Time-optimal Active Portfolio Selection Thomas Balzer [email protected] November 27, 2001

Abstract In a complete financial market model where the prices of the assets are modeled as Ito diffusion processes, we consider portfolio problems where the investor measures the result of his trading strategy with respect to some benchmark. The investor tries to track the risk that he might fail to outperform the benchmark down to some specified level within an optimal time horizon. Such problems of time-optimal portfolio selection are studied in cases where risk is measured as shortfall probability, expected shortfall or in general weighted shortfall. By means of duality methods on optimal stopping under constraints it is possible to identify the optimal portfolio with a hedging strategy for a suitable option with random expiration. For different risk measures the structure of the optimal time horizon can be derived as well.

1

Introduction

Faced with the problem of how to structure his portfolio an investor can follow several different objectives. Almost classical investment objectives are the mean-variance approach of Markowitz [19] and the problem of maximizing expected utility from wealth (cf. [20], [15] or [16]). In practice however, a portfolio manager is usually judged by the result of his strategy compared to a relevant benchmark, e.g. a stock index (for related studies cf. [5], [6], [7]). In the literature two different approaches to benchmark-oriented portfolio optimization are distinguished (cf. [24]): Passive portfolio management on the one hand consists in imitating the benchmark to acheive a comparable performance. In contrast, an active portfolio manager is trying to outperform the benchmark by an appropriate trading strategy. 1

1

INTRODUCTION

2

The aim of this article is to analyze problems of active portfolio management in the context of time-optimal portfolio selection where the time horizon is taken as the objective function of the optimization (cf. [2] or [3] for related studies). As the prime example of such a problem the following setup is considered: The investor aims at a relevant benchmark and measures at each instant the risk to fall short of the competing strategy (this risk, of course, can be measured in different ways). The investor’s goal is to track down this risk within an optimal time horizon to a level that he is willing to tolerate. Different time horizons are compared by means of the expected return of one optimally growing unit of wealth relative to the benchmark. The risk of falling short of the benchmark can be quantified as shortfall probability, expected shortfall or more general as the shortfall weighted by some loss function l (e.g. l(x) = xp /p for p > 1). These risk measures resemble those from the analyses of F¨ollmer and Leukert ([9] and [10]), Cvitanic and Karatzas [8] or Pham [22] in different settings with a fixed time horizon. The structure of the article is as follows: In Section 2 the main mathematical model for the financial market is laid out. The analysis is based on a standard setup for a complete financial market where the prices of the assets follow Ito diffusion processes (cf. [15] or [16]). Since there is no time horizon fixed in advance, the model is stated with an infinite time horizon. Notions as portfolios, wealth processes, benchmarks and num´eraires are introduced in this section as well. In Section 3 the basic idea of time-optimal portfolio selection and the approach on comparing different time intervals are stated. Furthermore, different risk measures as shortfall probability, expected shortfall and weighted shortfall are introduced such that the investment objective of outperforming the benchmark can be appropriately formalized. These preliminaries allow for an exact mathematical formulation of the relevant portfolio problem which is seen to involve a minimization over a class of pairs consisting of portfolios and stopping times satisfying suitable constraints. By using the concept of contingent claims with random expiration (cf. [3]), the portfolio problem is equivalently restated in Section 4 as a problem of combined stopping and deciding under constraints. If risk is measured as shortfall probability this equivalent problem coincides exactly with the problem of sequential testing for the drift of a Brow-

2

THE MARKET MODEL

3

nian Motion. This observation allows to state in Section 5 the optimal solution of this specific equivalent problem explicitly using results of [18]. For general problems of optimal stopping and deciding under constraints there exists a duality result such that the minimization problem can be transformed into a dual equivalent maximin procedure. In Section 6 this general duality theorem from [4] is recalled in the form which is needed for the analysis of general risk measures. Based on this duality method it is possible to analyze the case where risk is measured as expected shortfall in further detail in Section 7. It is possible to reduce the first step of the dual maximin-procedure into a problem of optimal stopping. For the special case of a constant coefficient setting the optimal stopping time can be partly identified. In Section 8 the analysis of the preceding section is extended to the case of a general loss function. Again the structure of the solution for the optimal stopping problem arising as a part of the dual problem can be partly obtained. The proofs of Propositions 1, 5 and 6 are deferred to the appendix.

2

The market model

Let there be given a complete probability space (Ω, F, P), endowed with a standard d-dimensional Brownian motion W = (W (1) , . . . , W (d) )∗ . The information structure is described by the filtration F := {F(t)}0≤t<∞ , the usual P-augmentation of F W (t) := σ(W (s), 0 ≤ s ≤ t) for t ∈ R+ . The financial market M consists of d + 1 financial assets, one riskless bond, whose price is modeled by the equation dP0 (t) = P0 (t)r(t)dt ,

P0 (0) = 1

(1)

and d risky stocks whose prices are governed by the equations dPi (t) = Pi (t)[bi (t)dt +

d X

σij (t)dW (j) (t)] , Pi (0) = pi > 0 ; i = 1, . . . , d.

j=1

(2) The market coefficients r (interest rate), b (vector of stock appreciation rates) and σ (volatility matrix) are all assumed to be progressively measurable with respect to F. Moreover, σ is assumed to be invertible and the

2

THE MARKET MODEL

4

processes r, b, σ as well as σ −1 are bounded uniformly in (t, ω) ∈ [0, T ] × Ω for each T ∈ R+ . Since σ is invertible, the d-dimensional process θ – the market price of risk – can be defined by θ(t) := (σ(t))−1 (b(t) − r(t)1)

(3)

for t ∈ R+ with 1 := (1, . . . , 1)∗ ∈ Rd . The process θ is bounded and F-progressively measurable as well, hence {Z0 (t)}0≤t<∞ defined as Z

t

Z0 (t) := exp{− 0

1 θ (s)dW (s) − 2 ∗

Z

t

kθ(s)k2 ds}

(4)

0

with Z0 (0) = 1 is a non-negative exponential martingale (cf. [14] p. 199). An investor who is equipped with an initial capital of x0 > 0 and who cannot influence the asset prices by his actions can decide at each instant t ∈ R+ which proportion πi (t) of his wealth X(t) to invest in the i-th stock. With the vector π(t) = (π1 (t), . . . , πd (t))∗ chosen, the proportion P 1 − di=1 πi (t) of the agent’s wealth is invested risklessly. Assuming a self-financing condition this leads to the wealth equation dX(t) = X(t){(r(t) + π ∗ (t)σ(t)θ(t))dt + π ∗ (t)σ(t)dW (t)}

(5)

with X(0) = x0 > 0. We define Definition 1 1. A F-progressively measurable process π = {π(t)}0≤t<∞ with values in RT Rd , fulfilling 0 X 2 (s)kπ(s)k2 ds < ∞ a.s. for each T ∈ R+ is called a portfolio process. 2. Given a portfolio process π, the solution X = X x0 ,π of (5) is called wealth process according to the initial wealth x0 und the portfolio π. 3. A portfolio process π is called admissible for an initial wealth x0 > 0, if a.s. X x0 ,π (t) ≥ 0 (6) holds for all 0 ≤ t < ∞. The set of all portfolios admissible for x0 is denoted A0 (x0 ).

