Optimal Cash ManagementControl for Processes with

two-sided JumpsNora Muler

Universidad Torcuato Di Tella (UTDT), Buenos AiresEAJ 2016, Lyon

Coauthor: Pablo Azcue (UTDT)

1

Consider a continuous time cash management problem of a firm where theuncontrolled money stock Xt is a compound Poisson process with two-sidedjumps and a negative drift m;

Xt x mt

Negative Jumps

∑i1

Nt1

−Ui1

Positive Jumps

∑j1

Nt2

Uj2.

We assume:

1 The arrival Poisson processes Nt1 corresponding to the negative jumps

and Nt2corresponding to positive jumps are independent of each other with

intensity 1 and 2 respectively.2 The sizes of the negative and positive jumps are iid random variables with

distributions F1 and F2 independent of each other as well.

A Brownian motion can also be added to the uncontrolled process. We considerin this presentation, for simplicity sake, the one without Brownian motion.

2

In this optimization problem the aim is to minimize theexpected discounted cost: The money managercontinuously monitors the cashflow and at any time he/shecan increase or decrease the amount of cash.

The cost has two components: the opportunity cost ofholding cash and the cost coming from decreasing andincreasing the amount of cash.

We consider the case that both the cost of increasing anddecreasing the amount of cash have proportional and fixedcomponents; the fixed components lead to an impulsecontrol problem.

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The control strategy should be adapted and consists on both the times andamounts of downward and upward adjustments of cash: d,u,∘ d n

d,Zndn≥0 , n

dare the times and Zn

d the amount of downwardadjustments.

∘ u nu,Zn

un≥0 and nu

are the times and Znu the amounts of upward

adjustments .

The corresponding controlled process is:

Xt Xt −

Downward Adjustments

∑n1

In

d≤tZnd

Upward Adjustments

∑n1

In

u≤tZnu.

We assume that it is mandatory to inject cash in order that Xt ≥ 0 for all t.

Let us call Kd0 0 and Kd

1 ≥ 0 the fixed and proportional cost coefficient ofdownward adjustment and Ku

0 0 and Ku1 ≥ 0 the ones of upward

adjustments.

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Let 0 be the discount rate. The expected discounted cost has two terms.

1. The total expected discounted opportunity cost of holding cash:

Hx Ex0

e−taXt

dt, where a 0.

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1. The total expected discounted opportunity cost Hx of holding cash:

Hx Ex 0

e−taXt

dt, where a 0.

2. The total expected discounted cost Ax coming from adjustments:An upward adjustment Zn

d of cash at time nd a cost Kd

0 Kd1Zn

d.A downward adjustment Zn

u of cash at time nu a cost Ku

0 Ku1Zn

u.

Ax Ex

Discounted cost of downward adjustments

∑n1

e−n

dKd

0 Kd1Zn

d

Discounted cost of upward adjustments

∑n1 e−n

uKu

0 Ku1Zn

u

So the value function of an strategy is

Cx Hx Ax.And our aim is to find Cx inf Cx and the optimal strategy ∗(if it exists).

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Properties of Cx:∘ Cx ≤ Kd

1x K0 (linear growth condition) and it is Lipschitz.∘ It is a viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB)

equation:

min LCx, min0≤w≤x

Kd1w Kd

0Cx − w−Cx,minz≥0Ku

1z Ku0Cz x − Cx 0,

where

LCxmC ′x −1 2 Cx 1 0

Cx − dF1 2 0

Cx dF2 ax

We will see that C ′ could not exists at some points.

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HJB equation and the optimal strategy.If the current cash reserve x satisfies the equality

min0≤w≤x

Kd1w Kd

0Cx − w − Cx 0,

the optimal strategy would be a downward adjustment.

Calling

Zdarg min0≤w≤x

Kd1w Kd

0Cx − w

the cash reserve after the adjustment becomes x − Zd paying Kd1ZdKd

0 as atransaction cost.

We also have,

x−Zd

xC ′s − Kd

1ds Kd0

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If the current cash reserve x satisfies

minz≥0Ku

1z Ku0Cz x − Cx 0,

the optimal strategy would be an upward adjustment.

Calling

Zuarg min0≤zKu

1z Ku0Cz x.

the cash reserve after the adjustment becomes x Zu paying Ku1ZuKu

0 as atransaction cost.

We also have,

x

xZu

C ′s Ku1 −Ku

0

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Otherwise, if the current cash reserve x satisfies

LCx 0,

where

LCx

Infinitesimal Generator of the Compound-Poisson Process with drift

mC ′x −1 2Cx 1 0

Cx − dF1 2 0

Cx dF2

Discounted Term

−Cx

Opportunity Cost Termax

then nothing should be done (non-intervention region).

So: the HJB equation suggests the optimal strategy.

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Important Results:

Cx is the largest viscosity solution satisfying the lineargrowth condition (characterization result).

If there is a cost function of an admissible strategy Cxthat is a viscosity solution of the HJB equation, then is theoptimal strategy and Cx Cx. (Verification Result).

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What we expect from the structure of the optimal strategy?

One of the simplest candidates for optimal strategies arethe so-called impulse band strategies (IBS) that arecharacterized by:∘ Two trigger levels Sd and Su .∘ Two targets levels sd and su, where Sd sd ≥ su Su.

Whenever the cash level is greater than Sd it isinstantaneously decreased to sd and whenever the cashlevel is less than Su the cash amount is instantaneouslyincreased to su.

