optimal consumption and investment for a large investor: an intensity-based control framework

Mathematical Finance, Vol. 0, No. 0 (2012), 1–31

OPTIMAL CONSUMPTION AND INVESTMENT FOR A LARGE INVESTOR:AN INTENSITY-BASED CONTROL FRAMEWORK

MICHAEL BUSCH, RALF KORN, AND FRANK THOMAS SEIFRIED

University of Kaiserslautern

We introduce a new stochastic control framework where in addition to controllingthe local coefficients of a jump-diffusion process, it is also possible to control theintensity of switching from one state of the environment to the other. Building upon thisframework, we develop a large investor model for optimal consumption and investmentthat generalizes the regime-switching approach of Bauerle and Rieder (2004).

KEY WORDS: optimal consumption and investment, large investor, market manipulation, regime-shift model.

1. INTRODUCTION

Motivation. We set up a general stochastic control framework in a regime-switchingenvironment. In addition to controlling the local dynamics (i.e., the coefficients of theunderlying jump-diffusion stochastic differential equation), in our model the controllercan also influence the intensity of the jump process that indicates the state of the envi-ronment. Following a classical stochastic control approach, we formulate the associatedHamilton–Jacobi–Bellman (HJB) equations as a coupled system of partial differentialequations and provide a suitable verification theorem. Our analysis in particular raisesthe issue of existence of a solution to the corresponding controlled jump-diffusion sys-tem; as one of our contributions we provide an explicit construction of the controlleddynamics based on change-of-measure arguments.

Our setting is related to the problems considered by Becherer and Schweizer (2005)where the authors establish existence and uniqueness of classical solutions to reaction-diffusion systems. This paper extends their analysis by including control decisions inboth the controlled jump-diffusion stochastic differential equation and the jump processgoverning regime shifts.

We also remark that our model provides a natural generalization of the regime-shiftmodel of Bauerle and Rieder (hereafter BR). In contrast to our setting, in their modelthere is no possibility to control the regime-switching intensities.

Large Investor Model. As a natural application of our stochastic control framework,we suggest a new class of models for a financial market with a large investor. The large

Michael Busch and Ralf Korn gratefully acknowledge financial support by the project “Alternative In-vestments” of the Bundesministerium fur Bildung und Forschung. Frank Seifried gratefully acknowledgesfinancial support by the Rheinland-Pfalz Cluster of Excellence “Dependable Adaptive Systems and Mathe-matical Modeling.”

Manuscript received March 2011; final revision received September 2011.Address correspondence to Frank Seifried, Department of Mathematics, University of Kaiserslautern,

67653 Kaiserslautern, Germany; e-mail: [email protected].

DOI: 10.1111/j.1467-9965.2012.00528.xC© 2012 Wiley Periodicals, Inc.

1

2 M. BUSCH, R. KORN, AND F. T. SEIFRIED

investor faces the classical optimal consumption and investment problem to maximizeexpected utility from terminal wealth and consumption; see, e.g., Merton (1969, 1971).The market environment switches between two regimes modeled via different coefficientsof the stock price evolution. The investor is large in the sense that the market takeshis actions as signals; thus the intensities for switching between market regimes dependon the investor’s asset holdings and/or consumption decisions. In particular, we thusgeneralize the framework of BR in an important dimension.

We show that the market impact of the large investor’s actions leads to significantdeviations of optimal portfolio and consumption decisions from the classical Merton-type optimal strategies of BR. Illustrative examples of this behavior are provided inSection 5. For instance, in a natural setting it can even be optimal for the large investorto consume although he obtains no utility from consumption at all! The reason forthis—and for seemingly too high investment in the stock—is the investor’s incentive toconvince the market to remain in the favorable state. Market manipulation strategies orpayments of bribes provide natural interpretations of this type of behavior.

Related Literature. There are large strands of literature concerning both regime-shiftmodels on the one hand and large investor models on the other. Models with possibleregime shifts related to optimal investment problems include the articles of BR, Sass andHaussmann (2004), Diesinger, Kraft, and Seifried (2010), or Kashiwabara and Nakamura(2011), among others; see also the monograph of Elliott, Aggoun, and Moore (1994).

Financial market models featuring an investor with market impact have been formu-lated in various settings and differ mainly in the way the large investor’s influence onthe market is formalized. In the literature the large investor often has a direct impacton asset prices, either via influencing drift and/or diffusion coefficients of the price dy-namics (see, e.g., the models of Cvitanic and Ma 1996; Basak 1997; Cuoco and Cvitanic1998; DeMarzo and Urosevic 2006) or via directly shifting the stock price (as in thediscrete-time setting of Jarrow 1992 or the general equilibrium model of Bank and Baum2004). Closed-form solutions are rare, a notable exception being Kraft and Kuhn (2011)who study a linear specification of the model of Cuoco and Cvitanic (1998).

The remainder of this paper is organized as follows. In Section 3, we set up the generalintensity-based stochastic control problem and provide the construction of the controlledprocess. In Section 4, we formulate the associated dynamic programming equation andestablish a suitable verification theorem. Section 4 introduces our large investor model.We investigate the large investor’s optimal consumption and investment problem andprovide explicit solutions in Section 5. Section 6 concludes with possible perspectives forfuture research. The Appendix contains proofs omitted from the main text.

2. GENERAL STOCHASTIC CONTROL FRAMEWORK

In this section, we study a general (not necessarily finance-related) stochastic controlproblem with different regimes where the choice of control affects the intensities ofregime shifts.

Informal Description. We study the following stochastic control problem: Consider acontrolled continuous-time stochastic system with state process Xu, where u denotes thecontrol process. At each point of time, the system is in one of two regimes i = 0 andi = 1. The dynamics of the controlled process Xu depend on both the control u and the

OPTIMAL CONSUMPTION AND INVESTMENT FOR A LARGE INVESTOR 3

current regime. The regime process I jumps from i to 1 − i with intensity ϑ i,1−i. Hereϑ i,1−i = ϑ i,1−i(u, Xu) may depend on the control u as well as the controlled process Xu,leading to a nontrivial mutual dependence between the dynamics of Xu and those of I .The controller’s task is to find a control u to maximize the expected payoff functional

E

[∫ T

0ψ I(t)(t, u(t), Xu(t)) dt + � I(T)(Xu(T))

].

In the following we provide a rigorous formulation of this stochastic control problem.

Mathematical Framework. To fix our framework, let (�,F,F(·)) be a filtered space withtime horizon [0, T ] that is endowed with a reference probability measure P such that F =F(T) and such that F(·) satisfies the usual conditions of right-continuity and completenesswith respect to P. Further, we suppose that (�,F, P) carries an m-dimensional standard(F(·), P)-Wiener process W , a p-dimensional (F(·), P)-Poisson process N with intensityλ ∈ (0, ∞) p, and two (F(·), P)-Poisson processes N0,1 and N1,0 with intensity 1, whichare jointly independent.

Dynamics of the Controlled Process. Let the state space O ⊆ Rd be open and thecontrol space U ⊆ Re be closed. Given a predictable U-valued control process u thecontrolled process, or state process, Xu is O-valued with dynamics governed by thestochastic differential equation

dXu = α I− (u, Xu−) dt + β I− (u, Xu

−).dW + γ I− (u, Xu−) · dN on [τk−1, τk)

Xu(0) = x0, Xu(τk) = F I(τk−)(u(τk), Xu(τk−)), k ≥ 1.

(2.1)

Here, the {0, 1}-valued regime process I has dynamics

dI = 1{I−=0} dNu0,1 − 1{I−=1} dNu

1,0(2.2)

and {τk}k∈N0 denotes the corresponding sequence of jump times, i.e.,

τk � inf{t ∈ [τk−1, T] : I(t) �= I(τk−1)

}for k ∈ N , τ0 � 0.

The coefficient functions αi, β i, γ i, and Fi, i = 0, 1 are assumed to be continuous andsatisfy standard linear growth conditions.

Turning to the key feature of the model, the driving process Nui ,1−i in (2.2) is a counting

process1 with intensity ϑ i,1−i(u, Xu) that triggers regime shifts from i to 1 − i, i = 0, 1.Here ϑ i ,1−i : U × O → R+

0 is a deterministic, bounded, and measurable function for i =0, 1, so that heuristically

P(I(t + dt) �= I(t) |F(t)) = ϑ I(t),1−I(t)(u(t), Xu(t)) dt.

Thus the control u affects the dynamics of the state process both directly via the coef-ficients in (2.1) and indirectly via the intensity of regime shifts. Similarly, the controlledprocess influences the intensity of its own jumps via the regime process. Given thesedynamics, the controller’s aim is to

(P) maximize E

[∫ T


]over all controls u.

1 Here and in the following, we use the terminology of Bremaud (1981).


However, because of the interdependencies mentioned above, it is not clear a priori thatthe processes in (2.1) and (2.2) exist. We will clarify this issue in the next subsection: Wegive an explicit construction via a suitable change of measure and provide an equivalentreformulation of Problem (P), see (P) below.

REMARK 2.1. Time-dependent coefficients and intensity functions in (2.1) and (2.2)are trivially subsumed by our framework. It suffices to add a component to the stateprocess Xu that records calendar time. The assumption that the intensity functions arebounded may not be necessary in specific applications (see, e.g., footnote 5), but simplifiesthe change-of-measure arguments in the general framework of this section.