2

THE MARKET MODEL The state-price-density process H0 , defined as Z t   Z t 1 Z0 (t) 2 (r(s) + kθ(s)k )ds − θ∗ (s)dW (s) H0 (t) := = exp − P0 (t) 2 0 0

5

(7)

for t ∈ R+ , plays a central role subsequently. Since for each admissible π the equation d(H0 (t)X x0 ,π (t)) = H0 (t)X x0 ,π (t){π ∗ (t)σ(t) − θ∗ (t)}dW (t)

(8)

holds true, the process H0 X x0 ,π is a non-negative, local P-martingale, hence a P-supermartingale. In particular, for two F-stopping times τ1 and τ2 satisfying P(τ1 ≤ τ2 ) = 1 E[H0 (τ2 )X x0 ,π (τ2 )|F(τ1 )] ≤ H0 (τ1 )X x0 ,π (τ1 )

(9)

holds with H0 (∞)X x0 ,π (∞) := 0. The choice τ1 = 0 leads to E(H0 (τ )X x0 ,π (τ )) ≤ x0

(10)

for each F-stopping time τ (cf. [23], Theorem 3.3, p. 70). Subsequently, the investor measures the performance of his actions with respect to a certain benchmark which is assumed to be the wealth process {Λ(t)}0≤t<∞ according to the initial capital x0 and a portfolio process πΛ . This portfolio is assumed to be bounded uniformly in (t, ω) ∈ [0, T ] × Ω for each T ∈ R+ . The process H0 Λ satisfies the equation ∗ d(H0 (t)Λ(t)) = H0 (t)Λ(t)(πΛ (t)σ(t) − θ∗ (t))dW (t) ∗ = H0 (t)Λ(t)θΛ (t)dW (t)

(11)

∗ (t) := π ∗ (t)σ(t) − θ ∗ (t) for t ∈ R . Due to the boundedness of θ with θΛ + Λ Λ the a.s. strictly positive process H0 Λ is a martingale. Furthermore, let {N (t)}0≤t<∞ , the num´eraire, be the wealth according ∗ := π ∗ σ − to a locally bounded portfolio πN and initial capital 1. We set θN N θ∗ . It is common to choose the num´eraire N as the wealth according to πN ≡ 0, i.e. the riskless bond, but this is a restriction which is not essential at this point of the analysis.

3

THE INVESTMENT OBJECTIVE

3 3.1

6

The investment objective The time horizon as the objective function

The basic idea of time-optimal portfolio selection consists in regarding the time horizon as the objective function that is to be minimized: The investor’s aim is to reach an a priori horizon-independent investment goal within an optimal time interval (for an introduction cf. [3]). Admissible periods are stochastic intervals of the form [0, [ τ]] := {(t, ω) ∈ R+ × Ω | 0 ≤ t ≤ τ (ω)}

(12)

for F-stopping times τ . The set of all a.s. finite F-stopping times is denoted T := {τ | τ is F-stopping time with P(τ < ∞) = 1}.

(13)

In the following different stochastic intervals are compared by the following criterion: with ζ defined as Z 1 t kθΛ (s)k2 ds (14) ζ(t) := 2 0 for t ∈ R+ , a stopping time τ1 ∈ T is preferred to some τ2 ∈ T if E(ζ(τ1 )) ≤ E(ζ(τ2 )) holds. Subsequently, we restrict the analysis to stopping times of the set T ∗ := {τ ∈ T | E(ζ(τ )) < ∞}. (15) Within the class T ∗ the objective function ζ has an inherent financial interpretation since for each stopping time τ ∈ T ∗ Z 1 τ H (τ )Λ(τ ))) = E( E(− ln(x−1 kθΛ (s)k2 ds) = E(ζ(τ )) (16) 0 0 2 0 holds (cf. [11], p. 62). K

Remark 1 The wealth process X 1,π according to the so-called growthoptimal portfolio π K , defined for t ∈ R+ as π K (t) = (σ ∗ (t))−1 θ(t) (cf. [13], pp. 42ff.), coincides with the process H0−1 . Hence the quantity E(ζ(τ )) = ∗ E(− ln(x−1 0 H0 (τ )Λ(τ ))) for τ ∈ T can be interpreted as the expected logarithmic return up to time τ of one unit of wealth invested growth-optimally relative to one unit of wealth invested in the benchmark Λ. Consequentially, the objective function ζ measures time in terms of lost profit which is an economically reasonable criterion.

3

THE INVESTMENT OBJECTIVE

3.2

7

Risk measures

The benchmark the investor is relating his actions to is the wealth according to an admissible portfolio. Hence, the investor always has the opportunity to imitate the benchmark exactly, a procedure which is known as passive portfolio management (cf. [24], pp. 799ff.). With such a strategy the investor never has the chance to acheive a better performance than the benchmark. In contrast to this, active portfolio management (cf. ibid) consists in choosing the trading strategy in such a way that the benchmark is outperformed. To formalize the relevant investment objective it has to be defined when the benchmark can be considered as outperformed. The benchmark at time t = 0 takes the same value x0 as the investor’s wealth. It is assumed that the aim of the investor is to reach a fixed multiple a > 1 of the benchmark wealth, hence to surpass the benchmark by a predetermined percentage. Since the investor and the “benchmark investor” act within the same model it is unrealistic to believe that the investment goal can be reached almost surely. Consequentially, the objective is formalized in the following way: the investor accepts a certain risk not to reach the aspired wealth and he is trying to track down this risk to a level of risk β which he is willing to accept. The aim is to choose the time horizon as favourable as possible (in terms of ζ) such that this tolerance level of risk can be reached by an appropriate strategy. If the investor pursues the portfolio π his risk with respect to the benchmark at time τ is given by Rπ (τ ) := E(l(N (τ )−1 (aΛ(τ ) − X x0 ,π (τ ))+ ))

(17)

where l : [0, ∞) → R+ is a loss function which is assumed to be increasing with l(0) = 0. This definition of risk covers several different risk criteria known in the literature. Example 1 1. risk = shortfall probability. The choice l(x) = 1(0,∞) (x) leads to Rπ (τ ) = P(X x0 ,π (τ ) < aΛ(τ )).

(18)

3

THE INVESTMENT OBJECTIVE

8

Only the probability of not having reached the aspired wealth at time τ is taken into account. Such a risk measure is considered in [5], [9] and [25] within different models and a fixed time horizon. In these contributions, risk arises from a contingent claim that does not allow for a perfect hedge. The idea of minimizing the probability to fall short of some stochastic target goes back to [17]. 2. risk = expected shortfall. With l(x) = x one obtains Rπ (τ ) := E(N (τ )−1 (aΛ(τ ) − X x0 ,π (τ ))+ ).

