Asset-Liability Management via Risk-Sensitive Control ... · Asset-Liability Management via Risk-Sensitive Control: Jump-Di usion Risk Sensitive Control: A De nition From Risk-Sensitive

Asset-Liability Management via Risk-Sensitive Control: Jump-Diffusion

Asset-Liability Management via Risk-Sensitive Control:Jump-Diffusion

Mark Davis1 and Sebastien Lleo2

ICSP 2013University of Bergamo, July 8, 2013

1Department of Mathematics, Imperial College London, London SW7 2AZ, England, Email:[email protected]

2Finance Department and Behavioral Sciences Research Centre, Reims Management School, 59rue Pierre Taittinger, 51100 Reims, France, Email: [email protected]


Outline

Outline

Asset and Liability Management

Risk Sensitive Control: A Definition

The Risk-Sensitive ALM Problem

Equity and Leverage

How to Solve a Stochastic Control Problem

Existence of a C 1,2 Solution to the HJB PDE

Verification Theorem

Concluding Remarks




Asset and liability management (ALM) is criticalfor funded investors such as endowment funds andpension funds, but also for investors who have theability to grow their asset base by borrowing: banksand hedge funds.

In this talk we solve an ALM problem usingRisk-Sensitive Control methods. Our work isrelated to the articles on surplus management byRudolf and Ziemba [13] and Benk [2].

In our approach, the investor’s objective is tojointly select an optimal amount of leverage and anoptimal asset allocation to maximise the expectedutility of the equity or surplus of his/her portfolio.Also, we allow securities prices and the value of theliability to be influenced by a number of underlyingfactors.



Balance Sheet of a Funded Investor

Assets at 'me t: •  Valued at V(t) •  Cons'tuted of a

por5olio of m securi'es.

Liabili'es at 'me t: •  Valued at L(t)

Equity at 'me t: •  Values at E(t)

E(t)=V(t)-‐L(t)

Leverage ra'o: ρ(t)=V(t)/E(t)




Risk-sensitive control is a generalization of classical stochastic control in whichthe degree of risk aversion or risk tolerance of the optimizing agent is explicitlyparameterized in the objective criterion and influences directly the outcome ofthe optimization.

In risk-sensitive control, the decision maker’s objective is to select a controlpolicy h(t) to maximize the criterion

J(x , t, h; θ) := −1

θlnE

[e−θF (t,x,h)

](1)

where t is the time, x is the state variable, F is a given reward function, andthe risk sensitivity θ ∈ (−1, 0) ∪ (0,∞) is an exogenous parameter representingthe decision maker’s degree of risk aversion.



An Intuitive View of the Criterion

A Taylor expansion of the previous expression around θ = 0 evidences the vitalrole played by the risk sensitivity parameter:

J(x , t, h; θ) = E [F (t, x , h)]− θ

2Var [F (t, x , h)] + O(θ2) (2)

I θ → 0, “risk-null”: corresponds to classical stochastic control;

I θ < 0: “risk-seeking” case corresponding to a maximization of theexpectation of a convex decreasing function of F (t, x , h);

I θ > 0: “risk-averse” case corresponding to a minimization of theexpectation of a convex increasing function of F (t, x , h).



From Risk-Sensitive Control to Risk-Sensitive Asset Management

If we chose F (t, x) = ln WealthT , the Taylor expression is tantamount to adynamic version of Markowitz’ mean-variance analysis, but with a built-incorrection for higher moments.

This leads us to the risk-sensitive asset management model:

J(x , t, h; θ) := −1

θlnE

[e−θ ln WealthT

]= −1

θlnE

[Wealth−θT

]= E [ln WealthT ]− θ

2Var [ln WealthT ] + O(θ2)



Emergence of a Risk-Sensitive Asset Management (RSAM) Theory

I Jacobson [9], Whittle [14], Bensoussan [3] led the theoretical developmentof risk sensitive control.