Construction of the Controlled Process. As explained above, there is a nontrivial de-pendence between (2.1) and (2.2) via the intensity processes of Nu

i ,1−i . Instead of arguingpathwise, we will define

Nu0,1 � N0,1, Nu

1,0 � N1,0(2.3)

and ensure that in this case (2.1), (2.2) admits a unique solution; then we construct anequivalent probability measure P

u ∼ P such that under Pu , N0,1 has intensity ϑ i,1−i(u, Xu),

while W and N remain a standard Wiener process and a Poisson process with intensity λ,respectively. For a similar change-of-measure construction in a context without stochasticcontrols, see, e.g., Kusuoka (1999) and Becherer and Schweizer (2005).

Thus suppose that (2.3) holds and note that then the regime-shift trigger processesin (2.2) are standard Poisson processes with unit jump intensity. Define the class A0 ofpre-admissible control strategies by

A0 �{u predictable U-valued : (2.1), (2.2) admits a unique O-valued solution

}.

For each u ∈ A0 we now define the probability measure Pu on (�,F) via the Girsanov

transformation

dPu

dP�∏

i=0,1

exp{∫ T

0[1 − ϑ i ,1−i (u(t), Xu(t))] dt

} ∏t∈[0,T]

�Ni ,1−i (t) �=0

ϑ i ,1−i (u(t), Xu(t)).(2.4)

For this construction to be well defined, we require

LEMMA 2.2. For any u ∈ A0 there exists a uniquely determined probability measure Pu

on F = F(T) such that (2.4) is satisfied.

Proof . Define Zu by

Zu �∏

i=0,1

exp{∫ ·

0[1 − ϑ i ,1−i (u(t), Xu(t))] dt

} ∏t∈[0,·]

�Ni ,1−i (t) �=0

ϑ i ,1−i (u(t), Xu(t)).(2.5)

We have to demonstrate that Zu is an (F(·), P)-martingale with E[Zu(T)] = 1. Notethat Zu is the stochastic exponential of

∑i=0,1

∫ ·0[ϑ i ,1−i (u(t), Xu(t)) − 1]dNi ,1−i (t), where

Ni ,1−i is the (F(·), P)-compensated process given by

Ni ,1−i (t) � Ni ,1−i (t) − t for t ∈ [0, T].


Hence, Zu is a local (F(·), P)-martingale, and since

supt∈[0,T]

|Zu(t)| ≤ e2T maxi=0,1

‖ϑ i ,1−i‖N0,1(T)+N1,0(T)∞ ∈ L1(P)(2.6)

it follows that Zu is in fact a uniformly integrable (F(·), P)-martingale. �REMARK 2.3. Since all measures P

u , u ∈ A0, are equivalent, the definition of stochasticintegrals, e.g., in (2.1), does not depend on u.

As a direct consequence of the preceding construction we have

PROPOSITION 2.4. Let u ∈ A0. Then for i = 0, 1 the process Ni,1−i is a counting processwith (F(·), P

u)-intensity ϑ i,1−i(u, Xu). Moreover, W is a standard (F(·), Pu)-Wiener process,

N is an (F(·), Pu)-Poisson process with intensity λ, and

[W, N] = [W, Ni ,1−i ] = [Ni ,1−i , N1−i ,i ] = [Ni ,1−i , N] = 0.

Proof . Recall that dPu

dP= Zu(T). As P

u is equivalent to P, quadratic covariation pro-cesses are the same under P and under P

u ; hence

[W, N] = [W, Ni ,1−i ] = [Ni ,1−i , N1−i ,i ] = [Ni ,1−i , N] = 0.

To show that W is an (F(·), Pu)-Wiener process, it suffices by Levy’s characterization of

Brownian motion to demonstrate that W is a local (F(·), Pu)-martingale, i.e., that ZuW

is a local (F(·), P)-martingale. However, this is an immediate consequence of the productformula

d(Zu W) = Zu−dW + WdZu + dZudW = Zu

−dW + WdZu

due to [Zu, W ] = 0. Next consider the counting processes Ni,1−i and N. A direct compu-tation via the product rule shows that with Ni ,1−i given by

Ni ,1−i � Ni ,1−i −∫ ·

0ϑ i ,1−i (u(t), Xu(t)) dt

we have

d(Zu Ni ,1−i ) = ϑ i ,1−i Zu−dNi ,1−i + Ni ,1−i

− dZu .

Thus Zu Ni ,1−i is a local (F(·), P)-martingale, so Ni ,1−i is a local (F(·), Pu)-martingale,

and Ni,1−i has (F(·), Pu)-intensity ϑ i,1−i(u, Xu). A similar argument shows that N has

constant (F(·), Pu)-intensity λ, so N is an (F(·), P

u)-Poisson process with intensity λ by aclassical result of S. Watanabe (see, e.g., theorem T5 in Bremaud 1981). �

Note from the proof of Lemma 2.2 that the (F(·), Pu)-compensated process Ni ,1−i

is in fact a square-integrable (F(·), Pu)-martingale. Indeed, since [Ni ,1−i ] = Ni ,1−i equa-

tion (2.6) yields

Eu[[Ni ,1−i ](T)

] = E[Zu(T)Ni ,1−i (T)] < ∞.


REMARK 2.5. Suppose Problem (P) is interpreted as a weak stochastic control problemin the sense of Fleming and Soner (2006) and only feedback controls

u(t) = ν(t, Xu(t−), I(t−)) are admissible.

Then, under weak regularity conditions, the distribution of (Xu, I) is uniquely determinedby the dynamics (2.1), (2.2) subject to ϑ i,1−i =ϑ i,1−i(u, Xu), see proposition 3.3 in Bechererand Schweizer (2005). Hence, in this situation it makes sense to regard (P) as a weakstochastic control problem if one wishes to do so.

In contrast to the assumptions in Remark 2.5, we continue to allow predictable controlprocesses. The class A we consider is defined in the next subsection.

Stochastic Control Problem. With the class of admissible strategies given by

A �{

u ∈ A0 : Eu[∫ T

0ψ I(t)(t, u(t), Xu(t))−dt + � I(T)(Xu(T))−

]< ∞

},

where x−�max { − x, 0} denotes the negative part of x, we now reformulate (P) as thestochastic control problem

(P) maximize Eu[∫ T


]over u ∈ A.

Problem (P) is an equivalent version of Problem (P) in the sense that the optimal controldecisions in (P) and (P) coincide. We wish to emphasize the nonstandard feature that theexpectation E

u is taken with respect to the measure Pu which depends explicitly on the

choice of control u.

3. HJB EQUATIONS AND THE VERIFICATION THEOREM

We present a dynamic programming approach to the stochastic control problem intro-duced in Section 2. In particular, we formulate the associated HJB equations and connectthem to Problem (P) via a verification result.

HJB System. Under suitable regularity conditions, the value function correspondingto Problem (P) is expected to satisfy a system of partial differential equations. Thus, wesay that a pair (v0, v1) of functions v0, v1 ∈ C1,2([0, T) × O) ∩ C([0, T] × O) is a solutionto the HJB system, or more briefly the HJB system, if it satisfies the following system ofcoupled partial differential equations:

0 = supu∈U

{ψ i (u, x) + vi

t (t, x) + αi (u, x)vix(t, x)

+ 12β i (u, x).β i (u, x) vi

xx(t, x) +p∑

p=1

λp[vi (t, x + γ i (u, x)) − vi (t, x)

]

+ ϑ i ,1−i (u, x)[v1−i (t, Fi (u, x)) − vi (t, x)

]}for (t, x) ∈ [0, T) × O

(3.1)


subject to the boundary conditions

vi (T, x) = � i (x), x ∈ O(3.2)

for i = 0, 1. Here we write vt � ∂v∂t , vx � ∂v

∂x , and vxx � ∂2v∂x2 .

Verification Theorem. We now provide a general verification result that allows us toidentify a solution of the HJB system as the value function of Problem (P).

THEOREM 3.1 (Verification Theorem). Suppose that (v0, v1) is a solution to the HJBsystem (3.1), (3.2), and assume that for every u ∈ A and p = 1, . . . , p, i = 0, 1 the local(F(·), P

u)-martingales∫ ·

0v I(t)

x (t, Xu(t)).β I(t)(u(t), Xu(t)).dW(t),

∫ ·

0[v I(t−)(t, Xu(t−) + γ I(t−)(u(t), Xu(t−))) − v I(t−)(t, Xu(t−))]dNp(t),

∫ ·

01{I(t−)=i}[v1−i (t, Fi (u(t), Xu(t−))) − vi (t, Xu(t−))]dNi ,1−1(t)

(3.3)

are in fact martingales.2 Then the following statements hold true:

(a) For every u ∈ A and all t0 ∈ [0, T ], x0 ∈ O, i = 0, 1 we have

Eu[∫ T

t0ψ I(t)(t, u(t), Xu(t)) dt + � I(T)(Xu(T)) | I(t0) = i , Xu(t0) = x0

]≤ vi (t0, x0).

