Using stochastic control in robust portfolio selection Martin - - PowerPoint PPT Presentation
Background The problem Results References And finally . . . Using stochastic control in robust portfolio selection Martin Schweizer Department of Mathematics ETH Z urich Stochastic Processes: Theory and Applications in honor of
Background The problem Results References And finally . . .
Martin Schweizer
Department of Mathematics ETH Z¨ urich
Stochastic Processes: Theory and Applications in honor of Wolfgang Runggaldier Bressanone, 20.07.2007 based on joint work with Giuliana Bordigoni (Milano) and Anis Matoussi (Le Mans)
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Well-known problem from economics:
maximize expected utility from investment and consumption lots of references . . . large majority assumes that underlying model is known
Here: what happens if model is not exactly known? Abstract formulation: consider EQ
U0,T(π, c) stands for utility over (0, T] from investment π and consumption c Q stands for probabilistic model maximize with respect to π, c minimize with respect to Q, to capture uncertainty about Q
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Formulation More details
Approach 1:
specify set Q of possible models Q consider sup
(π,c)
inf
Q∈Q EQ
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Formulation More details
Approach 1:
specify set Q of possible models Q consider sup
(π,c)
inf
Q∈Q EQ
Approach 2:
allow all Q as possible models penalize by some functional R(Q) consider sup
(π,c)
Q∈M1(P)
sup)
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Formulation More details
Approach 1:
specify set Q of possible models Q consider sup
(π,c)
inf
Q∈Q EQ
Approach 2:
allow all Q as possible models penalize by some functional R(Q) consider sup
(π,c)
Q∈M1(P)
sup)
Here: some results about partial problem inf
Q∈M1(P)
R(Q) is entropic penalty term U0,T represents abstract utility
Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Formulation More details
More precisely:
discount factor Sδ
t := exp
t
0 δs ds
t,T := α
T
t Sδ
s
Sδ
t Us ds + ¯
α Sδ
T
Sδ
t
¯ UT penalty term Rδ
t,T(Q) :=
T
t δs Sδ
s
Sδ
t log Z Q s
Z Q
t ds + Sδ T
Sδ
t log Z Q T
Z Q
t
Goal: minimize cost functional Q → Γ(Q) := EQ
0,T + βRδ 0,T(Q)
For δ ≡ 0, we get EQ
0,T
to minimize relative entropy with respect to suitable P U; solution is Q∗ = P U. But δ ≡ 0 is not so easy (for us . . . ).
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Assumptions:
(A1) 0 ≤ δ ≤ δ∞ < ∞ for some constant δ∞ (A2) T
0 |Us| ds has all exponential moments (subspace Mexp of
Orlicz space Lexp) (A3) |¯ UT| has all exponential moments
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Assumptions:
(A1) 0 ≤ δ ≤ δ∞ < ∞ for some constant δ∞ (A2) T
0 |Us| ds has all exponential moments (subspace Mexp of
Orlicz space Lexp) (A3) |¯ UT| has all exponential moments
First step: for constants depending only on α, α′, β, δ, T, U, ¯ UT (but not on Q), we have estimate c
RHS is easy; LHS exploits exponential moment conditions Consequences:
can work with Q ∈ Qf :=
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Theorem 1: Under (A1) – (A3), there exists unique Q∗ ∈ Qf minimizing Q → Γ(Q) over all Q ∈ Qf . Moreover, Q∗ ≈ P.