I Risk-sensitive control first applied to finance by Lefebvre andMontulet [11] in a corporate finance context and by Fleming [6] in aportfolio selection context.

I Bielecki and Pliska [4]: first to apply continuous time risk-sensitive controlas a practical tool to solve ‘real world’ portfolio selection problems.

I Major contribution by Kuroda and Nagai [10]: elegant solution methodbased on a change of measure argument.



Extensions to a Jump-Diffusion Setting

The Risk-Sensitive Asset Management (RSAM) theory was developed based ondiffusion models. In a jump-diffusion setting,

I Davis and Lleo [5] consider a finite time horizon problem with randomjumps in the asset prices. They prove the existence of an optimal controland showed that the value function is a smooth (strong) solution of theHamilton Jacobi Bellman Partial Differential Equation (HJB PDE).

I Davis and Lleo [6] consider a finite time horizon problem with randomjumps in both asset prices and factors. Under standard controlassumptions they prove the existence of an optimal control and showedthat the value function is a smooth (strong) solution of the HamiltonJacobi Bellman Partial Differential Equation (HJB PDE).

I Davis and Lleo [7] extend these results to a benchmarked assetmanagement problem.



The Risk-Sensitive ALM Problem - General ModelLet (Ω, Ft ,F ,P) be the underlying probability space. Take a market with

1. A money market asset S0 with dynamics

dS0(t)

S0(t)= a0 (t,X (t)) dt, S0(0) = s0 (3)

2. m risky assets following jump-diffusion SDEs

dSi (t)

Si (t−)=

[a(t,X (t−)

)]idt +

N∑k=1

Σik(t,X (t))dWk(t) +

∫Z

γi (t, z)N(dt, dz),

Si (0) = si , i = 1, . . . ,m (4)

3. A liability given by the same type of jump-diffusion process:

dL(t)

L(t)= c(t,X (t−))dt+ς ′(t,X (t))dW (t)+

∫Z

η(t, z)N(dt, dz), L(0) = l

(5)

4. A n-dimensional vector of factors X (t) following

dX (t) = b(t,X (t−)

)dt + Λ(t,X (t))dW (t) +

∫Z

ξ(t,X (t−), z

)N(dt, dz),

X (0) = x0 ∈ Rn. (6)



Note:

I W (t) is a Rm+n+1-valued (Ft)-Brownian motion with components Wk(t),k = 1, . . . , (m + n + 1).

I Np(dt, dz) is a Poisson random measure (see e.g. Ikeda andWatanabe [8]) defined as

Np(dt, dz)

=

Np(dt, dz)− ν(dz)dt =: Np(dt, dz) if z ∈ Z0

Np(dt, dz) if z ∈ Z\Z0



Two Implementations

We consider two different implementations of this general model:

1. Affine dynamics, with jumps in assets and liabilities only;

2. Standard control assumptions with jumps in assets, liabilities and factors.



Assumption 1: Affine dynamics with jumps in assets and liabilities

1. The money market asset S0 has a dynamics

dS0(t)

S0(t)=(a0 + A′0X (t)

)dt, S0(0) = s0 (7)


dSi (t)

Si (t−)= (a + AX (t))i dt +

N∑k=1

ΣikdWk(t) +

∫Z

γi (z)N(dt, dz),

Si (0) = si , i = 1, . . . ,m (8)


dL(t)

L(t)=(c + C ′X (t)

)dt+ς ′dW (t)+

∫Z

η(z)N(dt, dz), L(0) = l (9)

4. A n-dimensional vector of factors X (t) following

dX (t) = (b + BX (t)) dt + ΛdW (t),

X (0) = x0 ∈ Rn. (10)



Assumption 2: Standard control assumptionsUnder standard control assumptions, our model has the same form as thegeneral model:

1. A money market asset S0 with dynamics

dS0(t)

S0(t)= a0 (t,X (t)) dt, S0(0) = s0 (11)


dSi (t)