(b) If there exists a strategy u� ∈ A such that

u�(t) ∈ arg maxu∈U

Hi (t, X�(t), u(t)) a.s. for each t ∈ [0, T),

where X� � Xu�

and Hi : [0, T] × O × U → R is given by

Hi (t, x, u) � ψ i (u, x) + vit (t, x) + αi (u, x)vi

x(t, x)

+ 12β i (u, x).β i (u, x) vi

xx(t, x) +p∑

p=1

λp[vi (t, x + γ i (u, x)) − vi (t, x)

]

+ ϑ i ,1−i (u, x)[v1−i (t, Fi (u, x)) − vi (t, x)

]then vi is the value function for Problem (P) with initial regime i,

vi (t0, x0) = supu∈A

Eu[∫ T

t0ψ I(t)(t, u(t), Xu(t)) dt + � I(T)(Xu(T)) | I(t0) = i , Xu(t0) = x0

]

for all t0 ∈ [0, T ], x0 ∈ O and i = 0, 1. Moreover, u� is an optimal control for Problem (P).

Proof . Let u ∈ A be an arbitrary admissible control, and suppose without loss ofgenerality that t0 = 0. As above, we denote by N the compensated Poisson process

2 This holds, for instance, if all integrands are bounded. See also Remark 3.2.


associated to N,

N(t) � N(t) − λt, t ∈ [0, T],

and similarly for the (F(·), Pu)-compensated process Ni ,1−i . Ito’s formula yields3

v I(T)(T, Xu(T)) +∫ T

0ψ I(t)(t, u(t), Xu(t)) dt = v I(0)(0, x0)

+∫ T

0HI(t)(t, u(t), Xu(t)) dt +

∫ T

0v I(t)

x (t, Xu(t)).β I(t)(u(t), Xu(t)).dW(t)

+p∑

p=1

∫ T

0[v I(t−)(t, Xu(t−) + γ I(t−)(u(t), Xu(t−))) − v I(t−)(t, Xu(t−))]dNp(t)

+1∑

i=0

∫ T

01{I(t−)=i}[v1−i (t, Fi (u(t), Xu(t−))) − vi (t, Xu(t−))]dNi ,1−1(t).

Comparing with (3.3) it follows that the last three terms are terminal values of (F(·), Pu)-

martingales. Hence taking Pu-expectations on both sides yields

Eu[∫ T


]

= v I(0)(0, x0) +∫ T

0E

u[HI(t)(t, u(t), Xu(t))

]dt.

Since (v0, v1) solves the HJB system, we have Hi(t, u, x) ≤ 0 for all u ∈ U and (t, x) ∈[0, T) × O, which proves part (a) of the claim. On the other hand, in the situation of (b)we have HI(t)(t, u�(t), X�(t)) = 0 a.s. for all t ∈ [0, T) and hence

Eu�

[∫ T

0ψ I(t)(t, u�(t), X�(t)) dt + � I(T)(X�(T))

]= v I(0)(0, x0).

This completes the proof. �Note, in particular, that in the situation of part (b) the dynamic programming principle

holds, even though the relevant measure Pu depends explicitly on the control process u.

REMARK 3.2. To apply Theorem 3.1 it is necessary to check the martingale conditions(3.3) in the setting of a specific control problem. Depending on the problem at hand, thiscan be technically demanding.

4. THE LARGE INVESTOR MODEL

In this section we formulate, within the framework of Section 2, the problem of optimalconsumption and portfolio choice for a “large” investor whose consumption and invest-ment decisions affect asset prices. Based on the methodology of Section 3, we characterizethe value function of the large investor’s optimization problem via an HJB system.

3 For ease of notation, we write I(t) and Xu(t) instead of I(t − ) and Xu(t − ) where this does not make adifference.


Motivation and Informal Description. Consider a financial market that consists of ariskless money market account P0 and n risky assets Pn, n = 1, . . . , n. At each timeinstant, the market is either in the normal state i = 0 or in the alerted state i = 1. Inboth normal and alerted times, asset prices are given by a jump-diffusion model, withcoefficients depending on the current state of the economy.

Moreover, asset prices are also affected by the large investor’s wealth and his con-sumption and portfolio strategy (π , c): The state of the market jumps from i to 1 − iwith intensity ϑ i,1−i(π , c, Xπ,c). Thus the market interprets the large investor’s decisionsas signals revealing possibly relevant information. Note that in this setting the largeinvestor’s actions have an indirect, persistent effect on the financial market, but do nottrigger immediate price jumps. Therefore, our model is particularly suitable when themarket perceives the investor to have superior information, which is reflected in his port-folio allocation. Possible examples include managers of mutual funds, asset managersof insurance companies or sovereign wealth funds, and executives of firms whose stocksare in their own portfolio. Note that typically such investors do disclose their investmentdecisions (voluntarily or compulsorily).

Asset Price Dynamics and Large Investor. As in Section 2, the regime process is de-scribed by the dynamics in (2.2),

dI = 1{I−=0}dN0,1 − 1{I−=1}dN1,0

with associated jump times {τk}k∈N0 . The asset price dynamics are given by

dP0 = P0−r I−dt, P0(0) = p0

0(4.1)

dPn = Pn−

⎡⎣(r I− + η

I−n )dt +

m∑m=1

σI−

n,mdWm +p∑

p=1

γI−

n,pdNp

⎤⎦

Pn(0) = pn0 , Pn(τk) = [1 + �

I(τk−),1−I(τk−)n ]Pn(τk−), k ≥ 1.

(4.2)

Here for i = 0, 1 the numbers ri ≥ 0, the vectors ηi ∈ Rn , �i ,1−i ∈ (−1, ∞)n and thematrices σ i ∈ Rn×m and γ i ∈ (−1, ∞)n× p are given parameters and σ i.(σ i) is positivedefinite.

The large investor chooses a consumption-portfolio strategy (control) u = (π , c) whereπn(t) represents the fraction of wealth held in the risky asset Pn at time t and c(t) dt isthe fraction of wealth consumed between t and t + dt. Then his wealth X = Xπ,c evolvesaccording to the stochastic differential equation

dX = X−[(r I− + π .ηI− − c) dt + π .σ I− .dW + π .γ I− .dN

](4.3)

X(τk) = [1 + π (τk) .�I(τk−),1−I(τk−)]X(τk−)(4.4)

on [τ k−1, τ k). Thus, in the framework of Section 2, the large investor’s consumption-portfolio strategy u = (π , c) is the control and his wealth process Xu = Xπ,c is the stateprocess. Admissible consumption-portfolio policies are assumed to be predictable andbounded; to avoid bankruptcy, admissible investment strategies are further required totake values in the unit simplex � � {π ∈ Rn : πn ≥ 0,

∑nn=1 πn ≤ 1} whenever either


γ i �= 0 or � �= 0. We denote by B the class of admissible strategies. The large investor’sstrategy influences the regime process in the sense that

Ni ,1−i has (F(·), Pπ,c)-intensity ϑ i ,1−i (π, c, Xπ,c).

REMARK 4.1. Since the large investor’s market influence is indirect, it is intuitivelyclear that our model does not give rise to arbitrage opportunities. This intuition willbe vindicated below, where we establish existence of an optimal investment strategy forProblem (CPP).

Optimal Consumption-Portfolio Problem. The large investor’s goal is to maximize util-ity from terminal wealth as well as from intermediate consumption. We suppose that hisrisk preferences are captured by a family of utility functions U ∈ C2([0, T] × R+) suchthat, for each fixed t ∈ [0, T ], U(t, ·) is a utility function that is polynomially bounded at0,4 i.e., for some constants K, κ, δ > 0

|U(t, x)| ≤ K(1 + x−κ ) for all x ∈ (0, δ) and t ∈ [0, T].

Given the above dynamics, the large investor’s optimal consumption and investment prob-lem is to

(CPP) max(π,c)∈B

Eπ,c[∫ T

0U(t, c(t)Xπ,c(t))dt + U(T, Xπ,c(T))

].

Thus the investor tries to maximize expected utility from terminal wealth and inter-mediate consumption, while he is aware of the fact that his consumption and invest-ment decisions will affect asset prices by triggering regime shifts in the market. Thisis reflected in the expectation operator E

π,c which depends explicitly on the investor’sstrategy (π , c).

HJB System and Verification. As (CPP) is clearly a special case of the general controlproblem (P), we can apply the dynamic programming methodology presented in Section 3to solve (CPP); to establish the relevant martingale properties, we impose some additionalgrowth conditions.

LEMMA 4.2. Let (v0, v1) be a solution of the HJB system for Problem (CPP) andassume that there are constants K, k, κ > 0 such that

|xvix(t, x)|, |vi (t, x)| ≤ K(1 + xk + x−κ ) for all (t, x) ∈ [0, T] × R+ and i = 0, 1.

4 This is a weak technical condition that ensures that the expected utility in (4) is finite; see the proof ofLemma 4.2 in the Appendix.


