Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Portfolio Turnpikes for Incomplete Markets

Paolo Guasoni1,2

Kostas Kardaras1 Scott Robertson3 Hao Xing4

1 Boston University
2 Dublin City University
3 Carnegie Mellon University
4 London School of Economics

Princeton ORFE Seminar
September 22nd , 2010


Outline

• Turnpike Theorems:
for Long Horizons, use Constant Relative Risk Aversion.


Outline

• Results:
Abstract, Classic, and Explicit Turnpikes.


Outline

• Results:
Abstract, Classic, and Explicit Turnpikes.
• Consequences:
Risk Sensitive Control and Intertemporal Hedging.


Portfolio Turnpikes

• An investor with utility U...


Portfolio Turnpikes

• ...invests optimally for a terminal wealth at horizon T .


Portfolio Turnpikes

• As the horizon increases, today’s optimal portfolio...


Portfolio Turnpikes

• ...converges? To what?


Portfolio Turnpikes

• Turnpike theorems: (under some conditions)
as T increases, the optimal portfolio for U is close to the optimal
portfolio for either power or log utility (CRRA).


Portfolio Turnpikes

• The power depends on the properties of U at large wealth levels.


Portfolio Turnpikes

• Different papers ﬁnd different conditions.


Portfolio Turnpikes

• Conditions involve preferences and market structure.


Portfolio Turnpikes

• Conditions involve preferences and market structure.
• Literature:
conditions neither more nor less general that others.


Literature

Mossin (1968) JB IID Disc −U /U = ax + b
Leland (1972) Proc IID Disc −U /U = ax + f (x)
Ross (1974) JFE IID Disc U sum of powers
(x−a)p p
Hakansson (1974) JFE IID Disc p
−A(p)<U(x)< (x+a) +A(p)
p
Huberman Ross (1983) EC IID Disc p>0, bounded below, U’ reg. var
Cox Huang (1992) JEDC IID Compl Cont |U −1 − A1 y −1/b | ≤ A2 y −a
Jin (1997) JEDC IID Compl Cont |U −1 − A1 y −1/b | ≤ A2 y −a
U0 (x)
Dybvig et al. (1999) RFS Compl Cont U1 (x)
→K
U0 (x)
Huang Zariph. (1999) FS IID Compl Cont x p−1
→ K , U(0) = 0

• Either IID returns, or market completeness, or both.


Literature

Mossin (1968) JB IID Disc −U /U = ax + b
Leland (1972) Proc IID Disc −U /U = ax + f (x)
Ross (1974) JFE IID Disc U sum of powers
(x−a)p p
Hakansson (1974) JFE IID Disc p
−A(p)<U(x)< (x+a) +A(p)
p
Huberman Ross (1983) EC IID Disc p>0, bounded below, U’ reg. var
Cox Huang (1992) JEDC IID Compl Cont |U −1 − A1 y −1/b | ≤ A2 y −a
Jin (1997) JEDC IID Compl Cont |U −1 − A1 y −1/b | ≤ A2 y −a
U0 (x)
Dybvig et al. (1999) RFS Compl Cont U1 (x)
→K
U0 (x)
Huang Zariph. (1999) FS IID Compl Cont x p−1
→ K , U(0) = 0

• Either IID returns, or market completeness, or both.
• Disparate conditions on utility functions.


This Paper

• Relax assumptions on market completeness and IID returns.


This Paper

• Use condition on marginal utility ratio for U.


This Paper

• Abstract turnpike:
convergence of portfolios under myopic probabilities PT .


This Paper

• Holds under minimal conditions on market structure.


This Paper

• Classic turnpike:
convergence of portfolios under physical probability.


This Paper

• Abstract turnpike implies classic turnpike if myopic IID optimum.


This Paper

• More results for diffusion model with many assets but one state.


This Paper

• Classic turnpike for diffusions.


This Paper

• Classic turnpike for diffusions.
• Explicit turnpike:
limit portfolio is solution to ergodic HJB equation.


Preferences
• Two investors. One with utility U, the other with CRRA 1 − p.


