Abstract, Classic, and Explicit Turnpikes

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Abstract, Classic, and Explicit Turnpikes

  1. 1. 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
  2. 2. Problem Abstract Diffusions Outline • Turnpike Theorems: for Long Horizons, use Constant Relative Risk Aversion.
  3. 3. Problem Abstract Diffusions Outline • Turnpike Theorems: for Long Horizons, use Constant Relative Risk Aversion. • Results: Abstract, Classic, and Explicit Turnpikes.
  4. 4. 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.
  5. 5. Problem Abstract Diffusions Portfolio Turnpikes • An investor with utility U...
  6. 6. Problem Abstract Diffusions Portfolio Turnpikes • An investor with utility U... • ...invests optimally for a terminal wealth at horizon T .
  7. 7. 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...
  8. 8. 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?
  9. 9. 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 optimal portfolio for either power or log utility (CRRA).
  10. 10. 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 optimal portfolio for either power or log utility (CRRA). • The power depends on the properties of U at large wealth levels.
  11. 11. 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 optimal portfolio for either power or log utility (CRRA). • The power depends on the properties of U at large wealth levels. • Different papers find different conditions.
  12. 12. 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 optimal portfolio 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.
  13. 13. 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 optimal portfolio 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.
  14. 14. Problem Abstract Diffusions 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.
  15. 15. Problem Abstract Diffusions 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.
  16. 16. Problem Abstract Diffusions This Paper • Relax assumptions on market completeness and IID returns.
  17. 17. Problem Abstract Diffusions This Paper • Relax assumptions on market completeness and IID returns. • Use condition on marginal utility ratio for U.
  18. 18. 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 .
  19. 19. 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.
  20. 20. 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.
  21. 21. 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.
  22. 22. 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.
  23. 23. 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.
  24. 24. 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.
  25. 25. Problem Abstract Diffusions Preferences • Two investors. One with utility U, the other with CRRA 1 − p.
  26. 26. Problem Abstract Diffusions Preferences • Two investors. One with utility U, the other with CRRA 1 − p. • Marginal Utility Ratio measures how close they are: U (x) R(x) := , x >0 x p−1
  27. 27. Problem Abstract Diffusions Preferences • Two investors. One with utility U, the other with CRRA 1 − p. • Marginal Utility Ratio measures how close they are: U (x) R(x) := , x >0 x p−1AssumptionU : R+ → R continuously differentiable, strictly increasing, strictlyconcave, 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
  28. 28. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes.
  29. 29. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes. • (Ω, (Ft )t∈[0,T ] , F T , P) filtered probability space. Usual conditions.
  30. 30. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes. • (Ω, (Ft )t∈[0,T ] , F T , P) filtered probability space. Usual conditions.AssumptionFor T > 0, X T is a set of nonnegative semimartingales such that:
  31. 31. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes. • (Ω, (Ft )t∈[0,T ] , F T , P) filtered probability space. Usual conditions.AssumptionFor T > 0, X T is a set of nonnegative semimartingales such that: i) X0 = 1 for all X ∈ X T ;
  32. 32. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes. • (Ω, (Ft )t∈[0,T ] , F T , P) filtered probability space. Usual conditions.AssumptionFor 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 ]);
  33. 33. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes. • (Ω, (Ft )t∈[0,T ] , F T , P) filtered probability space. Usual conditions.AssumptionFor 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];
  34. 34. Problem Abstract Diffusions Market Structure • Investors choose from a common set X T of wealth processes. • (Ω, (Ft )t∈[0,T ] , F T , P) filtered probability space. Usual conditions.AssumptionFor 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 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 ]
  35. 35. Problem Abstract Diffusions Well Posedness and Growth • Use index 0 for the CRRA investor, and index 1 for investor with U.
  36. 36. Problem Abstract Diffusions 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
  37. 37. Problem Abstract Diffusions 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 • Well posedness:
  38. 38. Problem Abstract Diffusions 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 • Well posedness:Assumption−∞ < u i,T < ∞ and optimal payoffs X i,T exist for all T > 0 and i = 0, 1.
