Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Portfolio Turnpikes for Incomplete Markets

Paolo Guasoni1,2

Kostas Kardaras1 Scott Robertson3 Hao Xing4

1Boston University

2Dublin City University

3Carnegie Mellon University

4London School of Economics

Princeton ORFE SeminarSeptember 22nd , 2010

Outline

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

• Results:Abstract, Classic, and Explicit Turnpikes.

• Consequences:Risk Sensitive Control and Intertemporal Hedging.

Outline

Portfolio Turnpikes

• An investor with utility U...

• ...invests optimally for a terminal wealth at horizon T .• As the horizon increases, today’s optimal portfolio...• ...converges? To what?• Turnpike theorems: (under some conditions)

as T increases, the optimal portfolio for U is close to the optimalportfolio for either power or log utility (CRRA).

• The power depends on the properties of U at large wealth levels.• Different papers find different conditions.• Conditions involve preferences and market structure.• Literature:

conditions neither more nor less general that others.

Portfolio Turnpikes

• An investor with utility U...• ...invests optimally for a terminal wealth at horizon T .

• As the horizon increases, today’s optimal portfolio...• ...converges? To what?• Turnpike theorems: (under some conditions)

Portfolio Turnpikes

• An investor with utility U...• ...invests optimally for a terminal wealth at horizon T .• As the horizon increases, today’s optimal portfolio...

• ...converges? To what?• Turnpike theorems: (under some conditions)

Portfolio Turnpikes

• An investor with utility U...• ...invests optimally for a terminal wealth at horizon T .• As the horizon increases, today’s optimal portfolio...• ...converges? To what?

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

Portfolio Turnpikes

• An investor with utility U...• ...invests optimally for a terminal wealth at horizon T .• As the horizon increases, today’s optimal portfolio...• ...converges? To what?• Turnpike theorems: (under some conditions)

Portfolio Turnpikes

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

• Different papers find different conditions.• Conditions involve preferences and market structure.• Literature:

Portfolio Turnpikes

• The power depends on the properties of U at large wealth levels.• Different papers find different conditions.

• Conditions involve preferences and market structure.• Literature:

Portfolio Turnpikes

• The power depends on the properties of U at large wealth levels.• Different papers find different conditions.• Conditions involve preferences and market structure.

• Literature:conditions neither more nor less general that others.

Portfolio Turnpikes

Literature

Mossin (1968) JB IID Disc −U ′/U ′′ = ax + bLeland (1972) Proc IID Disc −U ′/U ′′ = ax + f (x)

Ross (1974) JFE IID Disc U sum of powersHakansson (1974) JFE IID Disc (x−a)p

p −A(p)<U(x)<(x+a)p

p +A(p)Huberman Ross (1983) EC IID Disc p>0, bounded below, U’ reg. var.

Cox Huang (1992) JEDC IID Compl Cont |U ′−1 − A1y−1/b| ≤ A2y−a

Jin (1997) JEDC IID Compl Cont |U ′−1 − A1y−1/b| ≤ A2y−a

Dybvig et al. (1999) RFS Compl Cont U′0(x)U′1(x)

Huang Zariph. (1999) FS IID Compl Cont U′0(x)xp−1 → K ,U(0) = 0

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

• Disparate conditions on utility functions.

Literature

Mossin (1968) JB IID Disc −U ′/U ′′ = ax + bLeland (1972) Proc IID Disc −U ′/U ′′ = ax + f (x)

Ross (1974) JFE IID Disc U sum of powersHakansson (1974) JFE IID Disc (x−a)p

p −A(p)<U(x)<(x+a)p

p +A(p)Huberman Ross (1983) EC IID Disc p>0, bounded below, U’ reg. var.

Cox Huang (1992) JEDC IID Compl Cont |U ′−1 − A1y−1/b| ≤ A2y−a

Jin (1997) JEDC IID Compl Cont |U ′−1 − A1y−1/b| ≤ A2y−a

Dybvig et al. (1999) RFS Compl Cont U′0(x)U′1(x)

Huang Zariph. (1999) FS IID Compl Cont U′0(x)xp−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.

