portfolio turnpikes in incomplete markets
TRANSCRIPT
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Portfolio turnpikes in incomplete markets
Hao Xing
London School of Economics
joint work with
Paolo Guasoni, Kostas Kardaras, and Scott Robertson
Stochastic Analysis in Finance and Insurance, Ann Arbor, May 18, 2011
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What is a turnpike?
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What is a turnpike? Express way
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What is the portfolio turnpike?Mertons problem:
uT(x) = max
XadmissibleE[U(XT)].
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What is the portfolio turnpike?Mertons problem:
uT(x) = max
XadmissibleE[U(XT)].
When U(x) = xp
/p, with p < 1, and the market is Black-Scholes,
=1
1 p1.
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What is the portfolio turnpike?Mertons problem:
uT(x) = max
XadmissibleE[U(XT)].
When U(x) = xp
/p, with p < 1, and the market is Black-Scholes,
=1
1 p1.
For more general utility function and market structure:
Existence: Karatzas-Shreve (98), Kramkov-Schachermayer (99)Explicit solutions: Kallsen (00), Zariphopoulou (01), Goll-Kallsen (03),
Hu-Imkeller-Muller (05)
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What is the portfolio turnpike?Mertons problem:
uT(x) = max
XadmissibleE[U(XT)].
When U(x) = xp
/p, with p < 1, and the market is Black-Scholes,
=1
1 p1.
For more general utility function and market structure:
Existence: Karatzas-Shreve (98), Kramkov-Schachermayer (99)Explicit solutions: Kallsen (00), Zariphopoulou (01), Goll-Kallsen (03),
Hu-Imkeller-Muller (05)
Portfolio Turnpike:When investment horizon is distant and market grows indefinitely,
Optimal portfolios for generic utilities
is close to
Optimal portfolios of power or logarithmic utilities.
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What is the portfolio turnpike?Mertons problem:
uT(x) = max
XadmissibleE[U(XT)].
When U(x) = x
p
/p, with p < 1, and the market is Black-Scholes,
=1
1 p1.
For more general utility function and market structure:
Existence: Karatzas-Shreve (98), Kramkov-Schachermayer (99)Explicit solutions: Kallsen (00), Zariphopoulou (01), Goll-Kallsen (03),
Hu-Imkeller-Muller (05)
Portfolio Turnpike:When investment horizon is distant and market grows indefinitely,
Optimal portfolios for generic utilities
is close to
Optimal portfolios of power or logarithmic utilities.
For long term investments, follow the CRRA turnpike!8 / 6 2
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A more mathematical formulation
Consider problems
u0,T = supX admissible
EP[XpT/p], u
1,T = supX admissible
EP[U(XT)].
Suppose that
Distant horizon: T.
Market grows indefinitely: X s.t. limT XT = .
Similar utilities at large wealth levels:
limx
U(x)
xp1= 1.
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A more mathematical formulation
Consider problems
u0,T = supX admissible
EP[XpT/p], u
1,T = supX admissible
EP[U(XT)].
Suppose that
Distant horizon: T.
Market grows indefinitely: X s.t. limT XT = .
Similar utilities at large wealth levels:
limx
U(x)
xp1= 1.
Then 1,Ts and 0,Ts are similar for s [0, t] with t is fixed.
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Literature
Discrete time:
Mossin (68): U(x)/U
(x) = ax + b
Leland (72), Ross (74), Hakansson (74)
Huberman-Ross (83): U(x) is regular varying at :
limx
U(ax)
U(x)= ap1, for any a > 0.
Continuous time:
Cox-Huang (92)
Jin (98): include consumptions
Huang-Zariphopouou (99): viscosity solutions
Dybvig-Rogers-Back (99): NO independent returns, completemarkets with Brownian filtration
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Literature
Discrete time:
Mossin (68): U(x)/U
(x) = ax + b
Leland (72), Ross (74), Hakansson (74)
Huberman-Ross (83): U(x) is regular varying at :
limx
U(ax)
U(x)= ap1, for any a > 0.
