Download pdf - The proof of Theorem 2.1 will be “guess and verify” and consist of … · 2012. 11. 28. · lim t →∞ Ew[e−δtv(wt ... then the same is true for all points along the ray

The proof of Theorem 2.1 will be “guess and verify” and consistof three steps:

1 formulate and solve HJB-equation,

2 an auxiliary technical result,

3 find appropriate supermartingale (needs Ito’s formula).





Step ??:

When π and c are constants, then the generator of wt acts onv ∈ C2 by

(Ac,πv)(w) =(

(r + (α− r)π)w − c)

v′(w) +1

2w2π2σ2v′′(w).





Step ??:

When π and c are constants, then the generator of wt acts onv ∈ C2 by

(Ac,πv)(w) =(

(r + (α− r)π)w − c)

v′(w) +1

2w2π2σ2v′′(w).

The HJB-equation reads

maxc,π

{(Ac,πv)(w) +cγ

γ− δv(w)} = 0 for all w > 0.

The maxima are achieved at

c = v′(w)−1

1−γ and π =−βv′(w)

wσv′′(w)


c = v′(w)−1


wσv′′(w)

and hence the HJB-equation is equivalent to

rwv′ −β2

2

(v′)2

v′′+

1− γ

γ(v′)−γ/(1−γ) − δv = 0.


c = v′(w)−1


wσv′′(w)

and hence the HJB-equation is equivalent to

rwv′ −β2

2

(v′)2

v′′+

1− γ

γ(v′)−γ/(1−γ) − δv = 0.

It is easy to see that v(w) = γ−1Cγ−1wγ is solution of thisdifferential equation.

Step ??:

Let (ct, πt) ∈ U be an arbitrary policy and define the process

xt :=

∫ t

0σπu dzu.

Then wt is given explicitly (proof: Ito’s formula) by

wt =

(

w −

∫ t

0csfs ds

)

E(xt) exp

(

rt+

∫ t

0(α− r)πu du

)

where E is the stochastic exponential of xt and

fs := exp

(

− rs−

∫ s

0

(

(α− r)πu −1

2σ2π2u

)

du−

∫ s

0σπu dzu

)

.

Step ??:

Let (ct, πt) ∈ U be an arbitrary policy and define the process

xt :=

∫ t

0σπu dzu.

Then wt is given explicitly (proof: Ito’s formula) by

wt =

(

w −

∫ t

0csfs ds

)

E(xt) exp

(

rt+

∫ t

0(α− r)πu du

)

where E is the stochastic exponential of xt and

fs := exp

(

− rs−

∫ s

0

(

(α− r)πu −1

2σ2π2u

)

du−

∫ s

0σπu dzu

)

.

⇒ wt has moments of all orders by Holder’s inequality andsince πt is bounded.

Step ??:

Define for any policy (ct, πt) the process

Mt :=

∫ t

0e−δsu(cs) ds + e−δtv(wt),

where v(w) = γ−1Cγ−1wγ .

Step ??:


Mt :=

∫ t


where v(w) = γ−1Cγ−1wγ . Ito’s formula then shows that

Mt = M0 +

∫ t

0e−δs

(

(Ac,πv)(ws) +cγsγ

− δv(ws))

ds

+σCγ−1

∫ t

0e−δsπsw

γs dzs.

Step ??:


Mt :=

∫ t


where v(w) = γ−1Cγ−1wγ . Ito’s formula then shows that

Mt = M0 +

∫ t

0e−δs

(

(Ac,πv)(ws) +cγsγ

− δv(ws))

ds

+σCγ−1

∫ t

0e−δsπsw

γs dzs.

⇒ Mt is a supermartingale and if (ct, πt) = (c∗t , π∗t ) it is a

martingale. Thus,

v(w) =M0 ≥ Ew[Mt] = Ew[

∫ t

0e−δsu(cs) ds] + Ew[e

−δtv(wt)].