Moreover let (π, c) ∈ B be an arbitrary consumption-portfolio strategy. Then the local(F(·), P

π,c)-martingales in (3.3),∫ ·

0π (t) .σ I(t) Xπ,c(t)v I(t)

x (t, Xπ,c(t)).dW(t),

∫ ·

0

[v I(t−)

(t,[1 + π (t) .γ

I(t−)·,p

]Xπ,c(t−)

)− v I(t−)(t, Xπ,c(t−))

]dNp(t),

∫ ·

01{I(t−)=i}

[v1−i (t, [1 + π (t) .�i ,1−i ]Xπ,c(t)) − vi (t, Xπ,c(t−))

]dNi ,1−i (t)

are (F(·), Pπ,c)-martingales for p = 1, . . . , p and i = 0, 1.

Proof . See the Appendix. �

5. OPTIMAL PORTFOLIOS FOR CRRA INVESTORS

This section continues the analysis of Section 4 and provides explicit solutions to thelarge investor’s consumption-portfolio problem with a constant relative risk aversion(CRRA) utility function for two concrete specifications of the intensity functions ϑ i,1−i

in Sections 5.1 and 5.2. Moreover, we provide a numerical approach to the consumption-portfolio problem for general intensity functions in Section 5.3.

CRRA Utility. Throughout Section 5 we focus on risk preferences with CRRA, i.e.,we assume that the large investor’s utility function takes the CRRA form

U(t, x) = εe−δt 11 − R

(x1−R − 1), U(T, x) = e−δT 11 − R

(x1−R − 1)(5.1)

for x ∈ R+, t ∈ [0, T). Here R > 0, R �= 1 denotes the investor’s relative risk aversion, δ > 0is his rate of time preference, and ε ≥ 0 is a weight factor for intermediate consumption.Note that this parametrization allows us to study the general problem of maximizingexpected utility from both terminal wealth and consumption (ε = 1) as well as thepure terminal wealth problem (ε = 0). Moreover observe that (5.1) clearly satisfies thecondition of Lemma 4.2, so the Verification Theorem 3.1 is applicable.

Model Specification. To simplify the analysis, we assume in Sections 5.1 and 5.2 thatthe stock price processes have no jumps. Moreover, we suppose that regime-shift intensitiesdepend only on the large investor’s portfolio proportions and consumption rate. In thissetting, the HJB system has the following form:

0 = sup(π,c)∈Rn×R+

0

{εe−δt 1

1 − R(x1−R − 1) + vi

t (t, x) + (r i + π .ηi − c)xvix(t, x)

+ 12π .σ i .(σ i ) .πx2vi

xx(t, x) + ϑ i ,1−i (π, c)[v1−i (t, x) − vi (t, x)]}

(5.2)

for (t, x) ∈ [0, T) × R+ and i = 0, 1, subject to the boundary conditions

vi (T, x) = e−δT 11 − R

(x1−R − 1), x ∈ R+ for i = 0, 1.(5.3)


We introduce the utility growth potentials of the market in states 0 and 1 by

� i � r i + 12

1R

(ηi ) .(σ i .(σ i ) )−1.ηi , i = 0, 1.

Without loss of generality, the labeling is chosen in such a way that a Merton investorwithout market impact would prefer state 0 to state 1, i.e., we assume

�0 > �1.

Reduced HJB System without Jumps. To solve the HJB system (5.2) explicitly, wesuggest the ansatz

v0(t, x) = 11 − R

f (t)((xeg(t))1−R − 1

),

v1(t, x) = 11 − R

f (t)((xeg(t)−h(t))1−R − 1

)(5.4)

for (t, x) ∈ [0, T] × R+ with C1-functions f , g, and h on [0, T ]. The inclusion of thefunction f is motivated by the presence of intertemporal consumption; the presence of hin the exponent of v1(t, x) reflects the difference in investment conditions.

Inserting (5.4) into (5.2), (5.3) yields


0

{1

1 − Rεe−δt(c1−R − x−(1−R))e−(1−R)(g(t)−1{i=1}h(t))

+ 11 − R

f ′(t)(1 − x−(1−R)e−(1−R)(g(t)−1{i=1}h(t)))

+ f (t)[

g′(t) − 1{i=1}h′(t) + r i + π .ηi − 12

Rπ .σ i .(σ i ) .π − c

+ ϑ i ,1−i (π, c)1

1 − R(e(−1)1−i (1−R)h(t) − 1)

]}

(5.5)

for (t, x) ∈ [0, T) × R+ and i = 0, 1, subject to the boundary conditions

f (T) = e−δT, g(T) = 0, h(T) = 0.(5.6)

Collecting the terms multiplying x−(1−R) we obtain the ordinary differential equation(hereafter ODE)

f ′(t) = −εe−δt, f (T) = e−δT

with the unique solution

f (t) = 1δ

e−δt(ε − (ε − δ)e−δ(T−t)).(5.7)


The remaining terms yield the reduced HJB system


0

{g′(t) − 1{i=1}h′(t) + r i + π .ηi − 1

2Rπ .σ i .(σ i ) .π

+ εδ

ε − (ε − δ)e−δ(T−t)

11 − R

(e−(1−R)(g(t)−1{i=1}h(t))c1−R − 1) − c

+ ϑ i ,1−i (π, c)1

1 − R(e(−1)1−i (1−R)h(t) − 1)

}

(5.8)

for t ∈ [0, T) and i = 0, 1, subject to the boundary conditions

g(T) = 0, h(T) = 0.

REMARK 5.1. Assuming h(t) = 0 for some t ∈ [0, T) and subtracting the two reducedHJB equations from each other leads to h′(t) < 0 due to �0 > �1; this, however, isinconsistent with the final condition h(T) = 0. Thus, we have

h(t) ≥ 0 for all t ∈ [0, T].

After the above preparatory considerations, we proceed to investigate two interestingspecifications of the regime-switching intensities in the following sections.

5.1. Affine Intensity Functions

Floored Affine Intensity. We first study floored affine intensity functions5

ϑ i ,1−i (π, c) = max{Ai + π .Biπ + Bi

cc, Ci }

with constants Ai ∈ R, Biπ ∈ Rn , B0

c ∈ R, B1c ≤ 0 and Ci ≥ 0 for i = 0, 1. The condition

B1c ≤ 0 precludes a possibly infinite optimal consumption rate in state 1 (see below).

A floor is natural in this setting as intensities have to be nonnegative. To interpret therole of the A and B parameters note that, for instance, positive entries in B1

π result inan increase of the intensity to switch to better market conditions if the large investorincreases his portfolio positions even in the bad market situation of state 1. Thus, themarket believes that the large investor anticipates a better performance of the stock in thefuture and hence also appreciates the stock investment. This mechanism can be seen asthe equivalent of switching back to state 0. Note further that the model of BR is obtainedas the special case Bi

π = 0, Bic = 0 for i = 0, 1.

5 Since admissible consumption-portfolio strategies (π , c) are assumed to be bounded, the fact that ϑ i,1−i

is not bounded does not cause any problems.


The reduced HJB system (5.8) now takes the form


0

{g′(t) − 1{i=1}h′(t) + r i + π .ηi − 1

2Rπ .σ i .(σ i ) .π

+ εδ

ε − (ε − δ)e−δ(T−t)

11 − R

(e−(1−R)(g(t)−1{i=1}h(t))c1−R − 1) − c

+ max{Ai + π .Biπ + Bi

cc, Ci } 11 − R

(e(−1)1−i (1−R)h(t) − 1)}

(5.9)

for t ∈ [0, T) and i = 0, 1, with the boundary conditions g(T) = 0, h(T) = 0.

5.1.1. No Influence of Consumption

Reduced HJB System. As the solution of the system (5.9) involves some heavy nota-tion, we first specialize to the case where only the investor’s portfolio decisions influenceintensities of regime shifts. Thus we assume B0

c = B1c = 0 so that

ϑ i ,1−i (π, c) = max{Ai + π .Biπ , Ci }.

To solve the reduced HJB system, we (formally) determine the maximizing portfolio andconsumption process. For this purpose, we introduce the functions Hi

port : R+0 × Rn →

R, i = 0, 1 and Hcons : [0, T] × R × R+0 → R given by

Hiport(h, π ) � r i + π .ηi − 1

2Rπ .σ i .(σ i ) .π

+ max{Ai + π .Biπ , Ci } 1

1 − R(e(−1)1−i (1−R)h − 1),

Hcons(t, g, c) � εδ

ε − (ε − δ)e−δ(T−t)

11 − R

(e−(1−R)gc1−R − 1) − c

with the convention 01−R�∞ if R > 1. Note that the function Hcons is independent of tand g if ε = 0. Now the reduced HJB system can be written briefly as


0

{g′(t) − 1{i=1}h′(t) + Hi

port(h(t), π ) + Hcons(t, g(t) − 1{i=1}h(t), c)}

for t ∈ [0, T), i = 0, 1 with the boundary conditions g(T) = 0, h(T) = 0.

Candidate Optimal Strategies. The first-order conditions and concavity of Hcons

(t, g, ·) yield the maximizers in the reduced HJB system.

LEMMA 5.2 (Maximizer of Hcons(t, g, ·)). For every (t, g) ∈ [0, T] × R the maximizer

ci ,�(t, g) � arg maxc∈R+

0

Hcons(t, g, c), i = 0, 1


is given by the Merton-type consumption rate6

ci ,�(t, g) = cM(t, g) �( εδ

ε − (ε − δ)e−δ(T−t)

) 1R e− 1−R

R g.