Continuous time:
Cox-Huang (92)
Jin (98): include consumptions
Huang-Zariphopouou (99): viscosity solutions
Dybvig-Rogers-Back (99): NO independent returns, completemarkets with Brownian filtration
Assume independent return, except Dybvig et al. = DPP.
Assume market completeness = duality method.
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Incomplete markets with stochastic investmentopportunities
The portfolio choice problem is least tractable, turnpike theorems aremost needed.
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Incomplete markets with stochastic investmentopportunities
The portfolio choice problem is least tractable, turnpike theorems aremost needed.
We prove three kind of turnpike theorems:
Abstract Turnpike: minimum market assumptions, converge undermyopic probabilities except for the log utility;
Classical Turnpike: a class of diffusion models with stochastic driftand volatility. Convergence is shown under the physical measure.
Explicit Turnpike: finite horizon optimal portfolios for genericutilities converge to the long-run optimal portfolio for power utility.
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Admissible wealth processes
For each T R+, wealth process is chosen from a set of nonneg.
semimartingales X
T
, such thati) X0 = 1 for all X X
T;
ii) XT contains some strictly positive X;
iii) XT is convex;
iv) XT is stable under compounding: for any X, X XT and a [0, T]
valued stopping time ,
X = XI[[0,[[ + XXX
I[[,T]] XT.
One can switch to another strategy at any stopping time in aself-financing way.
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Admissible wealth processes
For each T R+, wealth process is chosen from a set of nonneg.
semimartingales X
T
, such thati) X0 = 1 for all X X
T;
ii) XT contains some strictly positive X;
iii) XT is convex;
iv) XT is stable under compounding: for any X, X XT and a [0, T]
valued stopping time ,
X = XI[[0,[[ + XXX
I[[,T]] XT.
One can switch to another strategy at any stopping time in aself-financing way.
These assumptions are satisfied in general frictionless market withsemimartingale dynamics.
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Assumptions on utilities
Let R(x) := U(x)/xp1.
limxR(x) = 1,
lim infx0R(x) > 0, for 0 = p < 1,
lim supx0R(x) < , for p < 1.
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Assumptions on utilities
Let R(x) := U(x)/xp1.
limxR(x) = 1,
lim infx0R(x) > 0, for 0 = p < 1,
lim supx0R(x) < , for p < 1.
Assumptions in the literature:
1. limxU(x)xp1
= 1,
2. limxxU
(x)U(x) = 1 p,
3. limxU(ax)U(x) = a
p1 for any a > 0.
1. = 2. = 3., but not in the other direction.
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A i ili i
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Assumptions on utilities
Let R(x) := U(x)/xp1.
limxR(x) = 1,
lim infx0R(x) > 0, for 0 = p < 1,
lim supx0R(x) < , for p < 1.
Assumptions in the literature:
1. limxU(x)xp1
= 1,
2. limxxU
(x)U(x) = 1 p,
3. limxU(ax)U(x) = a
p1 for any a > 0.
1. = 2. = 3., but not in the other direction.
Dybvig-Rogers-Back (99) does not have assumptions on R(x) at x = 0.
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A i f li h i bl
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Assumptions on portfolio choice problem
We assume the portfolio choice problem is well-posed for each horizon:
For all T > 0 and i = 0, 1,
< ui,T < ;
optimal wealth processes Xi,T exist.
Together with the admissible assumption, this implies the no arbitragecondition:
Xs = 0 = Xt = 0, for any t s.
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M i b biliti
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Myopic probabilities
Define the myopic probabilities (PT)T0 as
dPT
dP=
X
0,T
Tp
EP
X0,TT
p .
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M i b biliti
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Myopic probabilities
Define the myopic probabilities (PT)T0 as
dPT
dP=
X
0,T
Tp
EP
X0,TT
p .
Appear in Kramkov-Sirbu (06, 07);
PT = P when log utility is considered; Power optimal under P is log optimal under PT.