• Use condition on marginal utility ratio for U.• Abstract turnpike:

convergence of portfolios under myopic probabilities PT .• Holds under minimal conditions on market structure.• Classic turnpike:

convergence of portfolios under physical probability.• Abstract turnpike implies classic turnpike if myopic IID optimum.• More results for diffusion model with many assets but one state.• Classic turnpike for diffusions.• Explicit turnpike:

limit portfolio is solution to ergodic HJB equation.

This Paper

• Relax assumptions on market completeness and IID returns.• Use condition on marginal utility ratio for U.

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

• Holds under minimal conditions on market structure.• Classic turnpike:

This Paper

• Relax assumptions on market completeness and IID returns.• Use condition on marginal utility ratio for U.• Abstract turnpike:

convergence of portfolios under myopic probabilities PT .

• Holds under minimal conditions on market structure.• Classic turnpike:

This Paper

convergence of portfolios under myopic probabilities PT .• Holds under minimal conditions on market structure.

• Classic turnpike:convergence of portfolios under physical probability.

• Abstract turnpike implies classic turnpike if myopic IID optimum.• More results for diffusion model with many assets but one state.• Classic turnpike for diffusions.• Explicit turnpike:

This Paper

convergence of portfolios under physical probability.

• Abstract turnpike implies classic turnpike if myopic IID optimum.• More results for diffusion model with many assets but one state.• Classic turnpike for diffusions.• Explicit turnpike:

This Paper

convergence of portfolios under physical probability.• Abstract turnpike implies classic turnpike if myopic IID optimum.

• More results for diffusion model with many assets but one state.• Classic turnpike for diffusions.• Explicit turnpike:

This Paper

convergence of portfolios under physical probability.• Abstract turnpike implies classic turnpike if myopic IID optimum.• More results for diffusion model with many assets but one state.

• Classic turnpike for diffusions.• Explicit turnpike:

This Paper

convergence of portfolios under physical probability.• Abstract turnpike implies classic turnpike if myopic IID optimum.• More results for diffusion model with many assets but one state.• Classic turnpike for diffusions.

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

This Paper

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

• Marginal Utility Ratio measures how close they are:

R(x) :=U ′(x)

xp−1 , x > 0

Assumption

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

limx↑∞

R(x) = 1, (CONV)

0 < lim infx↓0

R(x), 0 6= p < 1, (LB-0)

lim supx↓0

R(x) <∞, p < 1. (UB-0)

Preferences• Two investors. One with utility U, the other with CRRA 1− p.• Marginal Utility Ratio measures how close they are:

R(x) :=U ′(x)

xp−1 , x > 0

Assumption

limx↑∞

R(x) = 1, (CONV)

0 < lim infx↓0

R(x), 0 6= p < 1, (LB-0)

lim supx↓0

R(x) <∞, p < 1. (UB-0)

Preferences• Two investors. One with utility U, the other with CRRA 1− p.• Marginal Utility Ratio measures how close they are:

R(x) :=U ′(x)

xp−1 , x > 0

Assumption

limx↑∞

R(x) = 1, (CONV)

0 < lim infx↓0

R(x), 0 6= p < 1, (LB-0)

lim supx↓0

R(x) <∞, p < 1. (UB-0)

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

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

Assumption

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

i) X0 = 1 for all X ∈ X T ;ii) X T contains a strictly positive X (Xt > 0 a.s. for all t ∈ [0,T ]);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 theprocess X ′′ that compounds X with X ′ at τ :

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

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

Market Structure• Investors choose from a common set X T of wealth processes.• (Ω, (Ft )t∈[0,T ],FT ,P) filtered probability space. Usual conditions.

Assumption

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

Assumption

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

Assumption

For T > 0, X T is a set of nonnegative semimartingales such that:i) X0 = 1 for all X ∈ X T ;

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

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

Assumption

For T > 0, X T is a set of nonnegative semimartingales such that:i) X0 = 1 for all X ∈ X T ;ii) X T contains a strictly positive X (Xt > 0 a.s. for all t ∈ [0,T ]);

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

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

Assumption

For T > 0, X T is a set of nonnegative semimartingales such that:i) X0 = 1 for all X ∈ X T ;ii) X T contains a strictly positive X (Xt > 0 a.s. for all t ∈ [0,T ]);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 positiveand τ is a [0,T ]-valued stopping time, then X T contains theprocess X ′′ that compounds X with X ′ at τ :

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

Assumption

For T > 0, X T is a set of nonnegative semimartingales such that:i) X0 = 1 for all X ∈ X T ;ii) X T contains a strictly positive X (Xt > 0 a.s. for all t ∈ [0,T ]);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

X ′′ = X I[[0,τ [[+X ′XτX ′τ

I[[τ,T ]] =

Well Posedness and Growth

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

• Maximization problems:

u0,T = supX∈X T

EP [X p/p] , u1,T = supX∈X T

EP [U (X )] .