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𝑆𝑢=0

𝑠𝑢

𝑠𝑑

𝑆𝑑

Upward Random Jump

Controlled Downward Adjustment 𝑍1𝑑

Downward Random Jump

Controlled Upward Adjustment 𝑍1

𝑢

Controlled Upward Adjustment 𝑍2𝑢

Upward Random Jump Trigger

Target

Target

Trigger

Simulated Trajectory under an Impulse Band Strategy (𝑆𝑑, 𝑠𝑑, 𝑠𝑢, 𝑆𝑢)

x

Results about the optimal strategy The optimal strategy exists and it depends only on the current cashflow. There is always a unique upward trigger Su 0 and target su 0. Su is

always zero because there is a positive opportunity cost of holding cashand because the required minimum level is zero.

If Kd1≥ a/: there is no downward triggers; it is never optimal to make a

downward adjustment, too expensive!. If Kd

1 a/ there are (finite) downward targets sdi and triggers Sd

i . Thedistance between consecutive downward targets and triggers is never lessthan Kd

0/a/ − Kd1.

If Kd1 0 there is a unique downward target sd (that corresponds to the

point that the minimum of Cx is attained). This implies a uniquedownward trigger? NO!, we will see an example.

We are thinking conditions under which the optimal strategy is an impulse bandstrategies (IBS).

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Numerical Example with an IBS optimal strategy

Parameters of the example:* Downward and upward jumps with exponential distribution functions:

F1x 1 − e−0.05x, F2x 1 − e−0.02x.* Discount factor: 0.06.* drift: m −365.* Poisson intensity: 1 30,2 10.* a 0.05,Ku

1 0.2, Kd1 0.3, Ku

0 Kd0 1 .

We have that:∘ The optimal strategy is an impulse band strategies (IBS) with

Su 0, su 372.3, sd 5551.7,Sd 6172.4

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0 2000 4000 6000

1800

2000

2200

2400

2600

2800

No action Upward adjustment

Downward adjustment Target 𝑠𝑢 Target 𝑠𝑑

C(x)

Trigger 𝑆𝑑 Trigger 𝑆𝑢 = 0

Target 𝑠𝑢

(𝐶′ 𝑠 + 𝐾1𝑢𝑠𝑢0

)=-𝐾0𝑢

(𝐶′ 𝑠 − 𝐾1𝑑𝑆𝑑𝑠𝑑

)=𝐾0𝑑

Target 𝑠𝑑

-𝐾1𝑢

𝐾1𝑑

C’(x)

Jump of C´ at triggers

Jump of C´ at triggers

Numerical Example with two downward triggersParameters of the example:* Downward jumps with distribution function F1x Ix≥1 and exponential

distribution for the upward jumps F2x 1 − e−x.* Discount factor: 0.1.* drift: m −1.* Poisson intensity: 1 10,2 1/100.* a 20 ,Ku

1 Kd1 0 (no proportional cost for cash adjustment), Ku

0 Kd0 1.

In this case the Optimal Strategy is not IBS! It has a unique "upward" trigger Su 0, and a unique targetsu 0.2 (this is always the case). It has a unique downward target sd 0.2 (that is the minimum of

Cx because the proportional cost is zero) BUT two downward triggersSd

1 1.0 and Sd2 1.3

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0.5 0.5 1.0 1.5 2.0

142.4

142.6

142.8

143.0

143.2

C(x)

TARGET: 𝑠𝑑=𝑠𝑢

No Action No Action Upward Adj.

Down. Adj. Downward Adjustment

𝑻𝒓𝒊𝒈𝒈. 𝑺𝟏𝒅 𝑻𝒓𝒊𝒈𝒈. 𝑺𝟐𝒅

Integral=0

Integral=𝐾0𝑑

Integral=-𝐾0𝑢

C’(x)

TARGET

Jump of C´ at upward trigger

Jumps of C´ at triggers

Oksendal and Sulem . Applied Stochastic Control of Jump Diffusion.Springer: 2004 (first edition) and 2011 (second edition).

Baumol (1952). The transactions demand for cash: an inventory theoreticmodel. Quarterly Journal of Economics.

Bar-Ilan, Perry and Stadje (2004). A generalized impulse control model ofcash management. Journal of Economic Dynamics & Control.

Benkherouf and Bensoussan (2009) .Optimality of an (s; S) policy withcompound Poisson and diffusion demands: a quasi-variational inequalitiesapproach. SIAM J. Control Optim.

Bayraktar, Emmerling and Menaldi (2013). On the Impulse Control ofJump Diffusions. SIAM J. Control Optim.

Alvarez and Lippi (2013). The demand of liquid assets with uncertainlumpy expenditures. Journal of Monetary Economics.

Yamazaki (2016). Inventory Control For Spectrally Positive Levy Demandprocesses .To be published in Mathematics of Operations Research.

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MERCI BEAUCOUP!

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A function u : J → R is a viscosity subsolution at x ∈ J if anycontinuously differentiable function : J → R withx ux and such that u − reaches the maximum at xsatisfies

min Lx, min0≤w≤x

Kd1w Kd

0ux − w−ux,minz≥0Ku

1z Ku0uz x − ux ≤ 0,

u

y

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A function u : J → R is a viscosity supersolution at x ∈ J ifany continuously differentiable function : J → R withx ux and such that u − reaches the minimum at xsatisfies

min Lx, min0≤w≤x

Kd1w Kd

0ux − w−ux,minz≥0Ku

1z Ku0uz x − ux ≥ 0,

u

j

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If a function u : J → R is both a subsolution and asupersolution at x ∈ J it is called a viscosity solution at x

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