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M o ic obabilities
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Myopic probabilities
Define the myopic probabilities (PT)T0 as
dPT
dP=
X
0,T
Tp
EP
X0,TT
p .
Appear in Kramkov-Sirbu (06, 07);
PT = P when log utility is considered; Power optimal under P is log optimal under PT.
First order condition EP
X0,TT
p1(XT X
0,TT )
0 for any
X XT. Change the measure to PT,
EPT
XT/X0,TT
1, for any X XT.
Hence X0,TT has numeraire property under PT.
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Last assumption
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Last assumptionIn the Black-Scholes case:
dPT
dP= TX
0T,
where is the density of the unique martingale measure.
Note:
limT
dPT
dP= 0, P a.s. ifp = 0.
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Last assumption
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Last assumptionIn the Black-Scholes case:
dPT
dP= TX
0T,
where is the density of the unique martingale measure.
Note:
limT
dPT
dP= 0, P a.s. ifp = 0.
Assumption on market growth:There exists (XT)T0 such that X
T XT and
limT
PT(XTT N) = 1, for any N > 0.
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Last assumption
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Last assumptionIn the Black-Scholes case:
dPT
dP= TX
0T,
where is the density of the unique martingale measure.
Note:
limT
dPT
dP= 0, P a.s. ifp = 0.
Assumption on market growth:There exists (XT)T0 such that X
T XT and
limT
PT(XTT N) = 1, for any N > 0.
This condition is satisfied in
Safe rate bounded from below by a constant r > 0.
Black-Scholes market dSu/Su = du+ dWu,
X can be chosen as the power optimal X0,T when = 0.
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First turnpike theorem
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First turnpike theoremOptimal terminal wealths are unbounded as T.
Define
rT
u := X1,T
u /X0,T
u , T
u := u0 drTv /rTv, u [0,T].
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First turnpike theorem
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First turnpike theoremOptimal terminal wealths are unbounded as T.
Define
r
T
u := X1,T
u /X0,T
u , T
u := u0 drTv /rTv, u [0,T].Theorem (Abstract Turnpike)Under above assumptions, for any > 0,
a) limT PT
supu[0,T]rTu 1 = 0,
b) limT PT
T, TT
= 0.
Remark:
In the market dSu/Su = udu+ udWu,
T, T
=
0
1,Tu 0,Tu u2 du. When log utility is considered, the convergence is under P.
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An Example
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An ExampleIn Huberman-Ross (83), a discrete time market model with independentreturn was studied.
U is regular varying at is a necessary condition for turnpike.
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An Example
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An ExampleIn Huberman-Ross (83), a discrete time market model with independentreturn was studied.
U is regular varying at is a necessary condition for turnpike.
In continuous time with dependent return, this does not hold.
Consider a Black-Scholes model with r = 0 and dSt/St = dt+ dWt.
Let X be the log-optimal wealth process and n = inf{t 0 |Xt = n}.
For problem with horizon n, is the optimal wealth process for genericutility U.
E[n /Xn ] 1 = E[n ] n, for any wealth process .
E[U(n)] U(E[n ]) U(n) = E[U(Xn )].
Then change time from n to n, the return becomes dependent.
Turnpike trivially holds. But the generic utility may not be regularly
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Drawback
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Drawback
Convergence is under {PT}T0,
Limit of (PT)T0 is in general singular wrt P.
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Drawback
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Drawback
Convergence is under {PT}T0,
Limit of (PT)T0 is in general singular wrt P.
So we look at a time window [0, t]
limT
PT
sup
u[0,t]
rTu 1
= 0,
limTPT T, Tt = 0.
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Drawback
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Convergence is under {PT}T0,
Limit of (PT)T0 is in general singular wrt P.
So we look at a time window [0, t]
limT
PT
sup
u[0,t]
rTu 1
= 0,
limTPT T, Tt = 0.
Then consider the conditional densities
dPT
dP
Ft
.
Question:
Does limT dPT/dP|Ft exist?