• Well posedness:

Assumption

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

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

u0,T = supX∈X T

EP [U (X )] .

• Well posedness:

Assumption

u0,T = supX∈X T

EP [U (X )] .

• Well posedness:

Assumption

u0,T = supX∈X T

EP [U (X )] .

• Well posedness:

Assumption

Central Objects

• Ratio of optimal wealth processes and its stochastic logarithm:

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

rTv−, for u ∈ [0,T ].

• rT0 = 1 (investors have same initial capital).

• myopic probabilities(PT )

T≥0:

(X 0,T

X 0,TT

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

Central Objects

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

rTv−, for u ∈ [0,T ].

T≥0:

(X 0,T

X 0,TT

Central Objects

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

rTv−, for u ∈ [0,T ].

T≥0:

(X 0,T

X 0,TT

Central Objects

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

rTv−, for u ∈ [0,T ].

T≥0:

(X 0,T

X 0,TT

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

• Optimal payoff for xp/p under P equal to log optimal under P.

Central Objects

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

rTv−, for u ∈ [0,T ].

T≥0:

(X 0,T

X 0,TT

Growth

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

Assumption

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

limT→∞

PT (X T ≥ N) = 1 for any N > 0. (GROWTH)

• Assumption trivially satisfied with a positive safe rate.• Holds in more generality.• But note PT , not P!

Growth

Assumption

limT→∞

Growth

Assumption

limT→∞

• Assumption trivially satisfied with a positive safe rate.

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

Growth

Assumption

limT→∞

• Assumption trivially satisfied with a positive safe rate.• Holds in more generality.

• But note PT , not P!

Growth

Assumption

limT→∞

Abstract Turnpike

Theorem (Abstract Turnpike)

Let previous assumptions hold. Then, for any ε > 0,

a) limT→∞ PT(

supu∈[0,T ]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ PT ([ΠT ,ΠT ]T ≥ ε

• For log utility PT ≡ P, hence convergence holds under P.• For a familiar diffusion dSu/Su = µu du + σ′udWu, [ΠT ,ΠT ]

measures distance between portfolios π1,T and π0,T :[ΠT ,ΠT

∫ ·0

(π1,T

u − π0,Tu

)′Σu

(π1,T

u − π0,Tu

Abstract Turnpike

a) limT→∞ PT(

supu∈[0,T ]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

∫ ·0

(π1,T

u − π0,Tu

)′Σu

(π1,T

u − π0,Tu

Abstract Turnpike

a) limT→∞ PT(

supu∈[0,T ]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

∫ ·0

(π1,T

u − π0,Tu

)′Σu

(π1,T

u − π0,Tu

Abstract Turnpike

a) limT→∞ PT(

supu∈[0,T ]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

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

• For a familiar diffusion dSu/Su = µu du + σ′udWu, [ΠT ,ΠT ]measures distance between portfolios π1,T and π0,T :[

ΠT ,ΠT]·

∫ ·0

(π1,T

u − π0,Tu

)′Σu

(π1,T

u − π0,Tu

Abstract Turnpike

a) limT→∞ PT(

supu∈[0,T ]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

∫ ·0

(π1,T

u − π0,Tu

)′Σu

(π1,T

u − π0,Tu

IID Myopic Turnpike

Corollary (IID Myopic Turnpike)

If, in addition to previous assumptions:

i) X Tt = X S

t ≡ Xt a.s. for all t ≤ S,T (myopic optimality);ii) Xt and XT/Xt are independent for all t ≤ T (independent returns).

then, for any ε > 0 and t ≥ 0:

a) limT→∞ P(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

• If optimal wealth myopic with IID returns, abstract implies classic.• In practice, if assets have IID returns, optimal portfolio myopic.• For example, Levy processes.