Preferences
• Marginal Utility Ratio measures how close they are:
U (x)
R(x) := , x >0
x p−1


Preferences
• Marginal Utility Ratio measures how close they are:
U (x)
R(x) := , x >0
x p−1

Assumption
U : R+ → R continuously differentiable, strictly increasing, strictly
concave, satisﬁes Inada conditions U (0) = ∞ and U (∞) = 0.
Marginal utility ratio satisﬁes:

lim R(x) = 1, (CONV)
x↑∞

0 < lim inf R(x), 0 = p < 1, (LB-0)
x↓0

lim sup R(x) < ∞, p < 1. (UB-0)
x↓0


Market Structure
• Investors choose from a common set X T of wealth processes.


Market Structure
• (Ω, (Ft )t∈[0,T ] , F T , P) ﬁltered probability space. Usual conditions.


Market Structure

Assumption
For T > 0, X T is a set of nonnegative semimartingales such that:


Market Structure

Assumption
i) X0 = 1 for all X ∈ X T ;


Market Structure

Assumption
ii) X T contains a strictly positive X (Xt > 0 a.s. for all t ∈ [0, T ]);


Market Structure

Assumption
iii) X T is convex: ((1 − α)X + αX ) ∈ X T for X , X ∈ X T , α ∈ [0, 1];


Market Structure

Assumption
iii) X T is convex: ((1 − α)X + αX ) ∈ X T for X , X ∈ X T , α ∈ [0, 1];
iv) X T stable under compounding: if X , X ∈ X T with X strictly positive
and τ is a [0, T ]-valued stopping time, then X T contains the
process X that compounds X with X at τ :

Xτ Xt (ω), if t ∈ [0, τ (ω)[
X = X I[[0,τ [[ +X I =
Xτ [[τ,T ]] (Xτ (ω)/Xτ (ω)) Xt (ω), if t ∈ [τ (ω), T ]


Well Posedness and Growth

• Use index 0 for the CRRA investor, and index 1 for investor with U.



• Maximization problems:

u 0,T = sup EP [X p /p] , u 1,T = sup EP [U (X )] .
X ∈X T X ∈X T




X ∈X T X ∈X T

• Well posedness:




X ∈X T X ∈X T

• Well posedness:

Assumption
−∞ < u i,T < ∞ and optimal payoffs X i,T exist for all T > 0 and i = 0, 1.


Central Objects
• Ratio of optimal wealth processes and its stochastic logarithm:

1,T u T
T Xu drv
ru := 0,T
, ΠT :=
u T
, for u ∈ [0, T ].
Xu 0 rv −


Central Objects

1,T u T
T Xu drv
ru := 0,T
, ΠT :=
u T
, for u ∈ [0, T ].
Xu 0 rv −
T
• r0 = 1 (investors have same initial capital).


Central Objects

1,T u T
T Xu drv
ru := 0,T
, ΠT :=
u T
, for u ∈ [0, T ].
Xu 0 rv −
T
• myopic probabilities PT T ≥0 :

p
0,T
dPT XT
= p
.
dP 0,T
EP XT


Central Objects

1,T u T
T Xu drv
ru := 0,T
, ΠT :=
u T
, for u ∈ [0, T ].
Xu 0 rv −
T

p
0,T
dPT XT
= p
.
dP 0,T
EP XT

• Myopic probabilities PT boil down to P for log utility.


Central Objects

1,T u T
T Xu drv
ru := 0,T
, ΠT :=
u T
, for u ∈ [0, T ].
Xu 0 rv −
T

p
0,T
dPT XT
= p
.
dP 0,T
EP XT

• Myopic probabilities PT boil down to P for log utility.
• Optimal payoff for x p /p under P equal to log optimal under P.


Growth

• Growth. As horizon increases, increasingly large payoffs available:


Growth


Assumption
ˆ ˆ
There exists a family (X T )T ≥0 such that X T ∈ X T and:

ˆ
lim PT (X T ≥ N) = 1 for any N > 0. (GROWTH)
T →∞


Growth


Assumption
ˆ ˆ

ˆ
T →∞

• Assumption trivially satisﬁed with a positive safe rate.


Growth


Assumption
ˆ ˆ

ˆ
T →∞

• Holds in more generality.


Growth


Assumption
ˆ ˆ

ˆ
T →∞

• Holds in more generality.
• But note PT , not P!