  39. 39. Problem Abstract Diffusions 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 −
  40. 40. Problem Abstract Diffusions 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 − T • r0 = 1 (investors have same initial capital).
  41. 41. Problem Abstract Diffusions 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 − T • r0 = 1 (investors have same initial capital). • myopic probabilities PT T ≥0 : p 0,T dPT XT = p . dP 0,T EP XT
  42. 42. Problem Abstract Diffusions 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 − T • r0 = 1 (investors have same initial capital). • myopic probabilities PT T ≥0 : p 0,T dPT XT = p . dP 0,T EP XT • Myopic probabilities PT boil down to P for log utility.
  43. 43. Problem Abstract Diffusions 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 − T • r0 = 1 (investors have same initial capital). • myopic probabilities PT T ≥0 : 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.
  44. 44. Problem Abstract Diffusions Growth • Growth. As horizon increases, increasingly large payoffs available:
  45. 45. Problem Abstract Diffusions Growth • Growth. As horizon increases, increasingly large payoffs available:Assumption ˆ ˆThere 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 →∞
  46. 46. Problem Abstract Diffusions Growth • Growth. As horizon increases, increasingly large payoffs available:Assumption ˆ ˆThere 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 →∞ • Assumption trivially satisfied with a positive safe rate.
  47. 47. Problem Abstract Diffusions Growth • Growth. As horizon increases, increasingly large payoffs available:Assumption ˆ ˆThere 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 →∞ • Assumption trivially satisfied with a positive safe rate. • Holds in more generality.
  48. 48. Problem Abstract Diffusions Growth • Growth. As horizon increases, increasingly large payoffs available:Assumption ˆ ˆThere 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 →∞ • Assumption trivially satisfied with a positive safe rate. • Holds in more generality. • But note PT , not P!
  49. 49. Problem Abstract Diffusions Abstract TurnpikeTheorem (Abstract Turnpike)Let previous assumptions hold. Then, for any > 0,
  50. 50. Problem Abstract Diffusions Abstract TurnpikeTheorem (Abstract Turnpike)Let previous assumptions hold. Then, for any > 0, Ta) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,
  51. 51. Problem Abstract Diffusions Abstract TurnpikeTheorem (Abstract Turnpike)Let previous assumptions hold. Then, for any > 0, Ta) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,b) limT →∞ PT ΠT , Π T T ≥ =0
  52. 52. Problem Abstract Diffusions Abstract TurnpikeTheorem (Abstract Turnpike)Let previous assumptions hold. Then, for any > 0, Ta) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,b) limT →∞ PT ΠT , Π T T ≥ =0 • For log utility PT ≡ P, hence convergence holds under P.
  53. 53. Problem Abstract Diffusions Abstract TurnpikeTheorem (Abstract Turnpike)Let previous assumptions hold. Then, for any > 0, Ta) limT →∞ PT supu∈[0,T ] ru − 1 ≥ = 0,b) limT →∞ PT ΠT , Π T T ≥ =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
  54. 54. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions:then, for any > 0 and t ≥ 0:
  55. 55. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ Xt a.s. for all t ≤ S, T (myopic optimality);then, for any > 0 and t ≥ 0:
  56. 56. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ 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 > 0 and t ≥ 0:
  57. 57. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ 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 > 0 and t ≥ 0: Ta) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,
  58. 58. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ 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 > 0 and t ≥ 0: Ta) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,b) limT →∞ P Π T , ΠT t ≥ = 0.
  59. 59. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ 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 > 0 and t ≥ 0: Ta) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,b) limT →∞ P Π T , ΠT t ≥ = 0. • If optimal wealth myopic with IID returns, abstract implies classic.
  60. 60. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ 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 > 0 and t ≥ 0: Ta) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,b) limT →∞ P Π T , ΠT t ≥ = 0. • If optimal wealth myopic with IID returns, abstract implies classic. • In practice, if assets have IID returns, optimal portfolio myopic.