IID Myopic Turnpike

If, in addition to previous assumptions:i) X T

t = X St ≡ Xt a.s. for all t ≤ S,T (myopic optimality);

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

then, for any ε > 0 and t ≥ 0:

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

IID Myopic Turnpike

ii) Xt and XT/Xt are independent for all t ≤ T (independent returns).then, for any ε > 0 and t ≥ 0:

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

IID Myopic Turnpike

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

IID Myopic Turnpike

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

IID Myopic Turnpike

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

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

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

IID Myopic Turnpike

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

• If optimal wealth myopic with IID returns, abstract implies classic.• In practice, if assets have IID returns, optimal portfolio myopic.

• For example, Levy processes.

IID Myopic Turnpike

a) limT→∞ P(

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

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

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

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

= r(Yt ) dt + dR it , 1 ≤ i ≤ d ,

• Cumulative excess return R = (R1, · · · ,Rd )′ follows diffusion:

dR it = µi(Yt ) dt +

n∑j=1

σij(Yt ) dZ jt , 1 ≤ i ≤ d ,

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

= r(Yt ) dt + dR it , 1 ≤ i ≤ d ,

n∑j=1

σij(Yt ) dZ jt , 1 ≤ i ≤ d ,

= r(Yt ) dt + dR it , 1 ≤ i ≤ d ,

n∑j=1

σij(Yt ) dZ jt , 1 ≤ i ≤ d ,

= r(Yt ) dt + dR it , 1 ≤ i ≤ d ,

n∑j=1

σij(Yt ) dZ jt , 1 ≤ i ≤ d ,

Regularity ConditionsAssumption

Set Σ = σσ′, A = a2, and Υ = σρa. r ∈ Cγ(E ,R), b ∈ C1,γ(E ,R),µ ∈ C1,γ(E ,Rd ), A ∈ C2,γ(E ,R), Σ ∈ C2,γ(E ,Rd×d ), andΥ ∈ C2,γ(E ,Rd ). For all y ∈ E , Σ is positive and A is strictly positive.

Assumption

(Σ ΥΥ′ A

). Infinitesimal generator of (R,Y ):

L = 12∑d+1

i,j=1 Aij(ξ) ∂2

∂ξi∂ξj+∑d+1

i=1 bi(ξ) ∂∂ξi

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

δ (pr(y)− q2µ′Σ−1µ(y)) for y ∈ E , q := p

p−1 , and δ := 11−qρ′ρ .

Assumption

(Σ ΥΥ′ A

L = 12∑d+1

i,j=1 Aij(ξ) ∂2

∂ξi∂ξj+∑d+1

Assumption

p−1 , and δ := 11−qρ′ρ .

Assumption

(Σ ΥΥ′ A

L = 12∑d+1

i,j=1 Aij(ξ) ∂2

∂ξi∂ξj+∑d+1

Assumption

p−1 , and δ := 11−qρ′ρ .

HJB Assumption (finite horizon)Assumption

There exist (vT (y , t))T>0 and v(y) such that:

i) vT > 0, vT ∈ C1,2((0,T )× E), and solves reduced HJB equation:

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

where L := 12A ∂2

yy + B ∂y and B := b − qΥ′Σ−1µ.

ii) The finite horizon martingale problems (PT )T>0 are well posed:

dRt = 1

(µ+ δΥ

vTy (y ,t)

vT (y ,t)

)dt + σ dZt

(B + AvT

y (y ,t)vT (y ,t)

)dt + a dWt

where L := 12A ∂2

dRt = 1

(µ+ δΥ

vTy (y ,t)

vT (y ,t)

)dt + σ dZt

(B + AvT

y (y ,t)vT (y ,t)

)dt + a dWt

where L := 12A ∂2

dRt = 1

(µ+ δΥ

vTy (y ,t)

vT (y ,t)

)dt + σ dZt

(B + AvT

y (y ,t)vT (y ,t)

)dt + a dWt

HJB Assumption (long run)Assumption

iii) v > 0, v ∈ C2(E), and (v , λc) solves the ergodic HJB equation:

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

iv) The long run martingale problem (P) is well posed:

dRt = 11−p

(µ+ δΥ

vy (y)ˆv(y)

)dt + σ dZt

dYt =(

B + A vy (y)v(y)

)dt + a dWt

v) Setting m(y) := 1A(y) exp

(∫ yy0

2B(z)A(z) dz

), for some y0 ∈ E :

∫ y0α

1v2Am(y)dy =

∫ βy0

1v2Am(y)dy =∞,

∫ βα v2 m(y) dy ,

∫ βα v m(y) dy <∞,

dRt = 11−p

(µ+ δΥ

vy (y)ˆv(y)

)dt + σ dZt

dYt =(

B + A vy (y)v(y)

)dt + a dWt

(∫ yy0

2B(z)A(z) dz

∫ y0α

1v2Am(y)dy =

∫ βy0

1v2Am(y)dy =∞,

∫ βα v m(y) dy <∞,

dRt = 11−p

(µ+ δΥ

vy (y)ˆv(y)

)dt + σ dZt

dYt =(

B + A vy (y)v(y)

)dt + a dWt

(∫ yy0

2B(z)A(z) dz

∫ y0α

1v2Am(y)dy =

∫ βy0

1v2Am(y)dy =∞,

∫ βα v m(y) dy <∞,

dRt = 11−p

(µ+ δΥ

vy (y)ˆv(y)

)dt + σ dZt

dYt =(

B + A vy (y)v(y)

)dt + a dWt

(∫ yy0

2B(z)A(z) dz

∫ y0α

1v2Am(y)dy =

∫ βy0

1v2Am(y)dy =∞,

∫ βα v m(y) dy <∞,

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dP|Ft =

d PdP|Ft .

• Proposition allows to replace PT with P in abstract turnpike.• Classic turnpike theorem follows from equivalence of P and P.

Theorem (Classic Turnpike for Diffusions)

Let previous assumptions hold. Then, for 0 6= p < 1 and any ε, t > 0:

a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,b) limT→∞ P

([ΠT ,ΠT

]t ≥ ε

• Proposition

limT→∞

dP|Ft =

d PdP|Ft .

∣∣rTu − 1

∣∣ ≥ ε) = 0,b) limT→∞ P

([ΠT ,ΠT

]t ≥ ε

• Proposition

limT→∞

dP|Ft =

d PdP|Ft .

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

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

∣∣rTu − 1

∣∣ ≥ ε) = 0,b) limT→∞ P

([ΠT ,ΠT

]t ≥ ε

• Proposition

limT→∞

dP|Ft =

d PdP|Ft .

∣∣rTu − 1

∣∣ ≥ ε) = 0,b) limT→∞ P

([ΠT ,ΠT

]t ≥ ε

• Proposition

limT→∞

dP|Ft =

d PdP|Ft .

∣∣rTu − 1

∣∣ ≥ ε) = 0,b) limT→∞ P

([ΠT ,ΠT

]t ≥ ε

• Proposition

limT→∞

dP|Ft =

d PdP|Ft .

Let previous assumptions hold. Then, for 0 6= p < 1 and any ε, t > 0:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT]

t ≥ ε)

• Proposition

limT→∞

dP|Ft =

d PdP|Ft .

Let previous assumptions hold. Then, for 0 6= p < 1 and any ε, t > 0:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,b) limT→∞ P

([ΠT ,ΠT

]t ≥ ε

Classic vs. Explicit

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

• Theorem says they come close for large horizons...• ...but neither one has explicit solution. Portfolio for xp/p is:

πT (t , y) =1

1− pΣ−1

(µ+ δΥ

vTy (t , y)

vT (t , y)

• Explicit turnpike:compare portfolio for U with horizon T to long run portfolio:

π(y) =1

1− pΣ−1

(µ+ δΥ

vy (y)

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

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

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

πT (t , y) =1

1− pΣ−1

(µ+ δΥ

vTy (t , y)

vT (t , y)

π(y) =1

1− pΣ−1

(µ+ δΥ

vy (y)

πT (t , y) =1

1− pΣ−1

(µ+ δΥ

vTy (t , y)

vT (t , y)

π(y) =1

1− pΣ−1

(µ+ δΥ

vy (y)

πT (t , y) =1

1− pΣ−1

(µ+ δΥ

vTy (t , y)

vT (t , y)

π(y) =1

1− pΣ−1

(µ+ δΥ

vy (y)

πT (t , y) =1

1− pΣ−1

(µ+ δΥ

vTy (t , y)

vT (t , y)

π(y) =1

1− pΣ−1

(µ+ δΥ

vy (y)

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

• X wealth process of long-run portfolio π.