(19)

Here the investor considers the expected amount he falls short of the benchmark while this shortfall is discounted by means of the num´eraire N . In the case of a fixed time horizon, a similar risk measure is studied in [8] and [10]. 3. risk = lower partial moment of the shortfall. The choice l(x) = xp /p for some p > 1 leads to 1 Rπ (τ ) := E( (N (τ )−1 (aΛ(τ ) − X x0 ,π (τ ))+ )p ), p

(20)

a risk measure known as lower partial moment of the expected shortfall. A higher exponent p indicates an increasingly risk averse investor (cf. [10] or [22] for related studies). The relevant optimization problem can be now stated in the following form. Problem 1 Assume 0 < β < l(x0 (a − 1)). Minimize E(ζ(τ )) subject to (τ, π) ∈ G, with

G := {(τ, π) ∈ T ∗ × A0 (x0 ) | E(l(N (τ )−1 (aΛ(τ ) − X x0 ,π (τ ))+ )) ≤ β}.

(21)

For general loss functions, the risk measure as defined in (17) does not necessarily have coherence properties in the sense of [1]. However, the present form of measuring risk is general enough to include most of the common risk definitions including the coherent measure for the choice l(x) = x. It is important to notice that in contrast to portfolio optimization with a fixed time horizon, the solution of a problem of time-optimal portfolio

4

AN EQUIVALENT OPTIMIZATION PROBLEM

9

selection always consists of two components: An optimal stochastic horizon as well as a portfolio that allows for reaching the investment goal within the optimal period.

4

An equivalent optimization problem

Similar as in [3], Problem 1 can be tranformed into an equivalent problem that can more conveniently be analyzed. Let L(E) denote the set of all E-valued progressively measurable processes for E ⊆ R. Lemma 1 We have inf

(τ,ξ)∈G 0

E(ζ(τ )) =

inf E(ζ(τ ))

(22)

(τ,π)∈G

with the set G 0 defined as G 0 := {(τ, ξ) ∈ T ∗ × L([0, 1]) | E(H0 (τ )Λ(τ )ξ(τ )) ≤

x0 , a

E(l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))) ≤ β }.

(23)

Proof: It is sufficient to prove a one-to-one relation between the sets G and G0. For (τ, π) ∈ G define the process ξ by ξ(t) := 1 ∧ (X x0 ,π (t)/(aΛ(t))) for t ∈ R+ . Then ξ ∈ L([0, 1]) holds as well as aΛ(τ )(1 − ξ(τ )) = aΛ(τ ) − aΛ(τ ) ∧ X x0 ,π (τ ) = (aΛ(τ ) − X x0 ,π (τ ))+ . Thus, E(l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))) = E(l(N (τ )−1 (aΛ(τ ) − X x0 ,π (τ ))+ )) ≤ β follows. Furthermore, with (10), one obtains E(H0 (τ )Λ(τ )ξ(τ )) ≤ E(H0 (τ )

x0 X x0 ,π (τ ) )≤ a a

and hence (τ, ξ) ∈ G 0 . For each (τ, ξ) ∈ G 0 the random variable B := aΛ(τ )ξ(τ ) can be interpreted as a contingent claim with random expiration τ (cf. [3]). The hedging price of this option does not exceed x0 since E(H0 (τ )B) = E(H0 (τ )aΛ(τ )ξ(τ )) ≤ x0 . Hence due to [3] there exists a portfolio π ∈ A0 (x0 ) such that X x0 ,π (τ ) ≥

4

AN EQUIVALENT OPTIMIZATION PROBLEM

10

aΛ(τ )ξ(τ ) holds. For ξ ∈ L([0, 1]) we have trivially aΛ(τ ) ≥ aΛ(τ )ξ(τ ). Thus aΛ(τ ) ∧ X x0 ,π (τ ) ≥ aΛ(τ )ξ(τ ) ⇔ (aΛ(τ ) − X x0 ,π (τ ))+ ≤ aΛ(τ )(1 − ξ(τ )). Since l is assumed to be increasing, E(l(N (τ )−1 (aΛ(τ ) − X x0 ,π (τ ))+ )) ≤ E(l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))) ≤ β follows which implies (τ, π) ∈ G.  The preceding lemma justifies the analysis of the following optimization problem which is equivalent to Problem 1. Problem 10 Minimize E(ζ(τ )) subject to (τ, ξ) ∈ G 0 . If the process ξ is interpreted as the value of a decision, Problem 10 is a special case of a problem of optimal stopping and deciding under constraints. For a detailed analysis of such problems cf. [4]. The structure of Problem 10 implies immediately that the interplay between the num´eraire and the benchmark is essential for the chance to reach the investment goal. Example 2 Let the num´eraire N be the wealth according to the portfolio πN = (σ ∗ )−1 θ and let τ ∈ T be a stopping time with E(H0 (τ )Λ(τ )) = x0 . Let Λ be an arbitrary benchmark and l a strictly increasing loss function. Assume there exists some ξ ∈ L([0, 1]) with (τ, ξ) ∈ G 0 , i.e. E(l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))) ≤ β and E(H0 (τ )Λ(τ )ξ(τ )) ≤ x0 /a. Due to the special choice of the num´eraire, we have N −1 = H0 and an application of Jensen’s inequality yields E(H0 (τ )Λ(τ )(1 − ξ(τ ))) ≤ l−1 (β)/a, where l−1 denotes the inverse of l. Additionally, 1 E(H0 (τ )Λ(τ )(1 − ξ(τ ))) = x0 − E(H0 (τ )Λ(τ )ξ(τ )) ≥ x0 (1 − ) a holds. Combining the derived inequalities leads to 1 x0 (1 − ) ≤ E(H0 (τ )Λ(τ )(1 − ξ(τ ))) ≤ l−1 (β)/a. a Hence, if β < E(l(N (0)−1 (aΛ(0) − x0 )+ )) holds, by contradiction there does not exist a process ξ ∈ L([0, 1]) with (τ, ξ) ∈ G 0 .

5

5

RISK MEASURED AS SHORTFALL PROBABILITY

11

Risk measured as shortfall probability

In this section, Problem 10 is analyzed for the loss function l(x) = 1(0,∞) (x), i.e. for measuring risk as shortfall probability. Proposition 1 Let l be the loss function l(x) = 1(0,∞) (x) and assume β + 1/a < 1. Then the value of Problem 10 (and the value of Problem 1) satisfies inf

(τ,ξ)∈G 0

E(ζ(τ )) ≥ (1 − β) ln(

1−β 1 a

) + β ln(

β ). 1 − a1

(24)

Proof: See Appendix A. 

10

As the following result shows, the lower bound for the value of Problem in (24) is in fact attained.

Proposition 2 Assume P(ζ(∞) = ∞) = 1. The pair (τ1 , ξ1 ) with τ1 := inf{t ∈ R+ | ξ1 (t) := 1{

x0 β 1−β ∈ /( , 1 )}, 1 H0 (t)Λ(t) 1− a a

(25)

0≤t<∞

(26)

x0 ≥a(1−β)} H0 (t)Λ(t)

,

belongs to the set G 0 and satisfies inf

(τ,ξ)∈G 0

E(ζ(τ )) = E(ζ(τ1 )) = (1 − β) ln(

1−β 1 a

) + β ln(

β ). (27) 1 − a1

Proof: As a result of [18], Lemma 17.7., p. 250, the stopping time τ1 satisfies E(ζ(τ1 )) < ∞ as well as τ1 ∈ T . Furthermore, Lemma 17.8 of [18], p. 251, implies E(l(N (τ1 )−1 (1 − ξ1 (τ1 ))aΛ(τ1 ))) = E(l(1 − ξ1 (τ1 ))) = E(1 − ξ1 (τ1 )) = β and E(H0 (τ1 )Λ(τ1 )ξ1 (τ1 )) = x0 /a. Thus, (τ1 , ξ1 ) ∈ G 0 . The second equation of (27) follows from Theorem 17.8 of [18], p. 249. The first equation of (27) is trivial in connection with Proposition 1 since the lower bound for the value of Problem 10 is attained by (τ1 , ξ1 ) . 