  61. 61. Problem Abstract Diffusions IID Myopic TurnpikeCorollary (IID Myopic Turnpike)If, in addition to previous assumptions: i) XtT = XtS ≡ 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 > 0 and t ≥ 0: Ta) limT →∞ P supu∈[0,t] ru − 1 ≥ = 0,b) limT →∞ P Π T , ΠT t ≥ = 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.
  62. 62. 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 .
  63. 63. 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: dSti = r (Yt ) dt + dRti , 1 ≤ i ≤ d, Sti
  64. 64. 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: 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
  65. 65. 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: 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.
  66. 66. Problem Abstract Diffusions Regularity ConditionsAssumptionSet 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 ), andΥ ∈ C 2,γ (E, Rd ). For all y ∈ E, Σ is positive and A is strictly positive.
  67. 67. Problem Abstract Diffusions Regularity ConditionsAssumptionSet 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 ), andΥ ∈ C 2,γ (E, Rd ). For all y ∈ E, Σ is positive and A is strictly positive.Assumption˜ Σ Υ ˜ µA= b= . Infinitesimal generator of (R, Y ): Υ A b 2L = 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.
  68. 68. Problem Abstract Diffusions Regularity ConditionsAssumptionSet 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 ), andΥ ∈ C 2,γ (E, Rd ). For all y ∈ E, Σ is positive and A is strictly positive.Assumption˜ Σ Υ ˜ µA= b= . Infinitesimal generator of (R, Y ): Υ A b 2L = 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.Assumptionρ 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
  69. 69. Problem Abstract Diffusions HJB Assumption (finite horizon)AssumptionThere exist (v T (y , t))T >0 and v (y ) such that: ˆ
  70. 70. Problem Abstract Diffusions HJB Assumption (finite horizon)AssumptionThere exist (v T (y , t))T >0 and v (y ) such that: 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
  71. 71. Problem Abstract Diffusions HJB Assumption (finite horizon)AssumptionThere exist (v T (y , t))T >0 and v (y ) such that: 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)
  72. 72. Problem Abstract Diffusions HJB Assumption (long run)Assumption
  73. 73. Problem Abstract Diffusions HJB Assumption (long run)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
  74. 74. Problem Abstract Diffusions HJB Assumption (long run)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 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 )
  75. 75. Problem Abstract Diffusions HJB Assumption (long run)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 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 < ∞,
  76. 76. Problem Abstract Diffusions Myopic Probabilities and Classic Turnpike
  77. 77. Problem Abstract Diffusions 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
  78. 78. Problem Abstract Diffusions 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 ˆ • Proposition allows to replace PT with P in abstract turnpike.
  79. 79. Problem Abstract Diffusions 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 ˆ • Proposition allows to replace PT with P in abstract turnpike. ˆ • Classic turnpike theorem follows from equivalence of P and P.
  80. 80. Problem Abstract Diffusions 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 ˆ • Proposition allows to replace PT with P in abstract turnpike. ˆ • Classic turnpike theorem follows from equivalence of P and P. Theorem (Classic Turnpike for Diffusions) Let previous assumptions hold. Then, for 0 = p < 1 and any , t > 0:
  81. 81. Problem Abstract Diffusions 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 ˆ • Proposition allows to replace PT with P in abstract turnpike. ˆ • Classic turnpike theorem follows from equivalence of P and P. Theorem (Classic Turnpike for Diffusions) Let previous assumptions hold. Then, for 0 = p < 1 and any , t > 0: T a) limT →∞ P (supu∈[0,t] ru − 1 ≥ ) = 0,
  82. 82. Problem Abstract Diffusions 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 ˆ • Proposition allows to replace PT with P in abstract turnpike. ˆ • Classic turnpike theorem follows from equivalence of P and P. Theorem (Classic Turnpike for Diffusions) Let previous assumptions hold. Then, for 0 = p < 1 and any , t > 0: T a) limT →∞ P (supu∈[0,t] ru − 1 ≥ ) = 0, b) limT →∞ P ΠT , ΠT t ≥ = 0.