LEMMA 5.3 (Maximizer of Hiport(h, ·)). For every h ∈ R+

0 the maximizer

π i ,�(h) � arg maxπ∈Rn

Hiport(h, π ), i = 0, 1

is given by

π0,�(h) = 1R

(σ 0 · (σ 0) )−1 ·(

η0 + B0π

11 − R

(e−(1−R)(h∧h0,crit) − 1))

,

π1,�(h) = 1R

(σ 1 · (σ 1) )−1 ·(

η1 + 1{h≥h1,crit} B1π

11 − R

(e(1−R)h − 1))

where

h0,crit � − 11 − R

ln

⎛⎜⎜⎜⎝−(1 − R)

(A0 + 1

R(η0) · (σ 0 · (σ 0) )−1 · B0

π − C0)+

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

+ 1

⎞⎟⎟⎟⎠ ,

h1,crit � 11 − R

ln

⎛⎜⎜⎜⎝(1 − R)2

(A1 + 1

R(η1) · (σ 1 · (σ 1) )−1 · B1

π − C1)−

1R

(B1π ) · (σ 1 · (σ 1) )−1 · B1

π

+ 1

⎞⎟⎟⎟⎠ .

Proof . Due to the max-part in Hiport(h, ·) the maximization over π in the reduced HJB

system amounts to finding the maximum of two strictly concave quadratic functionalsFi, Gi over the two half-spaces separated by the hyperplane

Ai + π · Biπ = Ci .

The maximum must be either one of the unique unconstrained maxima of Fi and Gi

(provided they lie in the relevant half-space) or the unique maximum attained at theintersection of the half-spaces. To check this is locally simple but lengthy. We thereforeskip the verification of the explicit forms for the maximizing values of the portfolioprocesses. For a detailed account of the necessary computations, the reader is referred toBusch (2011). �

Note that the maximizing portfolios are in fact only functions of time as the variable hwill be replaced by the (yet undetermined) function h(t). Note further that the portfolioscan be decomposed into the classical Merton portfolios and an additional component

6 However, g = g(t) is not the same function as in the Merton or BR models.


that is due to the investor’s influence on the market, via

π0,�(h) = π0,M + 1R

(σ 0.(σ 0) )−1.B0π

11 − R

(e−(1−R)(h∧h0,crit) − 1),

π1,�(h) = π1,M + 1{h≥h1,crit}1R

(σ 1.(σ 1) )−1.B1π

11 − R

(e(1−R)h − 1).

Solution of the Consumption-Portfolio Problem. Inserting the above maximizers(π i,�(h(t)), ci,�(t, g(t) − 1{i =1}h(t))) into the reduced HJB system (5.9) yields a systemof coupled ODEs for the functions g and h:

g′(t) = −�0 + εδ

ε − (ε − δ)e−δ(T−t)

(1 − R

1 − R

((

εδ

ε − (ε − δ)e−δ(T−t))

1−RR e− 1−R

R g(t) − 1))

− C0 11 − R

(e−(1−R)h(t) − 1)(A0 + (π0,M) · B0π − C0)

11 − R

(e−(1−R)(h(t)∧h0,crit) − 1)

− 12

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

1(1 − R)2

(e−(1−R)(h(t)∧h0,crit) − 1)2,

(5.10)

h′(t) = −(�0 − �1) +(

εδ

ε − (ε − δ)e−δ(T−t)

) 1R

e− 1−RR g(t) R

1 − R(e

1−RR h(t) − 1)

− C0 11 − R

(e−(1−R)h(t) − 1) + C1 11 − R

(e(1−R)h(t) − 1)

− (A0 + (π0,M) · B0π − C0)

11 − R

(e−(1−R)(h(t)∧h0,crit) − 1)

− 12

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

1(1 − R)2

(e−(1−R)(h(t)∧h0,crit) − 1)2

+ (A1 + (π1,M) · B1π − C1)

11 − R

(e(1−R)(h(t)∨h1,crit) − 1)

+ 12

1R

(B1π ) · (σ 1 · (σ 1) )−1 · B1

π

1(1 − R)2

(e(1−R)(h(t)∨h1,crit) − 1)2

(5.11)

subject to the boundary conditions

g(T) = 0, h(T) = 0.(5.12)

LEMMA 5.4. The ODE-system (5.10), (5.11) subject to the boundary conditions (5.12)admits a unique nonexplosive solution.

Proof . See the Appendix.


We are now able to state the main result of this section, which yields a complete solutionto the large investor’s investment problem in the framework of the present section.

THEOREM 5.5 (Solution of the Optimal Consumption and Investment Problem) Sup-pose that B0

c = B1c = 0 and let g and h be given by (5.10), (5.11) subject to the boundary

conditions (5.12). Then the strategy

(π i ,�(t), ci ,�(t)) � (π i ,�(h(t)), ci ,�(t, g(t) − 1{i=1}h(t))), i = 0, 1

as given in Lemmas 5.2 and 5.3 is optimal for the investment problem (CPP).

Proof . As (π i,�(t), ci,�(t)) is a maximizer in the HJB system (5.9) for each t ∈ [0, T ],optimality of the strategy (π i,�, ci,�) follows from the Verification Theorem 3.1. �

5.1.2. Including Market Impact via Consumption

General Approach. We now provide the optimal portfolio and consumption processin the general setting where the investor’s consumption also influences the intensity for aregime switch. The method to construct the optimal pairs (π i,�, ci,�) in each market stateis similar to the preceding analysis:

• Determine the formal maximizers (π i,�, ci,�) of the HJB system (5.9).• Use them to reduce the HJB system to a system of ODEs.• Prove existence of a unique solution of the ODEs and use the Verification Theorem

3.1 to establish optimality of (π i,�, ci,�).

For simplicity of presentation, we focus on the case ε = 0, i.e., the investor obtainsutility from terminal wealth only. In this situation we get explicit formulae of moderatelength, while the model exhibits interesting effects caused by the influence of consumptionon the switching intensity.

Proceeding similarly as above, we first determine the candidate for the optimalconsumption-portfolio strategy; the proof is similar to that of Lemma 5.3.

LEMMA 5.6 (Maximizer of Hi(h, ·, ·)). For every h ∈ R+0 let

(π i ,�(h), ci ,�(h)) � arg max(π,c)∈Rn×R+

0

Hi (h, π, c), i = 0, 1

with

Hi (h, π, c) � −c + r i + π .ηi − 12

Rπ .σ i .(σ i ) .π

+ max{Ai + π .Biπ + Bi

cc, Ci } 11 − R

(e(−1)1−i (1−R)h − 1).


Then for ε = 0 the maximizers (π i,�, ci,�) are given by

π0,�(h) = π0,M + 1R

(σ 0 · (σ 0) )−1 · B0π

11 − R

(e−(1−R)(h∧h0,crit) − 1),

π1,�(h) = π1,M + 1{h≥h1,crit}1R

(σ 1 · (σ 1) )−1 · B1π

11 − R

(e(1−R)h − 1),

c0,�(h) = 1{h≥h0,crit}

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

(B0c )2

⎛⎜⎝ (A0 + (π0,M) · B0

π − C0)+

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

B0c + 1

⎞⎟⎠

−

,

c1,�(h) = 0

where π i,M denotes the Merton portfolio in market regime i and

h0,crit � − 11 − R

ln

⎛⎜⎝(1 − R)

1B0

c

⎛⎜⎝1 − (

(A0 + (π0,M) · B0π − C0)+

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

B0c + 1)+

⎞⎟⎠+ 1

⎞⎟⎠ ,

h1,crit � 11 − R

ln

⎛⎜⎝(1 − R)2

(A1 + (π1,M) · B1π − C1)−

1R

(B1π ) · (σ 1 · (σ 1) )−1 · B1

π

+ 1

⎞⎟⎠ .

REMARK 5.7. Let us comment on the form of (π i,�, ci,�). While the structure of theoptimal portfolio as a sum of the Merton portfolio and a hedging term has been discussedabove, the form of the optimal consumption rate deserves a closer look: Although thelarge investor does not obtain any (direct) utility from consumption, it can be beneficialfor him to consume if the market is in state 0 (the good one!). This occurs only for B0

c < 0,i.e., when consumption reduces the intensity to switch to the bad market regime. Theexact necessary and sufficient conditions for positive consumption without utility are

• A0 + (π0,M) .B0π > C0, i.e., the Merton strategy does not yield the minimal intensity

in the good market;

• B0c < − 1

R (B0π ) .(σ 0.(σ 0) )−1.B0

π

(A0+(π0,M) .B0π −C0)+ , i.e., consumption improves the intensity;

• h(t) ≥ − 11−R ln((1 − R) 1

B0c

+ 1), i.e., the difference between the two market regimesas perceived by the investor is sufficiently large.

This behavior leaves a lot of space for interpretation: As the large investor does notgain utility from consumption, we may regard the positive consumption as a paymentintended to deliberately manipulate the market. Note also that a positive value of B1

ccould lead to limc→∞Hi(h, π , c) = ∞ for sufficiently large values of h. This could againbe interpreted as a kind of manipulating payment; this situation, however, is excluded byour restriction B1

c ≥ 0 since we do not wish to consider infinite consumption rates.