Theorem (Explicit Turnpike)

Under the previous assumptions, for any ε, t > 0 and 0 6= p < 1:

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

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

Explicit Turnpike

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

Explicit Turnpike

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

Explicit Turnpike

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

Under the previous assumptions, for any ε, t > 0 and 0 6= p < 1:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

Explicit Turnpike

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

Explicit Turnpike

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

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

• Finite horizon portfolios converge to long run portfolio.

Explicit Turnpike

rTu :=

X 1,Tu

Xu, ΠT

rTv−

, for u ∈ [0,T ],

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

Conclusion• Portfolio turnpikes:

at long horizons, optimal portfolios approach those of CRRA class.

• Abstract turnpike:optimal portfolios for U and xp/p at horizon T become close.Under the myopic probabilities.

• Classic turnpike:optimal portfolios for U and xp/p at horizon T become close.Under the physical probability P.

• Abstract implies classic if optimal wealth myopic with IDD returns.• Class of diffusion models:

classic turnpike without myopic portfolios.Intertemporal hedging components converge.

• Explicit turnpike: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.

at long horizons, optimal portfolios approach those of CRRA class.• Abstract turnpike:

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

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

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

Abstract, Classic, and Explicit Turnpikes

Economy & Finance

Transcript of Abstract, Classic, and Explicit Turnpikes

The Romantic Period (1800-1870). 1803 Louisiana Purchase—westward expansion Transportation—canals, turnpikes, railroads, steamboats Gold Rush of 1849.

Explicit Dynamics Chapter 6 Explicit Meshing

EXPLICIT REALIZATION OF

Turnpikes, Canals, And Railroads

ESSENTIAL QUESTION: Improving the Roads An effort to improve transportation between states States chartered companies to operate turnpikes- roads that.

Knowledge – Explicit, implicit and tacit: Philosophical ...Explicit knowledge is knowledge that the knower can make explicit by means of a verbal statement: ‘Someone has explicit

Explicit STR in ANSYS Workbench · 2011. 8. 12. · Explicit STR in ANSYS Workbench Page 1. Explicit STR in ANSYS Workbench Page 2. Explicit STR in ANSYS Workbench Page 3

Indian Turnpikes (BOT Toll Roads)

Explicit Dynamics Explicit Meshing

explicit dynamics

Explicit Instruction

Explicit Substitutions Calculi with Explicit Eta Rules - UnBayala/venturaayalakamareddine.pdf · Explicit Substitutions Calculi with Explicit Eta Rules ... 2 Explicit Substitutions

Making Values and Ethics Explicit - Brian Williamsonbrianwilliamson.id.au/cit/Making Values and Ethics Explicit - ADCA.pdf · Making Values and Ethics Explicit ... implementation

WRED-Explicit Congestion Notification · WRED-Explicit Congestion Notification Currently,thecongestioncontrolandavoidancealgorithmsforTransmissionControlProtocol(TCP)are ...

Explicit Music

ANSYS Explicit Dynamics - Highvec.netcae.highvec.net/files/4_Tutorials/ANSYS TUTORIALS...Explicit Dynamics: Meshing ANSYS Explicit Dynamics Explicit Dynamics: Meshing 1-2 ANSYS, Inc.

EDM Explicit)

Making Mathematical Practices Explicit in Discourse: Sarah ...€¦ · MAKING MATHEMATICAL PRACTICES EXPLICIT IN DISCOURSE Abstract Making mathematical practices explicit while still

Lesson Title: Recursive and Explicit Formulas Course ...€¦ · Lesson Title: Recursive and Explicit Formulas Course: ... Recursive and Explicit Formulas Course: ... geometric sequences

Advanced Explicit Summary