abstract, classic, and explicit turnpikes

100
Problem Abstract Diffusions Portfolio Turnpikes for Incomplete Markets Paolo Guasoni 1,2 Kostas Kardaras 1 Scott Robertson 3 Hao Xing 4 1 Boston University 2 Dublin City University 3 Carnegie Mellon University 4 London School of Economics Princeton ORFE Seminar September 22 nd , 2010

Upload: guasoni

Post on 27-Jun-2015

1.413 views

Category:

Economy & Finance


1 download

TRANSCRIPT

Page 1: 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

Page 2: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Outline

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

• Results:Abstract, Classic, and Explicit Turnpikes.

• Consequences:Risk Sensitive Control and Intertemporal Hedging.

Page 3: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Outline

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

• Results:Abstract, Classic, and Explicit Turnpikes.

• Consequences:Risk Sensitive Control and Intertemporal Hedging.

Page 4: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Outline

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

• Results:Abstract, Classic, and Explicit Turnpikes.

• Consequences:Risk Sensitive Control and Intertemporal Hedging.

Page 5: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 6: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 7: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 8: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 9: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 10: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 11: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 12: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 13: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 14: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

→ K

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.

Page 15: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

→ K

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.

Page 16: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 17: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 18: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 19: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 20: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 21: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 22: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 23: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 24: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 25: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

Page 26: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

Page 27: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

Page 28: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 ]

Page 29: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 ]

Page 30: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 ]

Page 31: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 ]

Page 32: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 ]

Page 33: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 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 ]] =

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

Page 34: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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 ]

Page 35: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 36: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 37: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 38: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 39: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Central Objects

• Ratio of optimal wealth processes and its stochastic logarithm:

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

∫ u

0

drTv

rTv−, for u ∈ [0,T ].

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

• myopic probabilities(PT )

T≥0:

dPT

dP=

(X 0,T

T

)p

EP[(

X 0,TT

)p] .

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

Page 40: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Central Objects

• Ratio of optimal wealth processes and its stochastic logarithm:

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

∫ u

0

drTv

rTv−, for u ∈ [0,T ].

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

• myopic probabilities(PT )

T≥0:

dPT

dP=

(X 0,T

T

)p

EP[(

X 0,TT

)p] .

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

Page 41: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Central Objects

• Ratio of optimal wealth processes and its stochastic logarithm:

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

∫ u

0

drTv

rTv−, for u ∈ [0,T ].

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

• myopic probabilities(PT )

T≥0:

dPT

dP=

(X 0,T

T

)p

EP[(

X 0,TT

)p] .

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

Page 42: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Central Objects

• Ratio of optimal wealth processes and its stochastic logarithm:

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

∫ u

0

drTv

rTv−, for u ∈ [0,T ].

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

• myopic probabilities(PT )

T≥0:

dPT

dP=

(X 0,T

T

)p

EP[(

X 0,TT

)p] .

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

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

Page 43: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Central Objects

• Ratio of optimal wealth processes and its stochastic logarithm:

rTu :=

X 1,Tu

X 0,Tu

, ΠTu :=

∫ u

0

drTv

rTv−, for u ∈ [0,T ].

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

• myopic probabilities(PT )

T≥0:

dPT

dP=

(X 0,T

T

)p

EP[(

X 0,TT

)p] .

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

Page 44: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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!

Page 45: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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!

Page 46: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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!

Page 47: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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!

Page 48: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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!

Page 49: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

)= 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

)du,

Page 50: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

)= 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

)du,

Page 51: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

)= 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

)du,

Page 52: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

)= 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

)du,

Page 53: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

)= 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

)du,

Page 54: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

)= 0.

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

Page 55: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

Page 56: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

Page 57: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

Page 58: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

Page 59: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

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

Page 60: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

• For example, Levy processes.

Page 61: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

IID Myopic Turnpike

Corollary (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(

supu∈[0,t]∣∣rT

u − 1∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT ,ΠT ]t ≥ ε

)= 0.

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

Page 62: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

dSit

Sit

= 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.

Page 63: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

dSit

Sit

= 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.

Page 64: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

dSit

Sit

= 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.

Page 65: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

dSit

Sit

= 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.

Page 66: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

A =

(Σ ΥΥ′ A

)b =

(µb

). 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 .

Page 67: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

A =

(Σ ΥΥ′ A

)b =

(µb

). 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 .

Page 68: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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

A =

(Σ ΥΥ′ A

)b =

(µb

). 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 .

Page 69: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(PT )

dRt = 1

1−p

(µ+ δΥ

vTy (y ,t)

vT (y ,t)

)dt + σ dZt

dYt =

(B + AvT

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

)dt + a dWt

.

Page 70: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(PT )

dRt = 1

1−p

(µ+ δΥ

vTy (y ,t)

vT (y ,t)

)dt + σ dZt

dYt =

(B + AvT

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

)dt + a dWt

.

Page 71: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(PT )

dRt = 1

1−p

(µ+ δΥ

vTy (y ,t)

vT (y ,t)

)dt + σ dZt

dYt =

(B + AvT

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

)dt + a dWt

.

Page 72: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(P)

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 <∞,

Page 73: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(P)

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 <∞,

Page 74: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(P)

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 <∞,

Page 75: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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:

(P)

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 <∞,

Page 76: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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

)= 0.

Page 77: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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

)= 0.

Page 78: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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

)= 0.

Page 79: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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

)= 0.

Page 80: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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

)= 0.

Page 81: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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)

= 0.

Page 82: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Myopic Probabilities and Classic Turnpike

• Proposition

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

limT→∞

dPT

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

)= 0.

Page 83: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

v(y)

).

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

Page 84: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

v(y)

).

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

Page 85: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

v(y)

).

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

Page 86: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

v(y)

).

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

Page 87: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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)

v(y)

).

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

Page 88: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:

a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

Page 89: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:

a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

Page 90: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:

a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

Page 91: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

Page 92: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

Page 93: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

• Finite horizon portfolios converge to long run portfolio.

Page 94: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

Explicit Turnpike

• Ratio of optimal wealth processes, and stochastic logarithms:

rTu :=

X 1,Tu

Xu, ΠT

u :=

∫ u

0

drTv

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:a) limT→∞ P (supu∈[0,t]

∣∣rTu − 1

∣∣ ≥ ε) = 0,

b) limT→∞ P([

ΠT , ΠT]

t≥ ε)

= 0.

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

Page 95: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 96: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 97: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 98: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 99: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.

Page 100: Abstract, Classic, and Explicit Turnpikes

Problem Abstract Diffusions

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.