Solution of the Consumption-Portfolio Problem. Plugging the maximizers determinedin Lemma 5.6 into the HJB system (5.9) produces a system of coupled ODEs:

g′(t) = −�0 − C0 11 − R

(e−(1−R)h(t) − 1)

− (A0 + (π0,M) · B0π − C0)

11 − R

(e−(1−R)(h(t)∧h0,crit) − 1)

− 12

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

1(1 − R)2

(e−(1−R)(h(t)∧h0,crit) − 1)2,

(5.13)

h′(t) = −(�0 − �1) − C0 11 − R

(e−(1−R)h(t) − 1) + C1 11 − R

(e(1−R)h(t) − 1)

− (A0 + (π0,M) · B0π − C0)

11 − R

(e−(1−R)(h(t)∧h0,crit) − 1)

− 12

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

1(1 − R)2

(e−(1−R)(h(t)∧h0,crit) − 1)2

+ (A1 + (π1,M) · B1π − C1)

11 − R

(e(1−R)(h(t)∨h1,crit) − 1)

+ 12

1R

(B1π ) · (σ 1 · (σ 1) )−1 · B1

π

1(1 − R)2

(e(1−R)(h(t)∨h1,crit) − 1)2

(5.14)

subject to the boundary conditions g(T) = 0, h(T) = 0 where hi,crit, i = 0, 1 are given inLemma 5.6.

As this system is a special case of the system (5.10), (5.11), Lemma 5.4 ensures existenceand uniqueness of the desired solution. Thus we obtain the following complete solutionto the large investor’s consumption-portfolio problem:

THEOREM 5.8 (Solution of the Optimal Consumption and Investment Problem). Let gand h be the unique solutions of the ODEs (5.13), (5.14) subject to the boundary conditionsg(T) = h(T) = 0. Then the strategy

(π i ,�(t), ci ,�(t)) � (π i ,�(h(t)), ci ,�(h(t))), i = 0, 1

from Lemma 5.6 is optimal for the consumption-portfolio problem (CPP) with ε = 0.

REMARK 5.9. In the general case of ε > 0, the structure of the statements and proofsare similar to those above, but a very lengthy notation is required. For a detailed analysis,the interested reader is referred to Busch (2011).

Comparison with the Merton Strategy. Figure 5.1 illustrates a particular setting whereit is optimal for the large investor to consume even when ε = 0. Note that state 1 isabsorbing and the optimal portfolio process in state 0 tends toward the Merton strategyfrom above as the time to maturity decreases. The interpretation is that—given thecurrent parameter setting—the large investor has to convince the market that the stockwill perform well, thus inducing the market to stay in the good state 0. He therefore


FIGURE 5.1. Optimal strategies π0,�, c0,� versus Merton strategies π0,M , c0,M

(r0 = 0.035, r1 = 0.02, η0 = 0.08, η1 = 0.01, σ 0 = 0.3, σ 1 = 0.55, δ = 0.04, ε = 0, R = 1.25,T = 1, A0 = 23, B0

π = −3, B0c = −485, C0 = 0, A1 = 0, B1

π = 0, B1c = 0, C1 = 0).

chooses a seemingly too high portfolio position while he also “pays” consumption ashis second means to manipulate the market. As soon as the comparative advantage ofbeing in state 0 decreases below a certain threshold (due to the time to the investmenthorizon being not long enough), the large investor stops “paying” consumption, whilehe gradually decreases the portfolio position. In state 1 the optimal portfolio policy isgiven by the Merton strategy.

Comparison with BR. The financial market model of BR is subsumed by our setting.We compare the BR model to ours by studying a large investor that is unaware of hismarket influence and therefore follows the optimal BR strategies in each regime. Thus,he implements the classical Merton portfolios in each of the two market regimes andchooses not to consume. As this strategy is not optimal in the presence of market impact,we assess its underperformance by evaluating the time-t (initial) wealth xPCD(t) that thelarge investor would need to obtain the same expected utility as the BR-investor withinitial capital x. Since the value function in the BR model has the same form as in thelarge investor model, we can determine xPCD(t) from

viBR(t, x) = vi

PCD(t, xiPCD(t)).

Hence the relative wealth loss caused by the large investor’s misconception is

wiBR,PCD(t) � x − xi

PCD(t)x

= 1 − egBR(t)−1{i=1}hBR(t)−(gPCD(t)−1{i=1}hPCD(t)).

Figure 5.2 shows that the price of misconception is not negligible, and increasesapproximately linearly as a function of the time horizon.

5.2. Piecewise-Constant Intensity Functions

Piecewise-Constant Intensity. We now study piecewise-constant intensity functions

ϑ i ,1−i (π, c) = Ci11{Ai +π .Bi

π +Bicc≤Ci } + Ci

21{Ai +π .Biπ +Bi

cc>Ci }


FIGURE 5.2. Relative wealth loss w0BR,PCD in state 0 as a function of T − t

(r0 = 0.03, r1 = 0.02, η0 = 0.1, η1 = 0.01, σ 0 = 0.3, σ 1 = 0.6, δ = 0.035, ε = 0,R = 1.5, T = 2, A0 = 50, B0

π = −45, B0c = −50, C0 = 0, A1 = 0, B1

π = 100, B1c = 0,

C1 = 0).

with constants Ai ∈ R, Biπ ∈ Rn , Bi

c ∈ R, Ci ∈ R and Cij ≥ 0 for i = 0, 1 and j = 1, 2

where C02 > C0

1 and C11 > C1

2 . The interpretation of the A and B parameters is similarto that in the affine setting: For instance, positive entries of Bi

π yield adverse switchingintensities.

For piecewise-constant intensity functions, the reduced HJB system (5.8) reads


0

{g′(t) − 1{i=1}h′(t) + r i + π · ηi − 1

2Rπ · σ i · (σ i ) · π

+ εδ

ε − (ε − δ)e−δ(T−t)

11 − R

(e−(1−R)(g(t)−1{i=1}h(t))c1−R − 1) − c

+ (Ci2 + (Ci

1 − Ci2)1{Ai +π ·Bi

π +Bicc≤Ci })

11 − R

(e(−1)1−i (1−R)h(t) − 1)}

(5.15)

for t ∈ [0, T) and i = 0, 1, subject to the boundary conditions g(T) = 0, h(T) = 0.

Reduced HJB System. In the general setting where both the large investor’s portfoliostrategy and consumption impact on the regime-shift intensities, we focus on the case ofmaximizing utility from terminal wealth and thus suppose that ε = 0.

The three-step procedure presented in Section 5.1 applies here, too. Thus, we firstcompute the candidate for the optimal consumption-portfolio decision.

LEMMA 5.10 (Maximizer of Hi(h, ·, ·)). For every h ∈ R+0 let

(π i ,�(h), ci ,�(h)) � arg max(π,c)∈Rn×R+

0

Hi (h, π, c), i = 0, 1


with

Hi (h, π, c) � −c + r i + π · ηi − 12

Rπ · σ i · (σ i ) · π

+ (Ci

2 + (Ci1 − Ci

2

)1{Ai +π ·Bi

π +Bicc≤Ci }

) 11 − R

(e(−1)1−i (1−R)h − 1).

Then the maximizers (π i,�, ci,�) are given by

π i ,�(h) = π i ,M + 1{h≥hi ,crit}1R

(σ i · (σ i ) )−1 · Biπ

1Bi

c

(1 − (Di )+

),

ci ,�(h) = 1{h≥hi ,crit}1R(Bi

π ) · (σ i · (σ i ) )−1 · Biπ

(Bic)2

(Di )−

where

hi ,crit � (−1)1−i 11 − R

× ln

⎛⎜⎝− 1 − R

Ci2 − Ci

1

12

1R

(Biπ ) (σ i · (σ i ) )−1 · Bi

π

(Bic)2

[(1 − (Di )+

)2 + 2(Di )−]+ 1

⎞⎟⎠ ,

Di � (Ai + (π i ,M) · Biπ − Ci )+

1R

(Biπ ) · (σ i · (σ i ) )−1 · Bi

π

Bic + 1.

The proof is similar to that of Lemma 5.3 and is therefore omitted.

REMARK 5.11. As for affine intensity functions it can be advantageous for the largeinvestor to consume although he does not obtain direct utility from consumption. Asbefore, for this it is necessary that Bi

c < 0, i.e., consumption influences the switchingintensities in a favorable manner.

The HJB system (5.8) now becomes a system of ODEs:

g′(t) = −�0 − C02

11 − R

(e−(1−R)h(t) − 1)

−

⎡⎢⎣(C0

1 − C02

) 11 − R

(e−(1−R)h(t) − 1)

− 12

1R

(B0π ) (σ 0.(σ 0) )−1.B0

π

(B0c )2

[(1 − (D0)+

)2 + 2(D0)−]⎤⎥⎦

+

,

(5.16)


h′(t) = −(�0 − �1) − C02

11 − R

(e−(1−R)h(t) − 1) + C12

11 − R

(e(1−R)h(t) − 1)

−

⎡⎢⎣(C0

1 − C02

) 11 − R

(e−(1−R)h(t) − 1) − 12

1R

(B0π ) (σ 0.(σ 0) )−1.B0

π

(B0c )2

× [(1 − (D0)+)2 + 2(D0)−

]⎤⎥⎦

+

+

⎡⎢⎣(C1

1 − C12 )

11 − R

(e(1−R)h(t) − 1) − 12

1R

(B1π ) (σ 1.(σ 1) )−1.B1

π

(B1c )2

× [(1 − (D1)+)2 + 2(D1)−

]⎤⎥⎦

+

(5.17)

subject to the boundary conditions g(T) = 0, h(T) = 0 with Di, i = 0, 1 as given inLemma 5.10.