Does the limit induce a measure equivalent to P on Ft?
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IID Myopic Turnpike
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y p p
Corollary
1. X0,Tt = X0,St Xt a.s. for all t S, T (myopic optimality);
2. X0,TT /X0,Tt andFt are independent for all t T (independent
returns).
then, for any > 0 and t 0:
a) limT P
supu[0,t]rTu 1
= 0,
b) limT P
T, Tt
= 0.
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IID Myopic Turnpike
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y p p
Corollary
1. X0,Tt = X0,St Xt a.s. for all t S, T (myopic optimality);
2. X0,TT /X0,Tt andFt are independent for all t T (independent
returns).
then, for any > 0 and t 0:
a) limT P
supu[0,t]rTu 1
= 0,
b) limT P
T, Tt
= 0.
Remark: This contains exponential Levy markets, Kallsen (00).
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Markets with stochastic investment opportunities
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pp
Consider a diffusion model, investment opportunities are driven by
dYt = b(Yt)dt+ a(Yt)dWt,
whose state space is E = (, ), with < .
The market includes a safe rate r(Yt) and d risky assets
dSit
Sit
= r(Yt)dt+ dRit, with
dRt = (Yt)dt+ (Yt)dZt,
dZi,Wt = i(Yt) dt.
DefineA
=a
2
, , =
, and = a
.
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Markets with stochastic investment opportunities
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Consider a diffusion model, investment opportunities are driven by
dYt = b(Yt)dt+ a(Yt)dWt,
whose state space is E = (, ), with < .
The market includes a safe rate r(Yt) and d risky assets
dSit
Sit
= r(Yt)dt+ dRit, with
dRt = (Yt)dt+ (Yt)dZt,
dZi,Wt = i(Yt) dt.
DefineA
=a
2
, , =
, and = a
.
We assume the martingale problem associated to (Y, R) is well posed.
Assumption on correlation: is a constant.
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Finite horizon problem for power
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Zariphopoulou (01) introduced the power transform. The nonlinear HJBequation is linearized.
uT
(x, y, t) =
xp
p vT(y, t) .vT solves the reduced HJB equation:
tv + Lv + cv = 0, v(T, y) = 1,
where L = 12 A2yy + (b q
1)y and c =1
(pr q2 1).
The optimal portfolio is
T =1
1 p1
+
vTy
vT
evaluated on (t,Yt).
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Finite horizon problem for power
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Zariphopoulou (01) introduced the power transform. The nonlinear HJBequation is linearized.
uT
(x, y, t) =
xp
p vT(y, t) .vT solves the reduced HJB equation:
tv + Lv + cv = 0, v(T, y) = 1,
where L = 12 A2yy + (b q
1)y and c =1
(pr q2 1).
The optimal portfolio is
T =1
1 p1
+
vTy
vT
evaluated on (t,Yt).
Under PT, the dynamics of (R, Y) is
dRt =1
1p
+
vTy
vT
(t,Yt) dt+ (Yt) dZt
dYt =
B+ A
vTy
vT
(t,Yt) dt+ a(Yt) d
Wt
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Long-run measure T
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Our goal: study the limit behavior of
dPT/dP|FtT0
.
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Long-run measure T
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Our goal: study the limit behavior of
dPT/dP|FtT0
.
Consider the ergodic HJB equation:
Lv+ cv = v.This relates to the risk sensitive control
max
limT
1
pTlog(E [U(XT)]) .
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Long-run measureO f
T / |
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Our goal: study the limit behavior of
dPT/dP|FtT0
.
Consider the ergodic HJB equation:
Lv+ cv = v.This relates to the risk sensitive control
max
limT
1
pTlog(E [U(XT)]) .
DefineP
as the solution to the following martingale problem: dRt =1
1p
+
vyv
(Yt) dt+ dZt
dYt =B+ A
vyv
(Yt) dt+ a dWt
Guasoni-Robertson (10) proposed that the long-run optimal strategy is
=1
1 p1
+
vyv
.