Remark 2 In the case of measuring risk as shortfall probability this specific Problem 10 can be interpreted as a statistical decision problem: If the process

5

RISK MEASURED AS SHORTFALL PROBABILITY

12

H0 Λ is regarded as a likelihood ratio process, Problem 10 coincides with the problem of finding a sequential test for the drift of a Brownian Motion (cf. [18], Section 17.6). This is the reason why it is possible to solve the problem by adapting techniques from [12] and [18]. Problem 1 with this special risk measure corresponds to the problem of maximizing the probability of a perfect hedge (cf. [25], [9]) if the time horizon is specified in advance. For a fixed time horizon this problem can be solved using the Neyman-Pearson-Lemma for testing statistical hypotheses as in [9]. If the time horizon is not fixed ex ante but subject to being optimized, the use of the Sequential Probability Ratio Test (as introduced by [27]) – the dynamic version of the Neyman-Pearson-Test – arises naturally. Although the value of Problem 1 is known from (27), the portfolio that allows for outperforming the benchmark within the optimal time horizon [0, [ τ1]] cannot be stated explicitly. From Lemma 1 is is clear, that the optimal trading strategy is the hedge of the τ1 -option B := aΛ(τ1 )ξ1 (τ1 ). The structure of this hedging strategy however, remains undisclosed. It is nevertheless interesting to compare the optimal portfolio for Problem 1 – denoted by π1 in the following – with the so-called log-optimal portfolio π K = (σ −1 )∗ θ which often is optimal for time-optimal portfolio selection with static investment goals (cf. [3]). In Tabular 1 the wealth generated by these two portfolios at the stopping K time τ1 is compared in dependance of the value of ξ1 (τ1 ) (recall X x0 ,π = x0 (H0 )−1 ). Portfolio π1

Portfolio π K

ξ1 (τ1 ) = 1

X x0 ,π1 (τ1 ) = aΛ(τ1 )

X x0 ,π (τ1 ) = (1 − β)aΛ(τ1 )

ξ1 (τ1 ) = 0

X x0 ,π1 (τ1 ) = 0

K

K

X x0 ,π (τ1 ) =

β a−1 aΛ(τ1 )

Tab. 1. Terminal wealth for the optimal portfolio versus the log-optimal portfolio.

From Tabular 1 it is clear that the optimal strategy π1 inhibits a rather extreme behaviour. While in the case ξ1 (τ1 ) = 1 the investor acheives the aspired multiple of the benchmark wealth, he loses in the case ξ1 (τ1 ) = 0 his entire capital while pursueing π1 . The log-optimal – and hence growth-optimal – strategy generates at τ1 in the case ξ1 (τ1 ) = 1 a strictly smaller wealth than the unknown portfolio π1 .

6

DUALITY METHODS

13

β In the case ξ1 (τ1 ) = 0 the growth-optimal strategy yields a wealth a−1 aΛ(τ1 ) which is strictly less than what is acheivable by imitating the benchmark. Both columns in Tabular 1 coincide for β = 0. Thus the portfolio π K turns out to be optimal strategy in the limiting case of an entirely vanishing risk tolerance.

6

Duality methods

For general risk measures, the analysis of the equivalent Problem 10 requires refined methods. To pursue a duality approach to Problem 10 based on the results of [4], we assume from now on that l is a continuous loss function and that the following holds: Assumption There exists some ε > 0 such that E( sup l( 0≤t≤ε

aΛ(t) )) < ∞ N (t)

(28)

holds. Definition 2 Define the set J ⊂ T ∗ × L([0, 1]) as J := {(τ, ξ) ∈ T ∗ × L([0, 1]) | P(τ > 0) = 1, E(|H0 (τ )Λ(τ )ξ(τ )|) < ∞, E(|l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))|) < ∞ }(29) Under appropriate conditions, Problem 10 can be transformed into a dual problem involving an unconstrained minimization and a maximization over Lagrange multipliers. Proposition 3 If there exists a pair (τ, ξ) ∈ J with E(H0 (τ )Λ(τ )ξ(τ )) < x0 −1 a and E(l(N (τ ) (1 − ξ(τ ))aΛ(τ ))) < β then inf

(τ,ξ)∈G 0 ∩J

E(ζ(τ )) =

sup

inf

µ∈R2+ (τ,ξ)∈J

E(ζ(τ ) + µ1 (H0 (τ )Λ(τ )ξ(τ ) −

+µ2 (l(

x0 ) a

(1 − ξ(τ ))aΛ(τ ) ) − β))(30) N (τ )

holds. Furthermore, the supremum on the right hand side is attained by some µ ˜ ∈ R2+ .

6

DUALITY METHODS

14

Proof: Cf. [4], Theorem 1.  In fact, the left hand side of (30) coincides with the value of Problem 10 . Lemma 2 The assumption β < l(x0 (a − 1)) implies inf

(τ,ξ)∈G 0 ∩J

E(ζ(τ )) =

inf

E(ζ(τ )).

inf

E(ζ(τ ))

(τ,ξ)∈G 0

(31)

Proof: Only the inequality inf

(τ,ξ)∈G 0 ∩J

E(ζ(τ )) ≤

(τ,ξ)∈G 0

˜ ∈ G 0 \ (G 0 ∩ J ) with has to be proven. Assume there exists (˜ τ , ξ) E(ζ(˜ τ )) <

inf

(τ,ξ)∈G 0 ∩J

E(ζ(τ )).

