
+ 1 + h γ(z) − 1 + h γ(z)1Z0 (z) ν(dz)
Z θ
(9)



and the Dol´ans exponential χh is given by
e t

t t
1
χh := exp −θ
t h(s) ΣdWs − θ2 h(s) ΣΣ h(s)ds
0 2 0
t
+ ˜
ln (1 − G (z, h(s); θ)) Np (ds, dz)
0 Z
t
+ {ln (1 − G (z, h(s); θ)) + G (z, h(s); θ)} ν(dz)ds ,
0 Z
(10)

with
−θ
G (z, h; θ) = 1 − 1 + h γ(z) (11)


Solving the Stochastic Control Problem


The process involves
1 change of measure;
2 deriving the HJB PDE;
3 identifying a (unique) candidate optimal control;
4 proving a veriﬁcation theorem;
5 proving existence of a C 1,2 solution to the HJB PDE.


Change of Measure

Change of Measure
This step is due to Kuroda and Nagai [?]. Let Pθ be the measure
h
on (Ω, FT ) defined via the Radon-Nikod´m derivative
y
dPθ h
:= χh
T (12)
dP
For a change of measure to be possible, we must ensure that
G (z, h(s); θ) < 1, which is satisfied iff h (s)γ(z) > −1 a.s. dν.

t
Wth = Wt + θ Σ h(s)ds
0
is a standard Brownian motion under the measure Pθ and X (t)
h
satisfies the SDE:
dX (t) = b + BX (t) − θΛΣ h(t) dt + ΛdWth , t ∈ [0, T ]
(13)

Change of Measure

As a result, introduce two auxiliary criterion functions under Pθ :
h
the risk-sensitive control problem:
T
1
I (v , x; h; t, T ; θ) = − ln Eh,θ exp θ
t,x g (Xs , h(s); θ)ds − θ ln v
θ t
(14)
where Et,x [·] denotes the expectation taken with respect to
the measure Pθ and with initial conditions (t, x).
h
the exponentially transformed criterion
T
˜(v , x, h; t, T ; θ) := Eh,θ exp θ
I t,x g (s, Xs , h(s); θ)ds − θ ln v
t
(15)
Note that the optimal control problem has become a diﬀusion
problem.


The HJB PDE

The HJB PDEs
The HJB PDE associated with the risk-sensitive control
criterion (14) is
∂Φ
(t, x) + sup Lh Φ(t, x) = 0,
t (t, x) ∈ (0, T ) × Rn (16)
∂t h∈J

where
Lh Φ(t, x) =
t b + Bx − θΛΣ h(s) DΦ
1 θ
+ tr ΛΛ D 2 Φ − (DΦ) ΛΛ DΦ − g (x, h; θ)
2 2
(17)
and subject to terminal condition Φ(T , x) = ln v This is a
quasi-linear PDE with two sources of non-linearity:
the suph∈J ;
the quadratic growth term (DΦ) ΛΛ DΦ;

The HJB PDE

We can address the second linearity by considering instead the
semi-linear PDE associated with the exponentially-transformed
problem (15):

˜
∂Φ 1 ˜ ˜ ˜
(t, x) + tr ΛΛ D 2 Φ(t, x) + H(t, x, Φ, D Φ) = 0 (18)
∂t 2
˜
subject to terminal condition Φ(T , x) = v −θ and where

H(s, x, r , p) = inf b + Bx − θΛΣ h(s) p + θg (x, h; θ)r
h∈J
(19)

for r ∈ R, p ∈ Rn .

˜
In particular Φ(t, x) = exp {−θΦ(t, x)}.



The supremum in (16) can be expressed as
sup Lh Φ
t
h∈J
1 θ
= (b + Bx) DΦ + tr ΛΛ D 2 Φ − (DΦ) ΛΛ DΦ + a0 + A0 x
2 2
1
a ˆ
+ sup − (θ + 1) h ΣΣ h − θh ΣΛ DΦ + h (ˆ + Ax)
h∈J 2
1 −θ
− 1 + h γ(z) − 1 + θh γ(z)1Z0 (z) ν(dz) (20)
θ Z
Under Assumption 5 the supremum is concave in h ∀z ∈ Z
a.s. dν.
ˆ
The supremum is reached for a unique maximizer h(t, x, p).
ˆ can be taken as a Borel
By measurable selection, h
measurable function on [0, T ] × Rn × Rn .


Broadly speaking, the verification theorem states that if we have
a C 1,2 ([0, T ] × Rn ) bounded function φ which satisfies the
HJB PDE (16) and its terminal condition;
the stochastic differential equation

dX (t) = b + BX (t) − θΛΣ h(t) dt + ΛdWtθ

defines a unique solution X (s) for each given initial data
X (t) = x; and,
there exists a Borel-measurable maximizer h∗ (t, Xt ) of
h → Lh φ defined in (17);
then Φ is the value function and h∗ (t, Xt ) is the optimal Markov
control process.

˜
. . . and similarly for Φ and the exponentially-transformed problem.