Abstract Turnpike

Theorem (Abstract Turnpike)
Let previous assumptions hold. Then, for any > 0,


Abstract Turnpike

T
a) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,


Abstract Turnpike

T
a) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,
b) limT →∞ PT ΠT , Π T T
≥ =0


Abstract Turnpike

T
a) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,
≥ =0

• For log utility PT ≡ P, hence convergence holds under P.


Abstract Turnpike

T
a) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,
≥ =0

• For log utility PT ≡ P, hence convergence holds under P.
• For a familiar diffusion dSu /Su = µu du + σu dWu , [ΠT , ΠT ]
measures distance between portfolios π 1,T and π 0,T :
·
1,T 0,T 1,T 0,T
ΠT , ΠT = πu − πu Σu πu − πu du,
· 0


IID Myopic Turnpike

Corollary (IID Myopic Turnpike)
If, in addition to previous assumptions:

then, for any > 0 and t ≥ 0:


IID Myopic Turnpike

i) XtT = XtS ≡ Xt a.s. for all t ≤ S, T (myopic optimality);



IID Myopic Turnpike

ii) Xt and XT /Xt are independent for all t ≤ T (independent returns).


IID Myopic Turnpike

T
a) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,


IID Myopic Turnpike

T
a) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,
b) limT →∞ P Π T , ΠT t
≥ = 0.


IID Myopic Turnpike

T
a) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,
≥ = 0.

• If optimal wealth myopic with IID returns, abstract implies classic.


IID Myopic Turnpike

T
a) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,
≥ = 0.

• In practice, if assets have IID returns, optimal portfolio myopic.


IID Myopic Turnpike

T
a) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,
≥ = 0.

• In practice, if assets have IID returns, optimal portfolio myopic.
• For example, Levy processes.


Diffusion Model
• One state variable Y , with values in interval E = (α, β) ⊆ R, with
−∞ ≤ α < β ≤ ∞.

dYt = b(Yt ) dt + a(Yt ) dWt .


Diffusion Model
−∞ ≤ α < β ≤ ∞.


• Market includes safe rate r (Yt ) and d risky assets with prices:

dSti
= r (Yt ) dt + dRti , 1 ≤ i ≤ d,
Sti


Diffusion Model
−∞ ≤ α < β ≤ ∞.



dSti
= r (Yt ) dt + dRti , 1 ≤ i ≤ d,
Sti

• Cumulative excess return R = (R 1 , · · · , R d ) follows diffusion:
n
dRti = µi (Yt ) dt + σij (Yt ) dZtj , 1 ≤ i ≤ d,
j=1


Diffusion Model
−∞ ≤ α < β ≤ ∞.



dSti
= r (Yt ) dt + dRti , 1 ≤ i ≤ d,
Sti

• Cumulative excess return R = (R 1 , · · · , R d ) follows diffusion:
n
dRti = µi (Yt ) dt + σij (Yt ) dZtj , 1 ≤ i ≤ d,
j=1

• W and Z = (Z 1 , · · · , Z n ) are multivariate Wiener processes with
correlation ρ = (ρ1 , · · · , ρn ) , i.e. d Z i , W t = ρi (Yt ) dt for 1 ≤ i ≤ n.


Regularity Conditions
Assumption
Set Σ = σσ , A = a2 , and Υ = σρa. r ∈ C γ (E, R), b ∈ C 1,γ (E, R),
µ ∈ C 1,γ (E, Rd ), A ∈ C 2,γ (E, R), Σ ∈ C 2,γ (E, Rd×d ), and
Υ ∈ C 2,γ (E, Rd ). For all y ∈ E, Σ is positive and A is strictly positive.


Assumption

Assumption
˜ Σ Υ ˜ µ
A= b= . Inﬁnitesimal generator of (R, Y ):
Υ A b
2
L = 2 d+1 Aij (ξ) ∂ξ∂∂ξj + i=1 bi (ξ) ∂ξi
1
i,j=1
˜
i
d+1 ˜ ∂

Martingale problem for L well posed, in that unique solution exists.