The proof is complete if we can show that

limt→∞

Ew[e−δtv(wt)] = 0

for any (c, π) ∈ U .


limt→∞


for any (c, π) ∈ U . To this end, observe that by Ito’s formula wemay write

e−δtwγt = wγ

0E(γxt) exp

(∫ t

0as ds

)

,

where

as = γ(

r + (α− r)πs −csws

−1

2(1− γ)π2sσ

2)

− δ.


limt→∞


for any (c, π) ∈ U . To this end, observe that by Ito’s formula wemay write

e−δtwγt = wγ

0E(γxt) exp

(∫ t

0as ds

)

,

where

as = γ(

r + (α− r)πs −csws

−1

2(1− γ)π2sσ

2)

− δ.

Since as ≤ −(1− γ)C the claim follows. This completes theproof.

Guessing solution for problem with transaction costs.

Ansatz: try L and U absolutely continuous with boundedderivatives, that is,

Lt =

∫ t

0ls ds, Ut =

∫ t

0us ds, 0 ≤ ls, us ≤ κ

Guessing solution for problem with transaction costs.

Ansatz: try L and U absolutely continuous with boundedderivatives, that is,

Lt =

∫ t

0ls ds, Ut =

∫ t

0us ds, 0 ≤ ls, us ≤ κ

The HJB-equation reads

maxc,l,u

{

1

2σ2y2vyy + rxvx + αyvy +

1

γcγ − cvx

(

− (1 + λ)vx + vy)

l +(

(1− µ)vx − vy)

u− δv

}

= 0.

Since vx and vy are positive (extra wealth gives increasedutility), we see that the maxima are attained as follows:

c = (vx)1/(γ−1),

l =

{

κ, if vy ≥ (1 + λ)vx,

0, if vy < (1 + λ)vx,

u =

{

0, if vy > (1− µ)vx,

κ, if vy ≤ (1− µ)vx.

Since vx and vy are positive (extra wealth gives increasedutility), we see that the maxima are attained as follows:

c = (vx)1/(γ−1),

l =

{

κ, if vy ≥ (1 + λ)vx,

0, if vy < (1 + λ)vx,

u =

{

0, if vy > (1− µ)vx,

κ, if vy ≤ (1− µ)vx.

This indicates that the optimal transaction policies are“bang-bang”: buying and selling either take place at maximumrate or not at all, and the solvency region splits into threeregions

B, the region in which stocks are bought,

S, the region in which stocks are sold,

NT the region where no transactions take place.

Let us analyse the boundary

vy = (1 + λ)vx

between S and NT (a similar argument applies for theboundary between NT and B). To this end assume that v ∈ C1

and that it is homothetic which implies that

vx(ρx, ρy) = ργ−1vx(x, y).

Let us analyse the boundary

vy = (1 + λ)vx

between S and NT (a similar argument applies for theboundary between NT and B). To this end assume that v ∈ C1

and that it is homothetic which implies that

vx(ρx, ρy) = ργ−1vx(x, y).

It follows that if vy(x, y) = (1 + λ)vx(x, y) for some point (x, y),then the same is true for all points along the ray through (x, y).

Heuristic argument suggests so far:

boundaries between transaction and no-transaction regionsare straight lines through the origin,



in the transaction regions , transactions take place atmaximum, i.e. infinite, speed, which implies that theinvestor will make an instantaneous finite transaction tothe boundary of NT ,




the finite transaction in S or B moves the portfolio downor up a line of slope −1/(1 − µ) or −1/(1 + λ).





after the initial transaction, all further transactions musttake place at the boundaries, and this suggests a “localtime” type of transaction policy,





after the initial transaction, all further transactions musttake place at the boundaries, and this suggests a “localtime” type of transaction policy,

meanwhile, consumption takes place at rate (vx)1/(γ−1).

In NT the value function v(x, y) satisfies the HJB-equationwith l = u = 0:

maxc

{

1

2σ2y2vyy + (rx− c)vx + αyvy +

1

γcγ − δv

}

= 0,

i.e.,

1


1− γ

γv−γ/(1−γ)x − δv = 0.

In NT the value function v(x, y) satisfies the HJB-equationwith l = u = 0:

maxc

{

1


1

γcγ − δv

}

= 0,

i.e.,

1


1− γ

γv−γ/(1−γ)x − δv = 0.

The final step now consists of reducing this equation to anequation in one variable. In order to do so, define

ψ(x) := v(x, 1).

By the homothetic property it follows that v(x, y) = yγψ(x/y).