˜
To show that there exists a unique C 1,2 solution Φ to the HJB
PDE (18) for the exponentially transformed problem, we follow
similar arguments to those developed by Fleming and Rishel [?]
(Theorem 6.2 and Appendix E). Namely, we use an approximation
in policy space alongside functional analysis-related results on
linear parabolic partial diﬀerential equations.

The approximation in policy space algorithm was originally
proposed by Bellman in the 1950s (see Bellman [?] for details) as a
numerical method to compute the value function.



Our approach has two steps. First, we use the approximation in
policy space algorithm to show existence of a classical solution in a
bounded region. Next, we extend our argument to unbounded
state space.

To derive this second result we follow a diﬀerent argument than
Fleming and Rishel [?] which makes more use of the actual
structure of the control problem.



Zero Beta Policy: by reference to the definition of the function g
ˇ
in equation (9), a ‘zero beta’ (0β) control policy h(t) is an
admissible control policy for which the function g is independent
from the state variable x (see for instance Black [?]).

A zero beta policy exists as long as the coefficient matrix A has
full rank.

Without loss of generality, in the following we will fix a 0β control
ˇ
h as a constant function of time so that
ˇ
g (x, h; θ) = g
ˇ

where g is a constant.
ˇ



Functional analysis notation: denote by
Lη (K ) the space of η-th power integrable functions on K ⊂ Q;
· η
η,K the norm in L (K );
L η (Q), 1 < η < ∞ the space of all functions ψ such that for
∂ψ ∂ 2
ψ(t, x) and all its generalized partial derivatives ∂ψ , ∂xi , ∂xiψj ,
∂t x
i, j = 1, . . . , n are in Lη (K );
(2)
ψ η,K the Sobolev-type norm associated with
L η (Q), 1 < η < ∞ and deﬁned as
n n
(2) ∂ψ ∂ψ ∂2ψ
ψ η,K := ψ η,K + + +
∂t η,K ∂xi η,K ∂xi xj η,K
i=1 i,j=1



Step 1: Approximation in policy space - bounded space
Consider the following auxiliary problem: ﬁx R > 0 and let BR be
the open n-dimensional ball of radius R > 0 centered at 0 deﬁned
as BR := {x ∈ Rn : |x| < R}.

We construct an investment portfolio by solving the optimal
risk-sensitive asset allocation problem as long as X (t) ∈ BR for
R > 0. Then, as soon as X (t) ∈ BR , we switch all of the wealth
/
ˇ
into the 0β policy h from the exit time t until the end of the
investment horizon at time T .



The HJB PDE for this auxiliary problem can be expressed as
˜
∂Φ 1 ˜ ˜ ˜
+ tr ΛΛ (t)D 2 Φ + H(t, x, Φ, D Φ) = 0
∂t 2
∀(t, x) ∈ QR := (0, T ) × BR
subject to boundary conditions
˜
Φ(t, x) = Ψ(t, x)
∀(t, x) ∈ ∂ ∗ QR := ((0, T ) × ∂BR ) ∪ ({T } × BR )
and where
Ψ(T , x) = e −θ ln v ∀x ∈ BR ;
Ψ(t, x) := ψ(t) := e θˇ (T −t) ∀(t, x) ∈ (0, T ) × ∂BR and
g
ˇ
where h is a ﬁxed arbitrary 0β policy. ψ is obviously of class
C 1,2 (Q ) and the Sobolev-type norm
R
(2) (2)
Ψ η,∂ ∗ QR = ψ η,QR (21)
is ﬁnite.


˜ ˜ ˜
Define a sequence of functions Φ1 , Φ2 ,... Φk ,... on
QR = [0, T ] × BR and of bounded measurable feedback control
laws h0 , h1 ,... hk ,... where h0 is an arbitrary control. Assuming hk
˜
is defined, Φk+1 solves the boundary value problem:
˜
∂ Φk+1 1
+ tr ΛΛ (t)D 2 Φk+1˜
∂t 2
˜ ˜
+f (t, x, hk ) D Φk+1 + θg (t, x, hk )Φk+1 = 0 (22)

subject to boundary conditions
˜
Φk+1 (t, x) = Ψ(t, x)
∀(t, x) ∈ ∂ ∗ QR := ((0, T ) × ∂BR ) ∪ ({T } × BR )

Based on standard results on parabolic Partial Differential
Equations (Appendix E in Fleming and Rishel [?], Chapter IV in
Ladyzhenskaya, Solonnikov and Uralceva [?]), the boundary value
problem (22) admits a unique solution in L η (QR ).


Moreover, for almost all (t, x) ∈ QR , k = 1, 2, . . ., we deﬁne hk+1
by the prescription

hk+1 = Argminh∈J ˜ ˜
f (t, x, h) D Φk+1 + θg (t, x, h)Φk+1 (23)

so that
˜ ˜
f (t, x, hk+1 ) D Φk+1 + θg (t, x, hk+1 )Φk+1
= inf ˜ ˜
f (t, x, h) D Φk+1 + θg (t, x, h)Φk+1
h∈J
˜ ˜
= H(t, x, Φk+1 , D Φk+1 ) (24)



˜
Observe that the sequence Φk is globally bounded:
k∈N
bounded from below by 0 (by Feynman-Kac).
bounded from above (optimality principle and ‘zero beta’ (0β)
control policy)
These bounds do not depend on the radius R and are therefore
valid over the entire space (0, T ) × Rn .