Si (t−)=

[a(t,X (t−)

)]idt +

N∑k=1

Σik(t,X (t))dWk(t) +

∫Z

γi (t, z)N(dt, dz),

Si (0) = si , i = 1, . . . ,m (12)


dL(t)

L(t)= c(t,X (t−))dt+ς ′(t,X (t))dW (t)+

∫Z

η(t, z)N(dt, dz), L(0) = l

(13)4. A n-dimensional vector of factors X (t) following

dX (t) = b(t,X (t−)

)dt + Λ(t,X (t))dW (t) +

∫Z

ξ(t,X (t−), z

)N(dt, dz),

X (0) = x0 ∈ Rn. (14)



I The functions a0, a, b, c, Σ = [σij ], ς, Λ are Lipschitz continuous,bounded with bounded derivatives in terms of the variables t and x .

I Ellipticity condition:

ΣΣ′ > 0 (15)

I The jump intensities ξ(z) and γ(z) satisfies appropriate well-posednessconditions.

I Independence of systematic (factor-driven) and idiosyncratic (asset-drivenand liability-driven) jump: ∀(t, x , z) ∈ [0,T ]× Rn × Z,γ(t, z)ξ′(t, x , z) = η(t, z)ξ′(t, x , z) = 0.

...



... plus an extra condition:

Assumption

The vector valued function γ(t, z) satisfy:∫Z

|ξ(t, x , z)|ν(dz) <∞, ∀(t, x) ∈ [0,T ]× Rn (16)

Note that the minimal condition on ξ under which the factor equation (14) iswell posed is ∫

Z0

|ξ(t, x , z)|2ν(dz) <∞,

However, for this paper it is essential to impose the stronger condition (16) inorder to connect the viscosity solution of HJB partial integro-differentialequation (PIDE) with the viscosity solution of a related parabolic PDE.



Next Steps: Find the Dynamics of The Assets and Equity

Assets at 'me t: •  Valued at V(t) •  Cons'tuted of a

por1olio of m securi'es.

Liabili'es at 'me t: •  Valued at L(t)

Equity at 'me t: •  Values at E(t)

E(t)=V(t)-‐L(t)

Leverage ra'o: ρ(t)=V(t)/E(t)



Wealth Dynamics

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

dV (t)

V (t−)= (a0 (t,X (t))) dt + h′(t)a (t,X (t)) dt + h′(t)Σ(t,X (t))dWt

+

∫Z

h′(t)γ(t, z)Np(dt, dz)

(17)

with initial endowment V (0) = v , where a := a− a01 and 1 ∈ Rm denotes them-element unit column vector.


Equity and Leverage

Equity and Leverage

The time t equity or surplus, E(t), is the wealth belonging directly to theinvestor. It is is defined as the difference between the value of the assets and ofthe liabilities, i.e.

E(t) = V (t)− L(t), E(0) := e0 = v − l > 0

WLOG, we assume that e0 = 1.

The dynamics of the equity is given in differential form by

dE(t) = dV (t)− dL(t)


Equity and Leverage

The time t leverage ratio, ρ(t), is defined as the ratio of asset value to equityvalue:

ρ(t) =V (t)

E(t)

Thus,V (t) = ρ(t)E(t) L(t) = (ρ(t)− 1)E(t)

In our case, leverage is a control variable: the investor’s objective is to chooseboth an optimal level of leverage and an optimal investment strategy.