˜ τ ))aΛ(˜ ˜ τ )) ≤ x0 < ∞ and 0 ≤ E(l(N (˜ τ )−1 (1−ξ(˜ τ ))) ≤ Thus 0 ≤ E(H0 (˜ τ )Λ(˜ τ )ξ(˜ a 0 ˜ β < ∞ hold true. If P(˜ τ > 0) = 1 holds, this implies (˜ τ , ξ) ∈ G ∩ J and hence a contradiction. Assume P(˜ τ > 0) < 1. Thus P(˜ τ = 0) = 1 since F(0) is trivial. The ˜ inequality E(H0 (0)Λ(0)ξ(0)) ≤ x0 /a implies that the a.s. constant random ˜ does not exceed 1/a. Hence variable ξ(0) ˜ ˜ β ≥ E(l(N (0)−1 (1 − ξ(0))aΛ(0))) = E(l((1 − ξ(0))ax 0 )) ≥ l(x0 (a − 1)) which contradicts β < l(x0 (a − 1)).  It is sometimes possible to reduce the first stage of the dual of Problem to a problem of optimal stopping. The simple proof of the following result is omitted. 10

Lemma 3 Let µ ∈ R2+ be fixed. If there exists a process ξµ ∈ L([0, 1]) such that for all ξ ∈ L([0, 1]) and almost all (t, ω) ∈ R+ × Ω µ1 H0 (t, ω)Λ(t, ω)ξµ (t, ω) + µ2 l(

(1 − ξµ (t, ω))aΛ(t, ω) ) N (t, ω)

≤ µ1 H0 (t, ω)Λ(t, ω)ξ(t, ω) + µ2 l(

(1 − ξ(t, ω))aΛ(t, ω) ) N (t, ω)

(32)

7

RISK MEASURED AS EXPECTED SHORTFALL

15

holds, then inf (τ,ξ)∈J

=

inf (τ,ξ)∈J

E(ζ(τ ) + µ1 (H0 (τ )Λ(τ )ξ(τ ) −

E(ζ(τ ) + µ1 (H0 (τ )Λ(τ )ξµ (τ ) −

+µ2 (l(

7

x0 (1 − ξ(τ ))aΛ(τ ) ) + µ2 (l( ) − β)) a N (τ ) x0 ) a

(1 − ξµ (τ ))aΛ(τ ) ) − β)) N (τ )

(33)

Risk measured as expected shortfall

In this section the case l(x) = x is considered. For this loss function it can be checked (cf. [2] Lemma 5.3.1, p. 117 for details) by a simple application of the Burkholder-Davis-Gundy inequality (cf. [14] p. 166) that the assumption (28) is always satisfied. The risk level β is assumed to satisfy 0 < β < x0 (a − 1). For a special choice of the benchmark, the relevant optimization problem can be explicitly solved as in the case for shortfall probability without use of the duality methods of Section 6. Example 3 If num´eraire and benchmark are the wealth processes according to an identical portfolio πΛ = πN , then x0 N (t) = Λ(t) holds for all t ∈ R+ . Problem 10 then takes the form to minimize E(ζ(τ )) subject to the constraints E(H0 (τ )Λ(τ )ξ(τ )) ≤ x0 /a and E(1 − ξ(τ )) ≤

β . ax0

Unter the assumption P(ζ(∞) = ∞) = 1 it is an immediate consequence of Proposition 2 that the optimal solution for Problem 10 is given by the pair ¯ with (¯ τ , ξ) τ¯ := inf{t ∈ R+ | ¯ ξ(t) := 1{

x0 β β ∈ /( , a − )}, H0 (t)Λ(t) x0 (a − 1) x0

x0 ≥a−β/x0 } H0 (t)Λ(t)

,

The condition β < x0 (a − 1) assures equation (27) one obtains inf

(τ,ξ)∈G 0

E(ζ(τ )) = E(ζ(¯ τ )) = (1 −

0 ≤ t < ∞. β x0 (a−1)

< 1 < a−

(34) (35)

β x0 .

In analogy to

β β β β ) ln(a − ) + ln( ). ax0 x0 ax0 x0 (a − 1)

7

RISK MEASURED AS EXPECTED SHORTFALL

16

To apply Proposition 3, it has to be assumed that there exists a pair (τ, ξ) ∈ J with E(H0 (τ )Λ(τ )ξ(τ )) < xa0 and E(N (τ )−1 (1−ξ(τ ))aΛ(τ )) < β. Sufficient conditions under which this assumption holds true are derived in [2], Section 5.5 in the Markovian case of a constant coefficient framework. With this assumption and considering Lemma 2 the equation (30) takes the form x0 inf E(ζ(τ )) = sup inf E(ζ(τ ) + µ1 (H0 (τ )Λ(τ )ξ(τ ) − ) 0 a (τ,ξ)∈G µ∈R2 (τ,ξ)∈J +

+µ2 (

(1 − ξ(τ ))aΛ(τ ) − β)). N (τ ) (2)

(36)

(2)

It can now easily be checked that the process ξµ defined as ξµ (t) := 1{µ1 H0 (t)<µ2 aN (t)−1 } for µ ∈ R2+ and t ∈ R+ satisfies the prerequistes of Lemma 3 which leads to Λ(τ ) (µ1 H0 (τ )N (τ ) ∧ µ2 a)). inf E(ζ(τ )) = sup inf E(ζ(τ ) + κ(µ) + 0 N (τ ) (τ,ξ)∈G µ∈R2 (τ,ξ)∈J +

with the constant κ(µ) := −µ1 x0 /a − µ2 β. Hence, Problem 10 has a dual procedure which as a first step involves a problem of optimal stopping of the process L2 (t, µ) defined as L2 (t, µ) := ζ(t) + κ(µ) +

Λ(t) (µ1 H0 (t)N (t) ∧ µ2 a) N (t)

(37)

for µ ∈ R2+ and t ∈ R+ . It is clear that for µ1 = 0 or µ2 = 0 the optimal stopping time for L2 (·, µ) is τ = 0. In general, it is complicated to solve problems of optimal stopping explicitly. However, if we specialize on a Markovian framework it is possible to derive the structure of the optimal stopping time for the process L2 (·, µ). Therefore we assume for the remainder of this section that the market coefficients r, b and σ as well as the portfolios πN and πΛ are constant. Furthermore, σ is regular and for simplicity we set x0 = 1. The analysis of the stopping problem is based on the following result. Proposition 4 Let {Y (t)}0≤t<∞ be a real-valued, adapted stochastic process with a.s. continuous paths. Assume that Y is bounded from below and E(Y (t)) < ∞ for all t ∈ R+ and limt→∞ Y (t) = ∞ a.s. hold true. Then there exists a P-a.s. finite stopping time τˆ with inf E(Y (τ )) = E(Y (ˆ τ )).

τ ∈T

7

RISK MEASURED AS EXPECTED SHORTFALL

17

Proof: The assertion follows from Theorem 2.5.1, pp. 23ff. in conjunction with Theorem 2.8.6, p. 39 and Lemma 2.8.7, pp. 39f. in [26].  The process L2 (·, µ) can be written in the form L2 (t, µ) = f (H0 (t)N (t),

Λ(t) , ζ(t); µ) N (t)

for f (z1 , z2 , z3 ; µ) := κ(µ) + z3 + z2 (µ1 z1 ∧ µ2 a) and z ∈ R2+ × R. Hence, L2 (·, µ) is a function of a three-dimensional Markov process. Proposition 5 Assume P(ζ(∞) = ∞) = 1. For each µ ∈ R2+ there exists an a.s. finite optimal stopping time for the process L2 (·, µ). This stopping time is of the form τ2 (µ) := inf{t ∈ R+ | (H0 (t)N (t),

Λ(t) ∗ ) ∈ S2 (µ)} N (t)

(38)

for a set S2 (µ) ⊆ R2+ called the stopping region. The stopping region satisfies (1)

S2 (z1 ; µ) := {z2 ∈ R+ | (z1 , z2 ) ∈ S2 (µ)} = [0, A(1) (z1 ; µ)]

(39)

(1)

for some A(1) (z1 ; µ) ≥ 0 or S2 (z1 ; µ) = R+ as well as (2)

S2 (z2 ; µ) := {z1 ∈ R+ | (z1 , z2 ) ∈ S2 (µ)} =

(40)

[0, A(2) (z2 ; µ)] ∪ [B (2) (z2 ; µ), ∞)

for µ1 > 0 and 0 ≤ A(2) (z2 ; µ) ≤

µ2 a µ1

≤ B (2) (z2 ; µ) < ∞.