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Long-run measureO l d h li i b h i f
dPT /dP|
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Our goal: study the limit behavior of
dPT/dP|FtT0
.
Consider the ergodic HJB equation:
Lv+ cv = v.This relates to the risk sensitive control
max
limT
1
pTlog(E [U(XT)]) .
DefineP
as the solution to the following martingale problem: dRt =1
1p
+
vyv
(Yt) dt+ dZt
dYt =B+ A
vyv
(Yt) dt+ a dWt
Guasoni-Robertson (10) proposed that the long-run optimal strategy is
=1
1 p1
+
vyv
.
is the log optimal strategy under P.
Then dP/dP|Ft
is the natural candidate for limT
dPT/dP|Ft
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Classical Turnpike
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Proposition
When Y is positively recurrent under P, then P P on Ft. Moreover
P limT
dPT
dP|Ft = 1.
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Classical Turnpike
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Proposition
When Y is positively recurrent under P, then P P on Ft. Moreover
P limT
dPT
dP|Ft = 1.
This allows to replace PT in the abstract turnpike with P.
Theorem (Classical Turnpike)Under above assumptions, for 0 = p < 1 and any , t > 0:
a) limT P (supu[0,t]rTu 1
) = 0,
b) limT P
T, Tt
= 0.
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Explicit Turnpike
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Compare finite horizon optimal portfolio for a generic utility with thelong-run optimal for the power utility.
DefinerTu =
X1,Tu
Xu, Tu =
u0
drTv
rTv, for u [0,T].
Theorem (Explicit Turnpike)
Under above assumptions, for any , t > 0 and 0 = p < 1:a) limT P (supu[0,t]
rTu 1 ) = 0,b) limT P
T, T
t
= 0.
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Explicit Turnpike
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Compare finite horizon optimal portfolio for a generic utility with thelong-run optimal for the power utility.
DefinerTu =
X1,Tu
Xu, Tu =
u0
drTv
rTv, for u [0,T].
Theorem (Explicit Turnpike)
Under above assumptions, for any , t > 0 and 0 = p < 1:a) limT P (supu[0,t]
rTu 1 ) = 0,b) limT P
T, T
t
= 0.
Remark:
Finite horizon optimal for a generic utility long-run optimal forpower utility;
The long-run portfolio is a myopic strategy, expressed by thesolution of the ergodic HJB.
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Proof for the log caseThe first order condition:
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The first order condition:
E[U(XTT )(XT XTT )] 0, for any admissible X.
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Proof for the log caseThe first order condition:
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The first order condition:
E[U(XTT )(XT XTT )] 0, for any admissible X.
Applying the first order condition to log x and U respectively.
EP[rT 1] 0,
EP[U(X1,TT )(X
0,TT X
1,TT )] 0.
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Proof for the log caseThe first order condition:
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The first order condition:
E[U(XTT )(XT XTT )] 0, for any admissible X.
Applying the first order condition to log x and U respectively.
EP[rT 1] 0,
EP[U(X1,TT )(X
0,TT X
1,TT )] 0.
Sum previous two inequalities
EP
1
R(X1,TT )
rTT
(rTT 1)
0.
Lemma
EP1 R(X1,TT )rTT
(rTT 1)
2EP R(X1,TT ) 12 .R is bounded, P- limTR(X
1,TT ) = 1 = P limTr
TT = 1
= L1 limTrTT = 1.
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Convergence of portfolios
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Lemma (Kardaras (10))
Consider a sequence{YT
}T0 of cadlag processes and a sequence{PT}T0, such that:
i) For each T 0, YT0 = 1 and YT is strictly positive.
ii) Each YT is a PT-supermartingale.
iii) limT EPT
[|YT
T
1|] = 0.
Then, for any > 0,
a) limT PT(supu[0,T] |Y
Tu 1| ) = 0.
b) limT PT([LT, LT]T ) = 0, where Y
T is the stochasticlogarithm of YT.