Assumption

Assumption
˜ Σ Υ ˜ µ
A= b= . Inﬁnitesimal generator of (R, Y ):
Υ A b
2
L = 2 d+1 Aij (ξ) ∂ξ∂∂ξj + i=1 bi (ξ) ∂ξi
1
i,j=1
˜
i
d+1 ˜ ∂

Martingale problem for L well posed, in that unique solution exists.

Assumption
ρ ρ is constant (does not depend on y ), and supy ∈E c(y ) < ∞,
c(y ) := 1 (pr (y ) − q µ Σ−1 µ(y )) for y ∈ E, q := p−1 , and δ := 1−qρ ρ .
δ 2
p 1


HJB Assumption (ﬁnite horizon)
Assumption
There exist (v T (y , t))T >0 and v (y ) such that:
ˆ


Assumption
ˆ
i) v T > 0, v T ∈ C 1,2 ((0, T ) × E), and solves reduced HJB equation:

∂t v + Lv + c v = 0, (t, y ) ∈ (0, T ) × E,
v (T , y ) = 1, y ∈ E,

where L := 1 A ∂yy + B ∂y and B := b − qΥ Σ−1 µ.
2
2


Assumption
ˆ
i) v T > 0, v T ∈ C 1,2 ((0, T ) × E), and solves reduced HJB equation:

∂t v + Lv + c v = 0, (t, y ) ∈ (0, T ) × E,
v (T , y ) = 1, y ∈ E,

where L := 1 A ∂yy + B ∂y and B := b − qΥ Σ−1 µ.
2
2

ii) The ﬁnite horizon martingale problems (PT )T >0 are well posed:

T
vy (y ,t)
 dRt = 1 ˜
dt + σ d Zt
1−p µ + δΥ v T (y ,t)


T
(P ) T
.
 dYt = B + A vyT (y ,t) dt + a d Wt
 ˜
 v (y ,t)


HJB Assumption (long run)
Assumption


Assumption

iii) v > 0, v ∈ C 2 (E), and (v , λc ) solves the ergodic HJB equation:
ˆ ˆ ˆ

L v + c v = λ v, y ∈ E, for some λc ∈ R


Assumption

ˆ ˆ ˆ


ˆ
iv) The long run martingale problem (P) is well posed:

ˆ
 dRt = 1 µ + δΥ vy (y ) dt + σ d Ztˆ
1−p ˆ
ˆ
(P) v (y )
ˆ
 dYt = B + A vy (y ) dt + a d Wt
ˆ
ˆ
v (y )


Assumption

ˆ ˆ ˆ


ˆ
iv) The long run martingale problem (P) is well posed:

ˆ
 dRt = 1 µ + δΥ vy (y ) dt + σ d Ztˆ
1−p ˆ
ˆ
(P) v (y )
ˆ
 dYt = B + A vy (y ) dt + a d Wt
ˆ
ˆ
v (y )

1 y 2B(z)
v) Setting m(y ) := A(y ) exp y0 A(z) dz , for some y0 ∈ E:

y0 1 β 1 β β
α v 2 Am(y ) dy
ˆ = y0 v 2 Am(y ) dy
ˆ = ∞, α v 2 m(y ) dy ,
ˆ α
ˆ
v m(y ) dy < ∞,


Myopic Probabilities and Classic Turnpike



• Proposition
Let diffusions assumptions hold. Then, for any t ≥ 0:

dPT ˆ
dP
lim | Ft = |F .
T →∞ dP dP t



• Proposition

dPT ˆ
dP
lim | Ft = |F .
T →∞ dP dP t

ˆ
• Proposition allows to replace PT with P in abstract turnpike.



• Proposition

dPT ˆ
dP
lim | Ft = |F .
T →∞ dP dP t

ˆ
ˆ
• Classic turnpike theorem follows from equivalence of P and P.



• Proposition

dPT ˆ
dP
lim | Ft = |F .
T →∞ dP dP t

ˆ
ˆ

Theorem (Classic Turnpike for Diffusions)
Let previous assumptions hold. Then, for 0 = p < 1 and any , t > 0:



• Proposition

dPT ˆ
dP
lim | Ft = |F .
T →∞ dP dP t

ˆ
ˆ

T
a) limT →∞ P (supu∈[0,t] ru − 1 ≥ ) = 0,



• Proposition

dPT ˆ
dP
lim | Ft = |F .
T →∞ dP dP t

ˆ
ˆ

T
a) limT →∞ P (supu∈[0,t] ru − 1 ≥ ) = 0,
b) limT →∞ P ΠT , ΠT t ≥ = 0.