Proof: See Appendix A.  With two examples it can be seen that the result of Proposition 5 is in accordance with previous observations. Example 4 1. For the num´eraire N = H0−1 the first component of (H0 N, Λ/N ) is constant with value 1. Due to Proposition 5 the optimal stopping time is of the form τ2 (µ) = inf{t ∈ R+ | H0 (t)Λ(t) ∈ [0, A(1) (1; µ)]}

8

RISK MEASURED AS EXPECTED WEIGHTED SHORTFALL

18

for some A(1) (1; µ). If A(1) (1; µ) ≥ 1 then H0 (0)Λ(0) = 1 implies τ2 (µ) = 0 a.s. For A(1) (1; µ) < 1 however, τ2 (µ) is the first time a martingale crosses a boundary, a stopping time that in general does not belong to T (cf. [14], p. 197). In both cases there does not exist a µ such that (τ2 (µ), ξµ ) ∈ J . This corresponds to the result of Example 2. 2. If num´eraire and benchmark are chosen to coincide, the second component of (H0 N, Λ/N ) is constant. For each µ ∈ R2+ the optimal stopping time takes the form τ2 (µ) = inf{t ∈ R+ | H0 (t)Λ(t) ∈ / (A(2) (1; µ), B (2) (1; µ))} for 0 ≤ A(2) (1; µ) ≤ µµ21a ≤ B (2) (1; µ) < ∞. This structure of τ2 (µ) coincides with the result of Example 3 where the explicit solution is given for this special choice of benchmark and num´eraire.

8

Risk measured as expected weighted shortfall

In this section we analyze the case of a loss function l : [0, ∞) → R+ which is assumed the be strictly increasing, strictly convex and differentiable with l(0) = 0. Standing assumptions are furthermore 0 < β < l(x0 (a − 1)) as well as Assumption (28) (in the case l(x) = xp /p for p > 1 the latter assumption is again always satisfied). Similar as in the case with l(x) = x we start off with an example where benchmark and num´eraire are chosen to coincide. Example 5 [Example 3 continued] Benchmark Λ and num´eraire N are assumed to be wealth processes according to an identical portfolio πΛ = πN . With these choices Λ = x0 N holds. Each pair (τ, ξ) ∈ G 0 satifies E(H0 (τ )Λ(τ )ξ(τ )) ≤ x0 /a and E(l(ax0 (1 − ξ(τ )))) ≤ β. As in Example 2, Jensen’s inequality yields E(1 − ξ(τ )) ≤ l−1 (β)/(ax0 ). Following the proof of Proposition 1, it can be seen that inf

(τ,ξ)∈G 0

E(ζ(τ )) ≥ (1 −

l−1 (β) l−1 (β) l−1 (β) l−1 (β) ) ln(a − )+ ln( )>0 ax0 x0 ax0 x0 (a − 1)

holds for the value of Problem 10 in this special case.

8

RISK MEASURED AS EXPECTED WEIGHTED SHORTFALL

19

For general benchmarks and num´eraires the analysis of Problem 10 is carried out by the duality method of Section 6. To apply Proposition 3 the existence of a pair (τ, ξ) ∈ J with E(H0 (τ )Λ(τ )ξ(τ )) < xa0 and E(l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))) < β is again assumed. The application of Proposition 3 implies that instead of Problem 10 a dual procedure can be analyzed. Lemma 3 allows again for a reduction of this dual procedure to a problem of optimal stopping. Subsequently, I(y) := (l0 )−1 (y) denotes the inverse of the strictly increasing derivative of the loss function l. Lemma 4 For µ ∈ R2+ with µ1 , µ2 > 0 and (t, ω) ∈ R+ × Ω define ξµ(3) (t, ω)

:= 1 − 1 ∧

µ1 H0 (t, ω)N (t, ω)) I( aµ 2

aN (t, ω)−1 Λ(t, ω)

.

(41)

Then for each ξ ∈ L([0, 1]) and a.a. (t, ω) ∈ R+ × Ω µ1 H0 (t, ω)Λ(t, ω)ξ(t, ω) + µ2 l(aN (t, ω)−1 Λ(t, ω)(1 − ξ(t, ω))) ≥ µ1 H0 (t, ω)Λ(t, ω)ξµ(3) (t, ω) + µ2 l(aN (t, ω)−1 Λ(t, ω)(1 − ξµ(3) (t, ω))) holds. Proof: For fixed (t, ω) define the function h for all z ∈ (−∞, 1] as h(z) := µ1 H0 (t, ω)Λ(t, ω)z + µ2 l(aN (t, ω)−1 Λ(t, ω)(1 − z)). Obviously, 0

h (z0 ) = 0 ⇔ z0 = 1 −

µ1 I( aµ H0 (t, ω)N (t, ω)) 2

aN (t, ω)−1 Λ(t, ω)

follows. As a simple consequence of the strict convexity of l and the positivity of N −1 Λ the restricted function h|[0,1] takes its minimum at z0 = 1 − 1 ∧

µ1 H0 (t, ω)N (t, ω)) I( aµ 2

aN (t, ω)−1 Λ(t, ω)

which yields the assertion.  For a general loss function l the analogue of Problem 1 with a fixed time horizon is the concept efficient hedging (cf. [9]), where the investor’s aim is to minimize the risk of not being able to pay for a contingent claim at a

8

RISK MEASURED AS EXPECTED WEIGHTED SHORTFALL

20

specified date. In [9] this claim is modified by a suitable Neyman-PearsonTest such that hedging the modified claim minimizes the expected weighted (3) shortfall. The process ξµ of Lemma (4) at a fixed date T ∈ R+ coincides with the optimal test derived in [9], Theorem 5.1, p. 125. Hence, the process (3) ξµ can be seen as a dynamic version of the optimal test function in the case of a fixed time horizon. An application of Proposition 3 and Lemmas 3 and 4 yields inf

(τ,ξ)∈G 0

E(ζ(τ )) =

sup

inf

E(ζ(τ ) + κ(µ)

µ∈R2+ (τ,ξ)∈J

+µ1 H0 (τ )Λ(τ )(1 − 1 ∧

µ1 H0 (τ )N (τ )) I( aµ 2

+µ2 l(aN (τ )−1 Λ(τ ) ∧ I(

aN (τ )−1 Λ(τ ) µ1 H0 (τ )N (τ ))) aµ2

)