If our conjectured optimal policy is correct then v is constantalong lines of slope (1− µ)−1 in S and along lines of slope(1 + λ)−1 in B, and this implies by homothetic property that

ψ(x) =1

γ(x+ 1− µ)γ , x ≤ x0,

ψ(x) =1

γ(x+ 1 + λ)γ , x ≥ xT ,

for some constants A,B and x0 and xT as in the picture.

If our conjectured optimal policy is correct then v is constantalong lines of slope (1− µ)−1 in S and along lines of slope(1 + λ)−1 in B, and this implies by homothetic property that

ψ(x) =1

γ(x+ 1− µ)γ , x ≤ x0,

ψ(x) =1

γ(x+ 1 + λ)γ , x ≥ xT ,

for some constants A,B and x0 and xT as in the picture. Usingthe homothetic property again, one can show that ψ satisfies forx ∈ [x0, xT ],

β3ψ′′(x) + β2xψ

′(x) + β1ψ(x) +1− γ

γ(ψ′(x))−γ/(1−γ) = 0,

where β1 = −12σ

2γ(1− γ + αγ)− δ, β2 = σ2(1− γ) + r − α,β3 =

12σ

2.

Theorem (4.1, follows from [?])

Take 0 < x0 < xT and let NT be the closed wedge shown in the

picture, with upper and lower boundaries ∂S, ∂B respectively.

Let c : NT → [0,∞) be any Lipschitz continuous function and

let (x, y) ∈ NT . Then there exists a unique process s0, s1 and

continuous increasing processes L,U such that for

t < τ = inf{t ≥ 0 : (s0(t), s1(t)) = 0}

ds0(t) =(

rs0(t)− c(s0(t), s1(t)))

dt

−(1 + λ)dLt + (1− µ)dUt, s0(0) = x,

ds1(t) = αs1(t)dt+ σs1(t)dzt − dUt, s1(0) = y,

Lt =

∫ t

01{(s0(ξ),s1(ξ))∈∂B}dLξ,

Ut =

∫ t

01{(s0(ξ),s1(ξ))∈∂S}dUξ.

The process ct := c(s0(t), s1(t)) satisfies condition (2.1)(i).

Define the set of policies that do not involve short selling:

U′ = {(c, L, U) ∈ U : (s0(t), s1(t)) ∈ S

′µ for all t ≥ 0},

where S ′µ = {(x, y) ∈ R

2 : y ≥ 0 and x+ (1− µ)y ≥ 0}.

Theorem (4.2, proof in [?])

Le 0 < γ < 1 and assume Condition A holds. Suppose there are

constants A,B, x0, xT and a function ψ : [−1(1 − µ),∞) → R

such that

0 < x0 < xT <∞,

ψ is C2 and ψ′(x) > 0 for all x,

ψ(x) =1

γA(x+ 1− µ)γ for x ≤ x0,

β3ψ′′(x) + β2xψ

′(x) + β1ψ(x)

+1− γ

γ(ψ′(x))−γ/(1−γ) = 0 for x ∈ [x0, xT ],

ψ(x) =1

γB(x+ 1 + λ)γ for x ≥ xT .

Theorem

Let NT denote the closed wedge

{(x, y) ∈ R2+ : x−1

T ≤ yx−1 ≤ x−10 }

and let B and S denote the regions below and above NT as in

the picture. For (x, y) ∈ NT \ {(0, 0)} define

c∗(x, y) = yψ′(x/y)−1/(1−γ).

Let c∗t = c∗(s0(t), s1(t)) where (s0, s1, L∗, U∗) is the unique

solution of (4.1) with c := c∗. Then the policy

(c∗(t), L∗(t), U∗(t)) is optimal in the class U ′ for any initial

endowment (x, y) ∈ NT . If (x, s) /∈ NT then an immediate

transaction to the closest point in NT followed by application of

this policy is optimal in U ′. The maximal expected utility is

v(x, s) = yγψ(x/y).

Davis, M. H. A. and Norman, A. R. (1990). Portfolioselection with transaction costs. Mathematics of Operations

Research. 15 676–713.

Tanaka, H. (1978). Stochastic differential equations withreflecting boundary condition in convex regions. Hiroshima

Math. J. 9 163–177.