Step 2: Convergence Inside the Cylinder (0, T ) × BR
It can be shown using a control argument that the sequence
˜
Φk ˜
is non increasing and as a result converges to a limit Φ
k∈N
˜ (2)
as k → ∞. Since the Sobolev-type norm Φk+1 is bounded η,QR
for 1 < η < ∞, we can show that the H¨lder-type norm |Φk |1+µ is
o ˜
QR
also bounded by apply the following estimate given by equation
(E.9) in Appendix E of Fleming and Rishel
(2)
|Φk |1+µ ≤ MR Φk
˜
QR
˜
η,QR (25)
for some constant MR (depending on R) and where
n+2
µ = 1−
η
n
|Φk |1+µ = |Φk |µ R +
˜
QR
˜
Q |Φki |µ R
˜x
Q
i=1



˜ ˜
|Φk (t, x) − Φk (t, y )|
|Φk |µ R
˜
Q = ˜
sup |Φk (t, x)| + sup
(t,x)∈QR |x − y |µ
(x, y ) ∈ G
0≤t≤T
˜ ˜
|Φk (s, x) − Φk (t, x)|
+ sup
|s − t|µ/2
x ∈G
0 ≤ s, t ≤ T

As k → ∞,
˜ ˜
D Φk converges to D Φ uniformly in Lη (QR ) ;
˜ ˜
D 2 Φk converges to D 2 Φ weakly in Lη (QR ) ; and
˜
∂ Φk ˜
∂Φ
∂t converges to ∂t weakly in Lη (QR ).
˜
We can then prove that Φ ∈ C 1,2 (QR ).


Step 3: Convergence from the Cylinder [0, T ) × BR to the
State Space [0, T ) × Rn
Let {Ri }i∈N > 0 be a non decreasing sequence with
limi→∞ Ri → ∞ and let {τi }i∈N be the sequence of stopping times
deﬁned as

τi := inf {t : X (t) ∈ BRi } ∧ T
/

Note that {τi }i∈N is non decreasing and limi→∞ τi = T .
˜
Denote by Φ(i) the limit of the sequence Φk˜ on
k∈N
(0, T ) × BRi , i.e.

˜ ˜
Φ(i) (t, x) = lim Φk (t, x) ∀(t, x) ∈ (0, T ) × BRi (26)
k→∞



˜
Figure: Convergence of the Sequence Φ(i)
i∈N



˜
The sequence (Φ(i) )i∈N is bounded and non increasing: it
˜
converges to a limit Φ. This limit satisfies the boundary condition.
˜
We now apply Ascoli’s theorem to show that Φ is C 1,2 and satisfies
the HJB PDE. These statements are local properties so we can
restrict ourselves to a finite ball QR .



Using the following estimate given by equation (E8) in Appendix E
of Fleming and Rishel, we deduce that

˜ (2) (2)
Φ(i) η,QR ≤M Ψ η,∂ ∗ QR (27)

for some constant M.

˜ (2)
Combineing (27) with assumption (21) implies that Φ(i) η,QR is
bounded for η > 1. Critically, the bound M does not depend on k.
˜ ˜
Moreover, by Step 2 Φ(i) and D Φ(i) are uniformly bounded on any
˜ (2)
compact subset of Q0 . By equation (27) we know that Φ η,QR is
bounded for any bounded set QR ⊂ Q0 .



˜
On QR , Φ(i) also satisﬁes the H¨lder estimate
o
(2)
|Φ(i) |1+µ ≤ M1 Φ(i)
˜
QR
˜
η,QR

for some constant M1 depending on QR and η.
˜ (i) 2 ˜ (i)
We ﬁnd, that ∂ Φ and ∂ Φxj also satisfy a uniform H¨lder
∂t ∂xi o
condition on any compact subset of Q.



˜
By Ascoli’s theorem, we can ﬁnd a subsequence Φl of
l∈N
˜ l
˜
Φ(i) ˜
such that Φl , ∂Φ ˜
, D Φl and
i∈N l∈N ∂t l∈N
l∈N
˜ ˜ ˜
∂Φ ˜ ˜
D 2 Φl tends to respective limits Φ, ∂t D Φ and D 2 Φ
l∈N
uniformly on each compact subset of [0, T ] × Rn .

˜
Finally, the function Φ is the desired solution of equation (18) with
˜
terminal condition Φ(T , x) = e −θ ln v


Concluding Remarks

Concluding Remarks

We have seen that risk-sensitive asset management can be
extended to include the possibility of infinite activity jumps in
asset prices. In this case a unique optimal admissible control
policy and a unique classical C 1,2 ((0, T ) × Rn ) solution exists.
This approach extends naturally and with similar results to a
jump-diffusion version of the risk-sensitive benchmarked asset
management problem (see Davis and Lleo [?] for the original
paper on benchmarks in a diffusion setting).
We want to extend this approach to cover credit risk, for
which we needed asset price processes with jumps.
We are also working on extending this approach to include
jumps in the factor processes as well as holding constraints.


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