(42)

where I(0) := 0 and I(∞) := ∞ is set. For µ ∈ R2+ we introduce the process L3 (t, µ) = g(H0 (t)N (t), N (t)−1 Λ(t), ζ(t); µ) for t ∈ R+ , µ ∈ R2+ with 1 µ1 z1 µ1 z1 g(z1 , z2 , z3 ; µ) := κ(µ) + z3 + µ1 z1 (z2 − z2 ∧ I( )) + µ2 l(az2 ∧ I( )). a aµ2 aµ2 The first step of the dual to Problem 10 is a problem of optimal stopping for the process L3 (·, µ) which is solved by τ = 0 in case µ1 = 0 or µ2 = 0. The general analysis of this stopping problem is again restricted to the Markov case. Therefore, for the remainder of Section 8 all coefficients as well as benchmark and num´eraire portfolio are assumed to be constant, furthermore, x0 = 1 is set. Proposition 6 Assume P(ζ(∞) = ∞) = 1. For each µ ∈ R2+ with µ1 , µ2 > 0 there exists an a.s. finite stopping time for the process L3 (·, µ). This is of the form τ3 (µ) := inf{t ∈ R+ | (H0 (t)N (t), N (t)−1 Λ(t))∗ ∈ S3 (µ)}

(43)

for a stopping region S3 (µ) ⊆ R2+ . For this region (2)

S3 (z2 ; µ) := {z1 ∈ R+ | (z1 , z2 ) ∈ S3 (µ)} =

[0, A(3) (z2 ; µ)] ∪ [B (3) (z2 ; µ), ∞)

(3) 2 with 0 ≤ A(3) (z2 ; µ) ≤ l0 (az2 ) aµ µ1 ≤ B (z2 ; µ) < ∞ holds.

(44) (45)

8

RISK MEASURED AS EXPECTED WEIGHTED SHORTFALL

21

Proof: See Appendix A. 

Example 6 The benchmark is chosen to coincide with the num´eraire. Furthermore, P(ζ(∞) = ∞) = 1 is assumed. With these assumptions, Proposition 6 is applicable and the optimal stopping time τ3 (µ) can be completely characterized. Since the second component of (H0 N, Λ/N ) is constant the optimal stopping time simplifies to τ3 (µ) = inf{t ∈ R+ | H0 (t)Λ(t) ∈ / (A(3) (1; µ), B (3) (1; µ))} (3) 2 for some 0 ≤ A(3) (1; µ) ≤ l0 (a) aµ µ1 ≤ B (1; µ) < ∞. Proposition 3 implies that

inf

(τ,ξ)∈G 0

E(ζ(τ )) =

inf (τ,ξ)∈J

E(L3 (τ, µ ˜))

(46)

holds for some µ ˜ ∈ R2+ . From Example 5 it follows that µ ˜1 , µ ˜2 > 0 must hold (3) 0 since the value of Problem 1 strictly exceeds 0. If (τ3 (˜ µ), ξµ˜ ) ∈ J holds, inf

(τ,ξ)∈G 0

E(ζ(τ )) = E(L3 (τ3 (˜ µ), µ ˜))

follows. The inequality τ3 (˜ µ) > 0 implies A(3) (1; µ ˜) < l0 (a)a˜ µ2 /˜ µ1 < B (3) (1; µ ˜) µ ˜1 µ ˜1 (3) (3) which is equivalent to I( a˜µ2 A (1; µ ˜)) < a < I( a˜µ2 B (1; µ ˜)). Hence (cf. Lemma 4) ( µ ˜1 (3) ˜))/a if H0 (τ3 (˜ µ))Λ(τ3 (˜ µ)) = A(3) (1; µ ˜), 1 − I( a˜ (3) µ2 A (1; µ ξµ˜ (τ3 (˜ µ)) = 0 else. The portfolio π enabling the investor to reach the target risk β within the time period [0, [ τ3 (˜ µ)]] is the hedging strategy of the τ3 (˜ µ)-option B := (3) 1,π aΛ(τ3 (˜ µ))ξµ˜ (τ3 (˜ µ)). The wealth X at the time τ3 (˜ µ) only takes two values which are given in the following table in dependence of the value the process H0 Λ takes at τ3 (˜ µ). µ ˜1 (3) ˜)) > 0 it can be concluded that X 1,π (τ3 (˜ µ)) < Due to I( a˜µ2 A (1; µ aΛ(τ3 (˜ µ)) always holds. Although the portfolio π allows for reaching the target risk β, the aspired multiple of the benchmark is never reached.

Value of H0 (τ3 (˜ µ))Λ(τ3 (˜ µ))

Value of wealth X 1,π (τ3 (˜ µ))

H0 (τ3 (˜ µ))Λ(τ3 (˜ µ)) = A(3) (1; µ ˜)

µ ˜1 (3) X 1,π (τ3 (˜ µ)) = Λ(τ3 (˜ µ))(a − I( a˜ ˜))) µ2 A (1; µ

H0 (τ3 (˜ µ))Λ(τ3 (˜ µ)) = B (3) (1; µ ˜)

X 1,π (τ3 (˜ µ)) = 0

A

PROOFS OF PROPOSITIONS 1, 5 AND 6

22

Tab. 2. Structure of the terminal wealth for Problem 1 in case benchmark = num´eraire.

A

Proofs of Propositions 1, 5 and 6

Proof: (Proposition 1) The following proof is motivated by the proof of Theorem 1.2.8 in [12], pp. 19f. Without loss of generality, G 0 6= ∅ can be assumed. Let (τ, ξ) ∈ G 0 be ) ξ(τ )) ∈ (0, 1). a pair with c1 := E(1 − ξ(τ )) ∈ (0, 1) and c2 := E( H0 (τx)Λ(τ 0 Now, on (Ω, F(τ ), P) two probability measures Q1 , Q2 are defined by Q1 (A) := (1 − c1 )−1 E(1A ξ(τ )) and Q2 (A) := c−1 1 E(1A (1 − ξ(τ ))). With EQi denoting the expectation under Qi and Jensen’s inequality E(ζ(τ )) − ln(x0 ) = (1 − c1 )EQ1 (− ln(H0 (τ )Λ(τ ))) + c1 EQ2 (− ln(H0 (τ )Λ(τ ))) ≥ (1 − c1 ) ln(

1 − c1 )+ E(H0 (τ )Λ(τ )ξ(τ ))

c1 ln(

c1 ) E(H0 (τ )Λ(τ )(1 − ξ(τ ))

follows. Due to (10) we have E(H0 (τ )Λ(τ )) ≤ x0 and hence E(H0 (τ )Λ(τ )(1− ξ(τ ))) ≤ x0 (1 − c2 ). This leads to E(ζ(τ )) ≥ (1 − c1 ) ln(

c1 1 − c1 ) + c1 ln( ). c2 1 − c2

The mapping (u, v) 7→ (1 − u) ln(

u 1−u ) + u ln( ) v 1−v

is strictly decreasing in both arguments on the set {(u, v) | u, v > 0, u + v ≤ 1}. For each pair (τ, ξ) ∈ G 0 c2 ≤ 1/a holds. Furthermore, c1 = E(1 − ξ(τ )) ≤ E(l(1 − ξ(τ ))) = E(l(N (τ )−1 (1 − ξ(τ ))aΛ(τ ))) ≤ β can be seen easily such that E(ζ(τ )) ≥ (1 − β) ln(