  83. 83. Problem Abstract Diffusions Classic vs. Explicit • Abstract and Classic turnpikes: compare portfolios for U and x p /p at finite horizon T .
  84. 84. Problem Abstract Diffusions 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...
  85. 85. Problem Abstract Diffusions 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 )
  86. 86. Problem Abstract Diffusions 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 ) • Explicit turnpike: 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 )
  87. 87. Problem Abstract Diffusions 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 ) • Explicit turnpike: 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.
  88. 88. Problem Abstract Diffusions 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 −
  89. 89. Problem Abstract Diffusions 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 − ˆ • X wealth process of long-run portfolio π . ˆ
  90. 90. Problem Abstract Diffusions 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 − ˆ • X wealth process of long-run portfolio π . ˆTheorem (Explicit Turnpike)Under the previous assumptions, for any , t > 0 and 0 = p < 1:
  91. 91. Problem Abstract Diffusions 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 − ˆ • X wealth process of long-run portfolio π . ˆTheorem (Explicit Turnpike)Under the previous assumptions, for any , t > 0 and 0 = p < 1: rT a) limT →∞ P (supu∈[0,t] ˆu − 1 ≥ ) = 0,
  92. 92. Problem Abstract Diffusions 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 − ˆ • X wealth process of long-run portfolio π . ˆTheorem (Explicit Turnpike)Under the previous assumptions, for any , t > 0 and 0 = p < 1: rT a) limT →∞ P (supu∈[0,t] ˆu − 1 ≥ ) = 0, ˆ ˆ b) limT →∞ P ΠT , ΠT ≥ = 0. t
  93. 93. Problem Abstract Diffusions 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 − ˆ • X wealth process of long-run portfolio π . ˆTheorem (Explicit Turnpike)Under the previous assumptions, for any , t > 0 and 0 = p < 1: rT a) limT →∞ P (supu∈[0,t] ˆu − 1 ≥ ) = 0, ˆ ˆ b) limT →∞ P ΠT , ΠT ≥ = 0. t • Explicit turnpike nontrivial even for U(x) = x p /p.
  94. 94. Problem Abstract Diffusions 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 − ˆ • X wealth process of long-run portfolio π . ˆTheorem (Explicit Turnpike)Under the previous assumptions, for any , t > 0 and 0 = p < 1: rT a) limT →∞ P (supu∈[0,t] ˆu − 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.
  95. 95. Problem Abstract Diffusions Conclusion • Portfolio turnpikes: at long horizons, optimal portfolios approach those of CRRA class.
  96. 96. Problem Abstract Diffusions Conclusion • Portfolio turnpikes: at long horizons, optimal portfolios approach those of CRRA class. • Abstract turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the myopic probabilities.
  97. 97. Problem Abstract Diffusions Conclusion • Portfolio turnpikes: at long horizons, optimal portfolios approach those of CRRA class. • Abstract turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the myopic probabilities. • Classic turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the physical probability P.
  98. 98. Problem Abstract Diffusions Conclusion • Portfolio turnpikes: at long horizons, optimal portfolios approach those of CRRA class. • Abstract turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the myopic probabilities. • Classic turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the physical probability P. • Abstract implies classic if optimal wealth myopic with IDD returns.
  99. 99. Problem Abstract Diffusions Conclusion • Portfolio turnpikes: at long horizons, optimal portfolios approach those of CRRA class. • Abstract turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the myopic probabilities. • Classic turnpike: optimal portfolios for U and x p /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.
  100. 100. Problem Abstract Diffusions Conclusion • Portfolio turnpikes: at long horizons, optimal portfolios approach those of CRRA class. • Abstract turnpike: optimal portfolios for U and x p /p at horizon T become close. Under the myopic probabilities. • Classic turnpike: optimal portfolios for U and x p /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.

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