LEMMA 5.12. The ODE-system (5.16), (5.17) subject to the boundary conditionsg(T) = h(T) = 0 admits a unique nonexplosive solution.

The proof is similar to that of Lemma 5.4 and is thus omitted. We obtain the following.

THEOREM 5.13 (Solution of the Optimal Consumption and Investment Problem). Letg and h be given by (5.16), (5.17) subject to the boundary conditions g(T) = h(T) = 0.Then the strategy

(π i ,�(t), ci ,�(t)) � (π i ,�(h(t)), ci ,�(h(t))), i = 0, 1

from Lemma 5.10 is optimal for the consumption-portfolio problem (CPP) with ε = 0.

REMARK 5.14. Again, in the general case ε > 0, the structure of the statements andproofs are entirely analogous to their counterparts above. However, some lengthy nota-tion becomes necessary. For details, we refer to Busch (2011).

A Multi-Asset Application. To exhibit new interesting effects, we now illustrate theresult of Theorem 5.13 in a multi-asset setting; the comparison to the Merton and theanalysis of BR would be similar to that carried out in Section 5.1. Figure 5.3 contrasts theoptimal portfolio strategies for our model with the Merton policies in a financial marketmodel with two risky assets. The large investor’s position in the first risky asset impacts onthe regime-shift intensities, while his position in the second asset has no direct influence;the infinitesimal returns of the risky assets have correlation �. This induces a hedgingdemand that results in a significant deviation of the investor’s optimal investment policyfrom the corresponding Merton allocation. The effect increases with higher correlation(for the exact numbers see Figure 5.3).


FIGURE 5.3. Optimal strategy π0,�1 , π

0,�2 versus Merton strategy π

0,M1 , π

0,M2 with

correlation � = 0.25 (r0 = r1 = 0.03, η0 = (0.1, 0.07) , η1 = (0.02, 0.05) ,σ 0

1 = 0.3, σ 02 = 0.2, σ 1

1 = 0.6, σ 12 = 0.4, δ = 0.035, ε = 0, R = 3, T = 1, A0 = 16.5,

B0π = (−5, 0) , B0

c = 0, C0 = 14, C01 = 5, C0

2 = 10, A1 = 5.1, B1π = (−4, 0) , B1

c = 0,C1 = 5, C1

1 = 5, C12 = 1.25).

5.3. General Intensity Functions and Price Jumps

Finally, we briefly discuss the extension to more general price dynamics and intensityfunctions.

Solution of the HJB System. In the presence of jumps and with general regime-shiftintensity functions ϑ i = ϑ i(π , c), explicit solutions as in Sections 5.1 and 5.2 are in generalnot available. With the jump-diffusion asset price dynamics specified in (4.1), (4.2) theHJB system for Problem (CPP) reads

0 = sup(π,c)∈�×R+

0

{εe−δt 1

1 − R(x1−R − 1) + vi

t (t, x) + (r i + π · ηi − c)xvix(t, x)

+ 12π · σ i · (σ i ) · πx2vi

xx(t, x) +p∑

p=1

λp[vi (t, [1 + π · γ i

·,p]x) − vi (t, x)]

+ ϑ i ,1−i (π, c)[v1−i (t, [1 + π · �i ,1−i ]x) − vi (t, x)

]}

(5.18)

for (t, x) ∈ [0, T) × R+ with boundary condition vi (T, x) = e−δT 11−R(x1−R − 1), x ∈ R+

for i = 0, 1 and admissible investment strategies are required to take values in the unitsimplex � � {π ∈ Rn : πn ≥ 0,

∑nn=1 πn ≤ 1}. The separation ansatz

vi (t, x) = 11 − R

f (t)((xegi (t))1−R − 1

)for (t, x) ∈ [0, T] × R+


with a C1-function gi for i = 0, 1 again leads to a reduced HJB system that is equivalentto the following coupled ODE system:

(gi )′(t) = − sup(π,c)∈�×R+

0

⎧⎨⎩r i + π · ηi − 1

2Rπ · σ i · (σ i ) · π

+ εδ

ε − (ε − δ)e−δ(T−t)

11 − R

(e−(1−R)gi (t)c1−R − 1) − c

+p∑

p=1

λp1

1 − R

([1 + π · γ i

.,p]1−R − 1)

+ ϑ i ,1−i (π, c)1

1 − R

([1 + π · �i ,1−i ]1−Re−(1−R)[gi (t)−g1−i (t)] − 1

)⎫⎬⎭

(5.19)

for t ∈ [0, T) with boundary conditions gi(T) = 0 for i = 0, 1. Being a system of ODEs,(5.19) can be readily solved with numerical methods via, for instance, a simple Eulerapproximation. Note that each Euler step requires an implicit numerical maximization;the corresponding maximizers π i,� and ci,� then yield the optimal investment and con-sumption strategies. Also note that the maximization step becomes superfluous wheneverit is possible to obtain explicit solutions to the associated first-order conditions.

Illustrations. As a particular example featuring both jumps and nonlinearities, weconsider a model with a single risky asset and regime-shift intensity functions

ϑ0,1(π, c) � α0 + β0(1 − e−ω0(c−c0)2)(π0 − π ),

ϑ1,0(π, c) � α1 + β1e−ω1(c−c1)2√π.

Moreover, we suppose that with p = 1 the jump parameters are given by

λ = 3, γ 0 = −0.01, γ 1 = −0.015, �0,1 = −0.1, �1,0 = 0.

In particular, regime shifts from market 0 to 1 are accompanied by a 10% decrease in thestock price, thus exacerbating the adverse effect of switching. We further suppose thatε = 0, so the large investor obtains utility from terminal wealth only.

Figure 5.4 illustrates the optimal investment and portfolio strategies in regime 0. Thelarge investor’s optimal portfolio strategy is significantly lower than the correspondingMerton strategy that ignores regime changes and the associated losses. Although he doesnot obtain utility from intermediate consumption, he spends a significant fraction of hiswealth to achieve more favorable regime-shift intensities.

In Figure 5.5, we depict the large investor’s optimal policies in state 1. Note that inthis regime the Merton strategies coincide with the optimal BR strategies, and that theydiffer substantially from the large investor’s optimal policies.

6. CONCLUSION AND OUTLOOK

Conclusion. In this paper, we have introduced a new class of intensity-based stochas-tic control problems: In a regime-switching framework, in addition to controlling the



(r0 = 0.03, r1 = 0.025, η0 = 0.5, η1 = 0.08, σ 0 = 0.53, σ 1 = 0.5, p = 1, λ = 3,γ 0 = −0.01, γ 1 = −0.015, �0,1 = −0.1, �1,0 = 0, δ = 0.035, ε = 0, R = 1.5, T = 1, α0 =2.1, β0 = 0.9, ω0 = 2000, c0 = 0.05, π0 = 1.5, α1 = 0, β1 = 10, ω1 = 4000, c1 = 0.02).


(parameters as in Figure 5.4).

local parameters of a jump-diffusion state process, the controller can also influence theintensities of regime shifts. We have established existence of the controlled processes andprovided a general verification result for the HJB system that characterizes the valuefunction.

As an application of this framework, we have presented a novel financial marketmodel with a large investor that differs from related models in the literature. In varioussettings, we have been able to solve the large investor’s consumption-portfolio problemexplicitly. The optimal policies we obtained exhibit several remarkable properties suchas, for instance,

• positive consumption in a setting with utility from terminal wealth only,• seemingly too high (or too low) portfolio positions compared to the corresponding

Merton strategies, and• significant deviations from the optimal strategies of a large investor that is not aware

of his influence on the regime-shift intensities.

Thus, our large investor model can also produce important economic insights.


Outlook. Our new framework opens many possibilities for different directions of fu-ture research. First, the large investor setting can be generalized and refined in variousways. Relevant extensions include the following:

• The extension of our model to more than two regimes is straightforward from aconceptual point of view. Explicit solutions of concrete problems will likely becomenotationally involved.

• It would be interesting to supplement our theoretical analysis with empirical evidenceconcerning real-world regime shift intensities. A first step of such an analysis couldbe based on the building blocks provided in Section 5 of this paper.

• As a generalization of our full-information setting, it would be promising to combineour approach with filtering methods or change point analysis when the state processis not observable.

We wish to emphasize, however, that the application of our stochastic control frame-work is by no means limited to applications in finance. Whenever a technical, social,economic or other system can be in different states that are chosen or influenced by thedecision-maker’s actions, our setting provides a suitable modeling framework. Examplesinclude:

• In technical systems or engineering tasks (such as production lines, cars, computersystems, ...), running a system at extremely high speed significantly increases thedanger of a damage or failure and, thus, a change to a different “defective” state.The problem of balancing yield and reliability can be naturally formulated in ourcontrol framework.