Equity and Leverage

The dynamics of the equity in response to an ALM policy (h(t), ρ(t)) can beexpressed as

dE(t) = V (t−)[(a0 (t,X (t))) dt + h′(t)a (t,X (t)) dt + h′(t)Σ(t,X (t))dWt

+

∫Z

h′(t)γ(t, z)N(dt, dz)

]−L(t−)

[c(t,X (t−))dt + ς ′(t,X (t))dW (t) +

∫Z

η(t, z)N(dt, dz)

]


Equity and Leverage

Rewriting in terms of equity and leverage only, we get

dE(t)

E(t−)= c(t,X (t))dt

+ρ(t)[h′(t)a(t,X (t))− c(t,X (t))

]dt

+(ς ′(t,X (t)) + ρ(t)[h′(t)Σ(t,X (t))− ς ′(t,X ())]

)dW (t),

+

∫Z

η(t, z) + ρ(t)

[h′(t)γ(t, z)− η(t, z)

]N(dt, dz)

= α(t,X (t), h(t), ρ(t))dt + β(t,X (t), h(t), ρ(t))dW (t),

+

∫Z

ζ((t, z , h(t), ρ(t)))N(dt, dz) (18)

where

α(t, x , h, ρ) := c(t, x) + ρ[h′a(t, x)− c(t, x)

]β(t, x , h, ρ) := ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

ζ(t, z , h, ρ) := η(t, z) + ρ(t)[h′(t)γ(t, z)− η(t, z)

]and c := c − a0.


Equity and Leverage

Investment and Leverage Constraints

We also consider r ∈ N fixed investment constraints expressed in the form

Υ′h(t) ≤ υ (19)

where Υ ∈ Rm × Rr is a matrix and υ ∈ Rr is a column vector.

Assumption

The systemΥ′y ≤ υ

for the variable y ∈ Rm admits at least two solutions.

We also introduce the following constraints on leverage:

K :=ρ ∈ R : −∞ < ρ− ≤ ρ(t) ≤ ρ+ <∞

(20)

where ρ−, ρ+ are two real constants.

These assumptions guarantee that there will be at least one ALM policysatisfying the constraints.


Equity and Leverage

Problem Formulation

The investor’s objective is to maximise the risk-sensitive criterion J(h, ρ)

J(h, ρ; θ) := −1

θlnE

[e−θ ln ET

]= −1

θlnE

[ln E−θT

](21)

where ln ET (h) can be interpreted as the log return on equity.


Equity and Leverage

From (17) and the general Ito formula we find that the term e−θ ln E(T ) can beexpressed as

e−θ ln E(T ) = exp

θ

∫ T

0

g(t,Xt , h(t))dt

χh(T ) (22)

where

g(t, x , h)

=1

2(θ + 1)ββ′(t, x , h, ρ)− α(t, x , h, ρ)

+

∫Z

1

θ

[(ζ(t, z , h, ρ))−θ − 1

]+ ζ(t, z , h, ρ)1Z0 (z)

ν(dz)

=1

2(θ + 1)

[ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

] [ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

]′−c(t, x)− ρ

[h′a(t, x)− c(t, x)

]+

∫Z

1

θ

[(η(t, z) + ρ(t)

[h′(t)γ(t, z)− η(t, z)

])−θ − 1]

+(η(t, z) + ρ(t)

[h′(t)γ(t, z)− η(t, z)

])1Z0 (z)

ν(dz) (23)


Equity and Leverage

and the Doleans exponential χht is given by

χh(t) := exp

−θ∫ t

0

β(s,X (s), h(s), ρ(s))dWs

−1

2θ2

∫ t

0

β(s,X (s), h(s), ρ(s))β(s,X (s), h(s), ρ(s))′ds

+

∫ t

0

∫Z

ln (1− G(s, z , h(s), ρ(s))) N(ds, dz)

+

∫ t

0

∫Z

ln (1− G(s, z , h(s), ρ(s))) + G(s, z , h(s), ρ(s)) ν(dz)ds

,

(24)

with

G(t, z , h, ρ) = 1− (ζ(t, z , h, ρ))−θ

= 1−(η(t, z) + ρ(t)

[h′(t)γ(t, z)− η(t, z)

])−θ(25)