1−β 1 a

) + β ln(

β ) 1 − a1

A

PROOFS OF PROPOSITIONS 1, 5 AND 6

23

follows. Taking the infimum on the left hand side yields the assertion. The lower bound for the value of Problem 1 is strictly positive since β+1/a < 1.  Proof: (Proposition 5) Since the process L2 (·, µ) is bounded from below with continuous paths and limt→∞ L2 (t, µ) = ∞ holds a.s. due to P(ζ(∞) = ∞) = 1 Proposition 4 ensures the existence of an a.s. finite optimal stopping time. Furthermore L2 (·, µ) is a function of a Markov process such that according to [26], Theorem 3.5.3, p. 68, the optimal stopping time is the first entry time into the stopping region of the form {z ∈ R2+ × R | f ∗ (z1 , z2 , z3 ; µ) < f (z1 , z2 , z3 ; µ)}C with f ∗ (z1 , z2 , z3 ; µ) :=

=

inf E(f (z1 H0 (τ )N (τ ), z2

τ ∈T

Λ(τ ) , z3 + ζ(τ ); µ)) N (τ )

κ(µ) + z3 + inf E(ζ(τ ) + z2 τ ∈T

Λ(τ ) (µ1 z1 H0 (τ )N (τ ) ∧ µ2 a)). N (τ )

For z ∈ R2+ × R we have f ∗ (z1 , z2 , z3 ; µ) < f (z1 , z2 , z3 ; µ) ⇔ inf E(ζ(τ ) + z2 τ ∈T

Λ(τ ) (µ1 z1 H0 (τ )N (τ ) ∧ µ2 a)) < z2 (µ1 z1 ∧ µ2 a), N (τ )

such that the stopping region simplifies to a subset of R2+ namely S2 (µ) := {z ∈ R2+ | z2 (µ1 z1 ∧ µ2 a) ≤ f˜(z1 , z2 ; µ)}

(47)

with f˜(z1 , z2 ; µ) :=

inf E(ζ(τ ) + z2 N (τ )−1 Λ(τ )(µ1 z1 H0 (τ )N (τ ) ∧ µ2 a)).

τ ∈T

The first entry time into this region coincides with τ2 (µ) as defined in (38). Although this set can not be given explicitly it is possible to state properties (1) (2) of the z1 - resp. z2 -sections S2 (z1 ; µ) resp. S2 (z2 ; µ). The method to derive these properties resembles ideas of [21], Section VI-5., pp. 135ff. For fixed τ ∈ T the mappings z1 7→ E(ζ(τ ) + z2 N (τ )−1 Λ(τ )(µ1 z1 H0 (τ )N (τ ) ∧ µ2 a)) , (z2 ∈ R+ ) resp.

z2 7→ E(ζ(τ ) + z2 N (τ )−1 Λ(τ )(µ1 z1 H0 (τ )N (τ ) ∧ µ2 a)) , (z1 ∈ R+ )

A

PROOFS OF PROPOSITIONS 1, 5 AND 6

24

are increasing and concave. Hence f˜ is increasing and concave in each component as well. Furthermore, for all z ∈ R2+ we have f˜(z1 , z2 ; µ) ≤ z2 (µ1 z1 ∧ µ2 a). For each z1 > 0 the set of all z2 ∈ R+ satisfying z2 (µ1 z1 ∧ µ2 a) ≤ ˜ f (z1 , z2 ; µ) is either of the form [0, A(1) (z1 ; µ)] for some A(1) (z1 ; µ) ∈ R+ or equal to R+ . Thus (1)

(1)

S2 (z1 ; µ) = [0, A(1) (z1 ; µ)] for A(1) (z1 ; µ) ≥ 0 or S2 (z1 ; µ) = R+ . For fixed z2 > 0 the same method yields (2) S2 (z2 ; µ) = {z1 ∈ R+ | z2 (µ1 ∧ µ2 az1 ) ≤ f˜(z1 , z2 ; µ)}

= [0, A(2) (z2 ; µ)] ∪ [B (2) (z2 ; µ), ∞) for 0 ≤ A(2) (z2 ; µ) ≤

µ2 a µ1

≤ B (2) (z2 ; µ) < ∞ if µ1 > 0 or R+ else. 

Proof: (Proposition 6) Similar as in the proof of Proposition 5 it can be concluded that for µ ∈ R2+ with µ1 , µ2 > 0 the stopping region for the process L3 (·, µ) is of the form {z ∈ R2+ × R | g ∗ (z1 , z2 , z3 ; µ) < g(z1 , z2 , z3 ; µ)}C with g ∗ (z1 , z2 , z3 ; µ) :=

=

inf E(g(z1 H0 (τ )N (τ ), z2

τ ∈T

Λ(τ ) , z3 + ζ(τ ); µ)) N (τ )

κ(µ) + z3 + inf E(ζ(τ ) + gˆ(z1 H0 (τ )N (τ ), z2 τ ∈T

Λ(τ ) ; µ)) N (τ )

and the function gˆ : R2+ → R defined by 1 µ1 z1 µ1 z1 gˆ(z1 , z2 ; µ) := µ1 z1 (z2 − z2 ∧ I( )) + µ2 l(az2 ∧ I( )). a aµ2 aµ2 This stopping region simplifies to a subset of R2+ which takes the form S3 (µ) := {z ∈ R2+ | gˆ(z1 , z2 ; µ) ≤ g˜(z1 , z2 ; µ)} with g˜(z1 , z2 ; µ) := inf E(ζ(τ ) + gˆ(z1 H0 (τ )N (τ ), z2 N (τ )−1 Λ(τ ); µ)) τ ∈T

(48)

A

PROOFS OF PROPOSITIONS 1, 5 AND 6

25

Hence the optimal stopping time has the structure as defined in (43). It can easily be seen (cf. [2], Lemma 5.4.4, p. 130) that the function g1 : R+ → R defined as g1 (z1 ) := gˆ(z1 , z2 ; µ) for fixed z2 ∈ R+ is increasing and concave. Furthermore, the function g2 : R+ → R defined as g2 (z2 ) := gˆ(z1 , z2 ; µ) for z1 ∈ R+ is increasing and convex. Hence, the mapping z1 7→ E(ζ(τ ) + gˆ(z1 H0 (τ )N (τ ), z2 N (τ )−1 Λ(τ ); µ)) is increasing and concave whereas the function z2 7→ E(ζ(τ ) + gˆ(z1 H0 (τ )N (τ ), z2 N (τ )−1 Λ(τ ); µ)) is increasing and convex. Thus, it can be concluded that g˜ is increasing and concave in the first component. It is however not possible to make a similar statement for the second component since the infimum of convex functions is not convex in general. As in the proof of Proposition 5 for fixed z2 > 0 one obtains (2)

S3 (z2 ; µ) = {z1 ∈ R+ | gˆ(z1 , z2 ; µ) ≤ g˜(z1 , z2 ; µ)} = [0, A(3) (z2 ; µ)] ∪ [B (3) (z2 ; µ), ∞) (3) 2 for 0 ≤ A(3) (z2 ; µ) ≤ l0 (az2 ) aµ µ1 ≤ B (z2 ; µ) < ∞. 

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