• In political or social systems, forcing a population or workforce to work too hardor too long can lead to dissatisfaction, decreasing motivation, protest activities and,finally, even to the failure of the society or company.

• The use of fertilizers to increase short-term output from an agricultural activitybrings with it the danger of degrading soil or spoiling waters, jeopardizing thepotential of future crop yields. Again, there is a trade-off between short-term profitand the long-term state.

We believe that there are numerous areas of application beyond those mentioned abovethat fit naturally into our intensity-based control framework, and we hope to see manyof these applications in the future.

APPENDIX: PROOFS OMITTED FROM THE MAIN TEXT

Proof of Lemma 4.2. Since W , Np and Ni ,1−i , p = 1, . . . , p, i = 0, 1 are square-integrable (F(·), P

π,c)-martingales whose Doleans measures admit a bounded Radon–Nikodym derivative with respect to dt ⊗ P

π,c, by the Ito isometry it suffices to verify

Eπ,c[∫ T

0H(t)2dt

]< ∞

for each of the integrands H in (3.3). Since π , σ i, [1 + π .γ i·,p] and [1 + π .�i,1−i] are

bounded for i = 0, 1, for this condition it is sufficient to show that

Eπ,c[∫ T

0Xπ,c(t)kdt

]< ∞ for an arbitrary k ∈ R.(A.1)


We will now establish (A.1). First, the Ito formula yields

dXπ,c(t)k = Xπ,c(t−)k[A(t) dt + B(t).dW(t) +p∑

p=1

C p(t)dNp(t)

+1∑

i=0

Di (t)dNi ,1−i (t)],

where A, B, Cp and Di, p = 1, . . . , p, i = 0, 1 are bounded. It follows that

Xπ,c(t)k = exp{∫ t

0A(s) ds

}Et

(∫ ·

0B(s).dW(s)

)

·p∏

p=1

∏s∈(0,t],

�N p (s) �=0

(1 + C p(s))1∏

i=0

∏s∈(0,t],

�Ni ,1−i (s) �=0

(1 + Di (s)).

Choose κ > 0 such that |π (t)|, c(t), |A(t)|, |B(t)|, |Cp(t)|, |D0(t)|, |D1(t)| ≤ κ for t ∈[0, T ], p = 1, . . . , p. Then, by Novikov’s condition, Et(

∫ ·0 B(s).dW(s)) is an L2(Pπ,c)-

martingale. Further let ξp = 12( p+2) for p = 1, . . . , p and ξ i ,1−i = 1

2( p+2) for i = 0, 1 so

that 12 +∑ p

p=1 ξp +∑1i=0 ξ i ,1−i = 1. Now Holder’s and Doob’s inequality yield

Eπ,c[

supt∈[0,T]

Xπ,c(t)k]

≤ eκTE

π,c

[sup

t∈[0,T]Et

(∫ ·

0B(s).dW(s)

)2] 1

2 p∏p=1

Eπ,c[

supt∈[0,T]

(1 + κ)ξp Np(t)] 1

ξp

·1∏

i=0

Eπ,c[

supt∈[0,T]

(1 + κ)ξi ,1−i Ni ,1−i (t)

] 1ξ i ,1−i

≤ 2eκTE

π,c

[ET

(∫ ·

0B(s).dW(s)

)2] 1

2 p∏p=1

Eπ,c[(1 + κ)ξp Np(T)] 1

ξp

·1∏

i=0

Eπ,c[(1 + κ)ξ

i ,1−i Ni ,1−i (T)] 1ξ i ,1−i .

Since Np is an (F(·), Pπ,c)-Poisson process with intensity λp, we get

Eπ,c[(1 + κ)ξp Np(T)] = e((1+κ)ξp −1)Tλp < ∞ for p = 1, . . . , p.

Moreover we have

Eπ,c[(1 + κ)ξ

i ,1−i Ni ,1−i (T)] = E[Zπ,c(T)(1 + κ)ξ

i ,1−i Ni ,1−i (T)]with Zπ,c(T) = dP

π,c

dPas defined in (2.4), so

supt∈[0,T]

∣∣Zπ,c(t)(1 + κ)ξi ,1−i Ni ,1−i (t)

∣∣≤ e2T max

i=0,1‖ϑ i ,1−i‖N0,1(T)+N1,0(T)

∞ (1 + κ)ξi ,1−i Ni ,1−i (T)


where the right-hand side is in L1(P) as Ni,1−i is a (F(·), P)-Poisson process. Thus Eπ,c[(1 +

κ)ξi ,1−i Ni ,1−i (T)] < ∞ and we conclude that

Eπ,c[

supt∈[0,T]

Xπ,c(t)k]

< ∞.

This in particular implies (A.1) and thus completes the proof. �Proof of Lemma 5.4. We prove the claim by making use of existence and uniqueness

of solutions in the corresponding Merton model without regime shifts.Step 1 (Decomposition). We set

FM(t, g, h) � −(�0 − �1) +(

εδ

ε−(ε−δ)e−δ(T−t)

) 1R

e− 1−RR g R

1−R(e1−R

R h − 1),

FiM(t, g) � −� i + εδ

ε−(ε−δ)e−δ(T−t)

×(

1 − R1−R

(( εδ

ε−(ε−δ)e−δ(T−t) )1−R

R e− 1−RR g − 1

)).

FM and FiM represent the right-hand sides of the corresponding ODEs in the standard

Merton setting; note that these correspond exactly to the respective first lines in (5.10)and (5.11). To rewrite the complete ODE system in compact form, we define the functionsχ i : R+

0 → R, F, Fi : [0, T] × R × R+0 → R by

χ0(h) � −C0 11 − R

(e−(1−R)h − 1) − d0(π0,M)1

1 − R(e−(1−R)(h∧h0,crit) − 1)

− 12

1R

(B0π ) · (σ 0 · (σ 0) )−1 · B0

π

1(1 − R)2

(e−(1−R)(h∧h0,crit) − 1)2,

χ1(h) � C1 11 − R

(e(1−R)h − 1) + d1(π1,M)1

1 − R(e(1−R)(h∨h1,crit) − 1)

+ 12

1R

(B1π ) · (σ 1 · (σ 1) )−1 · B1

π

1(1 − R)2

(e(1−R)(h∨h1,crit) − 1)2,

di (π ) � Ai + π · Biπ − Ci ,

F(t, g, h) � FM(t, g, h) + χ0(h) + χ1(h),

Fi (t, g, h) � FiM(t, g) + (−1)iχ i (h).

Then it follows that for i = 0, 1 the functions g and h satisfy

h′(t) = F(t, g(t), h(t)), g′(t) − 1{i=1}h′(t) = Fi (t, g(t) − 1{i=1}h(t), h(t)).

As the corresponding ODEs in the Merton model admit unique nonexplosive solutions,it suffices to verify that the extra terms χ i in the ODEs for the large investor model arelocally Lipschitz continuous and nonnegative.

Step 2 (Local Lipschitz Continuity of χ i). Since the χ i’s are compositions of continu-ously differentiable functions, the max { ·, ·}- and the min { ·, ·}-function, they are locallyLipschitz. Hence, the ODE-system (5.10) and (5.11) has a unique local solution (g, h).


Step 3 (Nonnegativity of χ0). To check that χ0 ≥ 0 we distinguish the two cases h0,crit =0 and h0,crit > 0. In the first case, we have

χ0(h) = −C0 11 − R

(e−(1−R)h − 1) ≥ 0.

For h0,crit > 0, we have to consider the cases h < h0,crit and h ≥ h0,crit. As h < h0,crit isequivalent to

0 < −d0(π0,M)1

1 − R(e−(1−R)h − 1) − 1

R(B0

π ) .(σ 0.(σ 0) )−1.B0π

1(1 − R)2

(e−(1−R)h − 1)2

we obtain χ0(h) ≥ 0 for every h ≥ 0. On the other hand, h ≥ h0,crit implies

χ0(h) = −C0 11 − R

(e−(1−R)h − 1) + 12

((d0(π0,M))+)2

1R

(B0π ) .(σ 0.(σ 0) )−1.B0

π

which is nonnegative for every h ≥ 0, too.Step 4 (Nonnegativity of χ1). The proof of nonnegativity of χ1 is similar to that of

nonnegativity of χ0. We thus skip it, referring the reader to Busch (2011) for details.Step 5 (Global Existence). Nonnegativity of the χ i’s implies that

F(t, g, h) ≥ FM(t, g, h) ≥ CM for every (t, g, h) ∈ [0, T] × R × R+0 ,(A.2)

F0(t, g, h) ≥ F0M(t, g) ≥ C0

M for every (t, g, h) ∈ [0, T] × R+0 × R+

0 ,(A.3)

F1(t, g, h) ≤ F1M(t, g) ≤ C1

M for every (t, g, h) ∈ [0, T] × R−0 × R+

0(A.4)

with constants CM , CiM that result from the solvability of the Merton ODEs. Inequality

(A.2) implies that h is linearly bounded from above, whereas (A.3) and (A.4) guaranteethat g is linearly bounded from above as well as from below. Hence, the unique localsolution (g, h) is nonexplosive and exists globally. �

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optimal consumption and investment for a large investor: an intensity-based control framework

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