Equity and Leverage

Change of Measure

For h ∈ A, ρ ∈ R and θ > 0 let Ph be the measure on (Ω,FT ) defined via theRadon-Nikodym derivative

dPh

dP= χh(T ), (26)

and let Eh denote the corresponding expectation. Then

J(h, ρ) = −1

θlnEh

[exp

(θ

∫ ′0

g(t,Xt , h(t))dt

)]. (27)

Moreover, under Ph,

W ht = Wt + θ

∫ t

0

β(s,X (s), h(s), ρ(s))ds

= Wt + θ

∫ t

0

ς ′(s,X (s)) + ρ(s)(h′(s)Σ(s,X (s))− ς ′(s,X (s)))ds

is a standard Brownian motion and...


Equity and Leverage

... the Ph-compensated Poisson random measure is given by∫ t

0

∫Z0

Nh(ds, dz)

=

∫ t

0

∫Z0

N(ds, dz)−∫ t

0

∫Z0

1− G(s, z , h(s)) ν(dz)ds

=

∫ t

0

∫Z0

N(ds, dz)−∫ t

0

∫Z0

(ζ(s, z , h(s), ρ(s)))−θ

ν(dz)ds

=

∫ t

0

∫Z0

N(ds, dz)−∫ t

0

∫Z0

(η(s, z) + ρ(s)

[h′(s)γ(s, z)− η(s, z)

])−θν(dz)ds


Equity and Leverage

As a result, under Ph the factor process X (s), 0 ≤ s ≤ t satisfies the SDE:

dX (s) = f (s,X (s), h(s))ds+Λ(s,X (s))dW θs +

∫Z

ξ(s,X (s−), z

)Nh(ds, dz), X (0) = x0

(28)where

f (t, x , h)

:= b(t, x)− θΛβ(t,X (t), h, ρ) +

∫Z

ξ(t, x , z)[(ζ(t, z , h, ρ))−θ

]ν(dz)

= b(t, x)− θΛ[ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

]+

∫Z

ξ(t, x , z)[(η(t, z) + ρ(t)

[h′(t)γ(t, z)− η(t, z)

])−θ − 1Z0 (z)]ν(dz)

(29)

and b is the P-measure drift of the factor process.


Equity and Leverage

Following the change of measure we introduce two auxiliary criterion functionsunder Pθh :

I the risk-sensitive control problem:

I (v , x ; h; t,T ; θ) = −1

θlnEh,θ

t,x

[exp

θ

∫ T

t

g(Xs , h(s); θ)ds

](30)

where Et,x [·] denotes the expectation taken with respect to the measurePθh and with initial conditions (t, x).

I the exponentially transformed criterion

I (v , x , h; t,T ; θ) := Eh,θt,x

[exp

θ

∫ T

t

g(s,Xs , h(s); θ)ds

](31)




Our objective is to solve the control problem in a classical sense.

The process involves

1. deriving the HJB P(I)DE;

2. identifying a (unique) candidate optimal control;

3. proving existence of a C 1,2 (classical) solution to the HJB P(I)DE.

4. proving a verification theorem;



The HJB P(I)DEs

The HJB PIDE associated with the risk-sensitive control criterion (30) is

∂Φ

∂t+ sup

h∈J ,ρ∈KLh(

t, x ,Φ,DΦ,D2Φ)

= 0 (32)

where

Lh (t, x , u, p,M) = f (t, x , h)′p +1

2tr(ΛΛ′(t, x)M

)− θ

2p′ΛΛ′(t, x)p

−g(t, x , h) + INL [t, x , u, p] (33)

with

INL [t, x , u, p] =

∫Z

−1

θ

(e−θ[u(t,x+ξ(t,x,z))−u(t,x)] − 1

)− ξ(t, x , z)′p

ν(dz)(34)

and subject to the terminal condition (recall our normalization e0 = 1)

Φ(T , x) = 0, x ∈ Rn. (35)

Condition (16) ensures that INL is well defined, at least for bounded u.