Classic vs. Explicit
• Abstract and Classic turnpikes:
compare portfolios for U and x p /p at ﬁnite horizon T .


• Theorem says they come close for large horizons...


• ...but neither one has explicit solution. Portfolio for x p /p is:

T
vy (t, y )
1
π T (t, y ) = Σ−1 µ + δΥ
1−p v T (t, y )



T
vy (t, y )
1
π T (t, y ) = Σ−1 µ + δΥ
1−p v T (t, y )

compare portfolio for U with horizon T to long run portfolio:

1 ˆ
vy (y )
π (y ) =
ˆ Σ−1 µ + δΥ .
1−p ˆ
v (y )



T
vy (t, y )
1
π T (t, y ) = Σ−1 µ + δΥ
1−p v T (t, y )

compare portfolio for U with horizon T to long run portfolio:

1 ˆ
vy (y )
π (y ) =
ˆ Σ−1 µ + δΥ .
1−p ˆ
v (y )

• Long run portfolio solve ergodic HJB equation. ODE, not PDE.


Explicit Turnpike
• Ratio of optimal wealth processes, and stochastic logarithms:

1,T u
Xu ˆu rT
d ˆv
rT
ˆu := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −


Explicit Turnpike

1,T u
Xu ˆu rT
d ˆv
rT
ˆu := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −

ˆ
• X wealth process of long-run portfolio π .
ˆ


Explicit Turnpike

1,T u
Xu ˆu rT
d ˆv
rT
ˆu := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −

ˆ
ˆ

Theorem (Explicit Turnpike)
Under the previous assumptions, for any , t > 0 and 0 = p < 1:


Explicit Turnpike

1,T u
Xu û rT
d ˆv
rT
û := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −

ˆ
ˆ

rT
a) limT →∞ P (supu∈[0,t] û − 1 ≥ ) = 0,


Explicit Turnpike

1,T u
Xu û rT
d ˆv
rT
û := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −

ˆ
ˆ

rT
a) limT →∞ P (supu∈[0,t] û − 1 ≥ ) = 0,
ˆ ˆ
b) limT →∞ P ΠT , ΠT ≥ = 0.
t


Explicit Turnpike

1,T u
Xu û rT
d ˆv
rT
û := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −

ˆ
ˆ

rT
a) limT →∞ P (supu∈[0,t] û − 1 ≥ ) = 0,
ˆ ˆ
b) limT →∞ P ΠT , ΠT ≥ = 0.
t

• Explicit turnpike nontrivial even for U(x) = x p /p.


Explicit Turnpike

1,T u
Xu û rT
d ˆv
rT
û := , ΠT := , for u ∈ [0, T ],
ˆ
Xu 0 rT
ˆv −

ˆ
ˆ

rT
a) limT →∞ P (supu∈[0,t] û − 1 ≥ ) = 0,
ˆ ˆ
b) limT →∞ P ΠT , ΠT ≥ = 0.
t

• Explicit turnpike nontrivial even for U(x) = x p /p.
• Finite horizon portfolios converge to long run portfolio.


Conclusion
• Portfolio turnpikes:
at long horizons, optimal portfolios approach those of CRRA class.


Conclusion
optimal portfolios for U and x p /p at horizon T become close.
Under the myopic probabilities.


Conclusion
Under the physical probability P.


Conclusion
• Abstract implies classic if optimal wealth myopic with IDD returns.


Conclusion
• Class of diffusion models:
classic turnpike without myopic portfolios.
Intertemporal hedging components converge.


Conclusion
• Class of diffusion models:
classic turnpike without myopic portfolios.
Intertemporal hedging components converge.
portfolios for U at horizon T approaches long run portfolio.
Long run portfolio has explicit solutions in several models.
Links risk-sensitive control to expected utility.

Abstract, Classic, and Explicit Turnpikes

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Abstract, Classic, and Explicit Turnpikes