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Convergence of portfolios
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Lemma (Kardaras (10))
Consider a sequence{YT
}T0 of cadlag processes and a sequence{PT}T0, such that:
i) For each T 0, YT0 = 1 and YT is strictly positive.
ii) Each YT is a PT-supermartingale.
iii) limT EPT
[|YT
T
1|] = 0.
Then, for any > 0,
a) limT PT(supu[0,T] |Y
Tu 1| ) = 0.
b) limT PT([LT, LT]T ) = 0, where Y
T is the stochasticlogarithm of YT.
Note that rT is a supermartingale and EP[|rTT 1|] = 0,
the abstract turnpike follows from the above lemma.
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Proof for the diffusion caseDefine
T
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hT(t, y) :=
vT(t, y)
ec(Tt)v(y).
It satisfies
thT + L hT = 0, (t, y) (0,T) E,
hT(T, y) = (v)1 (y), y E,
where
L := L + Avy
v
y.
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Proof for the diffusion caseDefine
T ( )
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hT(t, y) :=
vT(t, y)
ec(Tt)v(y).
It satisfies
thT + L hT = 0, (t, y) (0,T) E,
hT(T, y) = (v)1 (y), y E,
where
L := L + Avy
v
y.
This is Doobs h-transform using v, the generator of Y under P.
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Proof for the diffusion caseDefine
T ( )
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hT(t, y) :=
vT(t, y)
ec(Tt)v(y).
It satisfies
thT + L hT = 0, (t, y) (0,T) E,
hT(T, y) = (v)1 (y), y E,
where
L := L + Avy
v
y.
This is Doobs h-transform using v, the generator of Y under P.
It has the stochastic representation hT(t, y) = EPyt
(v(YT))1
.
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Proof for the diffusion caseDefine
T ( )
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hT(t, y) :=
vT(t, y)
ec(Tt)v(y).
It satisfies
thT + L hT = 0, (t, y) (0,T) E,
hT(T, y) = (v)1 (y), y E,
where
L := L + Avy
v
y.
This is Doobs h-transform using v, the generator of Y under P.
It has the stochastic representation hT(t, y) = EPyt
(v(YT))1
.
vT(t, y) = ec(Tt)v(y)E
Py
t
(v(YT))
1
= EPy
t
exp
Tt
c(Ys)ds
.
This confirms a remark in Zariphopoulou(01) via h-transform.56/62
Convergence of densities
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dPT
dP
Ft
=hT(t,Yt)
hT(0, y) .
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Convergence of densities
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dPT
dP
Ft
=hT(t,Yt)
hT(0, y) .
When Y is positive ergodic under P,
(this can be characterized via the scale function of Y)
limT
hT(t, y) = limT
EPy
t
(v(YT))
1
= K, for all (t, y).
.
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Verification
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1. Assume Lv + cv = v has a strictly positive C2 solution v.
2. Define hT(t, y) = EPyt
(v(YT))1
.
3. Show hT is a C1,2 solution to the associated PDE.
No uniform ellipticity assumption.
4. Define vT(t, y) = ec(Tt)v(y)hT(t, y).
5. Verify that uT(t, x, y) = 1p
xp
vT(t, y)
.
uT(t, x, y) = 1p
xp
vT(t, y)
E[ETt ].
Show ET is a martingale.
We use the result in Cheriditio-Filipovic-Yor (05).
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Conclusion and future works
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We show three turnpike theorems in incomplete markets withstochastic investment opportunities.
The goal is to provide a tractable portfolio for a generic utility whenthe horizon is long.
This can be viewed as a stability result for portfolio choice problem
when the horizon is long.
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Conclusion and future works
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We show three turnpike theorems in incomplete markets withstochastic investment opportunities.
The goal is to provide a tractable portfolio for a generic utility whenthe horizon is long.
This can be viewed as a stability result for portfolio choice problem
when the horizon is long.
Future works:
Diffusion model with multiple state variables?
Market with transaction cost?
Forward utility?
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Thanks for your attention!
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