To remove the quadratic growth term, we consider the PIDE associated withthe exponentially-transformed problem (31):

∂Φ

∂t(t, x) +

1

2tr(

ΛΛ′(t, x)D2Φ(t, x))

+ H(t, x , Φ,DΦ)

+

∫Z

Φ (t, x + ξ(t, x , z))− Φ(t, x)− ξ(t, x , z)′DΦ(t, x)

ν(dz) = 0(36)

subject to terminal condition

Φ(T , x) = 1 (37)

where for r ∈ R, p ∈ Rn

H(s, x , r , p) = infh∈J

f (s, x , h)′p + θg(s, x , h)r

(38)

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



Some Insights...

The presence (or absence) of jumps plays a crucial role in our problem. If wethink about our general model as a meta model for diffusion and jump-diffusionproblems we observe that:

1. Pure diffusion: we have a pure diffusion problem and the hope of findingan analytical solution for the optimal policy pair (h∗, ρ∗). The HJBequation is a parabolic PDE;

2. Jumps in asset and liabilities only: because of the jumps we will notgenerally be able to analytical solution for the optimal policy pair (h∗, ρ∗).However, the HJB equation remains a parabolic PDE;

3. Jumps in factors only: we can potentially find an analytical solution forthe optimal policy pair (h∗, ρ∗), but now the jumps in the factor level havetransformed the HJB equation is a parabolic PIDE;

4. Full jump diffusion: because of the jumps in assets and liabilities we willnot generally be able to analytical solution for the optimal policy pair(h∗, ρ∗). Moreover, the jumps in the factor level have transformed theHJB equation is a parabolic PIDE;



Identifying a (Unique) Candidate Optimal Control

The supremum in (32) can be expressed as

suph∈J ,ρ∈K

Lh (t, x , u, p,M)

= b′(t, x)p +1

2tr(ΛΛ′(t, x)M

)− θ

2p′ΛΛ′(t, x)′p + c(t, x) + INL [t, x , u, p]

suph∈J ,ρ∈K

−1

2(θ + 1)

[ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

] [ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

]′−θΛ

[ς ′(t, x) + ρ(h′Σ(t, x)− ς ′(t, x))

]p + ρ

[h′a(t, x)− c(t, x)

]−1

θ

∫Z

(1− θξ(t, x , z)′p

) [(η(t, z) + ρ(t)

[h′(t)γ(t, z)− η(t, z)

])−θ − 1]

+(ρ(t)

[h′(t)γ(t, z)− η(t, z)

])1Z0 (z)

ν(dz)

(39)



This equation looks messy, but it is actually well structured:

I Because ΣΣ′ > 0 and because systematic jumps are independent fromidiosyncratic jumps, this problem corresponds to the maximization of aconcave function on a convex set of constraints.

I By the Lagrange Duality (see for example Theorem 1 in Section 8.6in [12]), we conclude that the supremum in (22) admits a uniquemaximizing pair (h, ρ)(t; x ; p) for (t; x ; p) ∈ [0; T ]× Rn × Rn.

I By measurable selection, (h, ρ) can be taken as a Borel measurablefunction on [0; T ]× Rn × Rn.


Existence of a C1,2 Solution to the HJB PDE

Existence of a C 1,2 Solution to the HJB PDE

Once we have cleared the hurdle of having two controls, we are back to PDEterritory:

I The number and properties of the controls do not matter (much);

I We can rely on our earlier results (see [5, 6]

Proving the existence of a strong, C 1,2, solution is the most difficult andintricate step in the process.

However, this is a necessary step if we want to use the Verification Theorem toconclusively solve our optimal investment problem.




Broadly speaking, the verification theorem states that if we have

I a C 1,2 ([0,T ]× Rn) bounded function φ which satisfies the HJB PDE (32)and its terminal condition;

I the stochastic differential equation

dX (s) = f (s,X (s), h(s), ρ(s); θ)ds + Λ(s,X (s))dW θs

+

∫Z

ξ(s,X (s−), z

)Nθ

p (ds, dz)

defines a unique solution X (s) for each given initial data X (t) = x ; and,

I there exists a Borel-measurable maximizing pair (h∗(t,Xt), ρ∗(t,Xt)) of

h 7→ Lhφ defined in (33);

then Φ is the value function and (h∗(t,Xt), ρ∗(t,Xt)) is the optimal pair of

Markov control processes.

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


Concluding Remarks

Concluding Remarks

I Factors give us a lot of flexibility in the way we setup and parametrize theproblem. We could even consider behavioural factors in our analysis(see [1])!

I Even with liabilities, leverage and jumps, we still manage to get a smoothvalue function and a convex optimisation problem for the controls. This ispromising from a numerical perspective.

I The final major question we face to implement the model is how topopulate the set of parameters sensibly.

I Coming soon: “Black-Litterman” for ALM (diffusion and affinejump-diffusion).


Concluding Remarks

Thank you!

3

Any question?

3Adapted from W. Krawcewicz, University of Alberta


Concluding Remarks

Thank you!

XXXXXX

Espresso

4

Any question?

4Adapted from W. Krawcewicz, University of Alberta


Concluding Remarks

G. Andruszkiewicz, M.H.A. Davis, and S. Lleo.Taming animal spirits: risk management with behavioural factors.Annals of Finance, 9:145–166, 2013.

Mareen Benk.Intertemporal surplus management with jump risk.In Horand I. Gassmann and William T. Ziemba, editors, StochasticProgramming: Applications in Finance, Energy, Planning and Logistics,volume 4 of World Scientific Series in Finance, pages 69–96. WorldScientific Publishing Company, 2012.

A. Bensoussan.Stochastic Control of Partially Observable Systems.Cambridge University Press, 2004.

T.R. Bielecki and S.R. Pliska.Risk-sensitive dynamic asset management.Applied Mathematics and Optimization, 39:337–360, 1999.

M.H.A. Davis and S. Lleo.Jump-diffusion risk-sensitive asset management I: Diffusion factor model.SIAM Journal on Financial Mathematics, 2:22–54, 2011.

M.H.A. Davis and S. Lleo.


Concluding Remarks

Jump-diffusion risk-sensitive asset management ii: Jump-diffusion factormodel.SIAM Journal on Control and Optimization (forthcoming), 2013.

M.H.A. Davis and S. Lleo.Jump-diffusion risk-sensitive benchmarked asset management.In H.I. Gassmann and W.T. Ziemba, editors, Stochastic Programming:Applications in Finance, Energy, Planning and Logistics, number 4 inWorld Scientific Series in Finance, pages 97–128. World, 2013.

N. Ikeda and S. Watanabe.Stochastic Differential Equations and Diffusion Processes.North-Holland Publishing Company, 1981.

D.H. Jacobson.Optimal stochastic linear systems with exponential criteria and theirrelation to deterministic differential games.IEEE Transactions on Automatic Control, 18(2):114–131, 1973.

K. Kuroda and H. Nagai.Risk-sensitive portfolio optimization on infinite time horizon.Stochastics and Stochastics Reports, 73:309–331, 2002.

M. Lefebvre and P. Montulet.


Concluding Remarks

Risk-sensitive optimal investment policy.International Journal of Systems Science, 22:183–192, 1994.

D. Luenberger.Optimization by Vector Space Methods.John Wiley & Sons, 1969.

Markus Rudolf and William T. Ziemba.Intertemporal surplus management.Journal of Economic Dynamics & Control, 28:975–990, 2004.

P. Whittle.Risk Sensitive Optimal Control.John Wiley & Sons, New York, 1990.

Documents

Asset-Liability Management via Risk-Sensitive Control ... · Asset-Liability Management via Risk-Sensitive Control: Jump-Di usion Risk Sensitive Control: A De nition From Risk-Sensitive