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Theorem 1: Under (A1) – (A3), there exists unique Q∗ ∈ Qf minimizing Q → Γ(Q) over all Q ∈ Qf . Moreover, Q∗ ≈ P. Sketch of proof:
Uniqueness from strict convexity For minimizing sequence of density processes Z n, Koml´
convex combinations ¯ Z n
T −
→ ¯ Z ∞
T P-a.s. Use entropy estimate
(1) (to get UI) to show that ¯ Q∞ with density ¯ Z ∞
T is in Qf
Γ(Q) as functional on density processes Z Q is convex and “morally lower semicontinuous” with respect to P-a.s. convergence of Z Q
T ; so ¯
Q∞ must attain infimum Proving lsc for (linear, but unbounded term) EQ
0,T
tricky; uses full strength of exponential moment conditions Equivalence adapts Frittelli (2000), using FOC for optimality
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Dynamic version: value of problem started at time τ is Vτ := P - ess inf
Q′ ∈ Qf , Q′ ≈ P
τ,T|Fτ] + βEQ′[Rδ τ,T(Q′)|Fτ]
→ stochastic control
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Dynamic version: value of problem started at time τ is Vτ := P - ess inf
Q′ ∈ Qf , Q′ ≈ P
τ,T|Fτ] + βEQ′[Rδ τ,T(Q′)|Fτ]
→ stochastic control Properties of JQ
τ := P -
ess inf
Q′ = Q on Fτ
0,T|Fτ] + βEQ′[Rδ 0,T(Q′)|Fτ]
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Dynamic version: value of problem started at time τ is Vτ := P - ess inf
Q′ ∈ Qf , Q′ ≈ P
τ,T|Fτ] + βEQ′[Rδ τ,T(Q′)|Fτ]
→ stochastic control Properties of JQ
τ := P -
ess inf
Q′ = Q on Fτ
0,T|Fτ] + βEQ′[Rδ 0,T(Q′)|Fτ]
Martingale optimality principle:
JQ is a Q-submartingale for each Q Q∗ is optimal iff JQ∗ is a Q∗-martingale
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Dynamic version: value of problem started at time τ is Vτ := P - ess inf
Q′ ∈ Qf , Q′ ≈ P
τ,T|Fτ] + βEQ′[Rδ τ,T(Q′)|Fτ]
→ stochastic control Properties of JQ
τ := P -
ess inf
Q′ = Q on Fτ
0,T|Fτ] + βEQ′[Rδ 0,T(Q′)|Fτ]
Martingale optimality principle:
JQ is a Q-submartingale for each Q Q∗ is optimal iff JQ∗ is a Q∗-martingale
JQ = SδV + α
s Us ds + β
s log Z Q s ds + βSδ log Z Q
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Theorem 2: Under (A1) – (A3) and if F is continuous, (V , MV ) is unique solution in Dexp × M0,loc(P) of backward stochastic differential equation (BSDE) dYt = (δtYt − αUt) dt + 1 2β dMt + dMt, (2) YT = ¯ α¯ UT. Moreover, E
βMV
is true P-martingale, and MV lies in martingale space Mp
0(P) for every p ∈ [1, ∞).
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Theorem 2: Under (A1) – (A3) and if F is continuous, (V , MV ) is unique solution in Dexp × M0,loc(P) of backward stochastic differential equation (BSDE) dYt = (δtYt − αUt) dt + 1 2β dMt + dMt, (2) YT = ¯ α¯ UT. Moreover, E
βMV
is true P-martingale, and MV lies in martingale space Mp
0(P) for every p ∈ [1, ∞).
Usefulness: can find optimal Q∗ by solving BSDE because Z Q∗ = E
βMV
Remark: For general filtration F? We do not know yet.
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
Remark: (2) is BSDE with quadratic term in its driver and unbounded coefficients and terminal condition. Most known
Sketch of proof:
Use Itˆ
involves V , hence MV and AV , as well as Z Q. By MOP, FV part is increasing for each Q and constant for Q∗. This allows to express AV in terms of MV ; MV comes in via Girsanov through MV , Log Z Q because we work under Q, not P. Integrability is tricky: similarly as in Schroder/Skiadas (1999), rewrite BSDE as recursive relation for Y and exploit that. Dexp is space of all progressive processes whose ess sup has all exponential moments (i.e., again Orlicz space Lexp comes up). Uniqueness adapted from SS99. Integrability of MV adapts and sharpens SS99.
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
For fixed strategy (π, c), dynamic value process is given by BSDE dVt =
2β dMt + dMt, (3) YT = ¯ α¯ UT(π, c). Intuition: This describes outcome of chosen strategy under worst case measure.
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
For fixed strategy (π, c), dynamic value process is given by BSDE dVt =
2β dMt + dMt, (3) YT = ¯ α¯ UT(π, c). Intuition: This describes outcome of chosen strategy under worst case measure. Second step: now need to maximize this by choice of (π, c) — for instance if Ut(π, c) = u(ct) and ¯ UT(π, c) = ¯ u
T
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Preliminary results Existence Dynamic formulation BSDE What next?
For fixed strategy (π, c), dynamic value process is given by BSDE dVt =
2β dMt + dMt, (3) YT = ¯ α¯ UT(π, c). Intuition: This describes outcome of chosen strategy under worst case measure. Second step: now need to maximize this by choice of (π, c) — for instance if Ut(π, c) = u(ct) and ¯ UT(π, c) = ¯ u
T
Structure: (3) describes a stochastic differential utility — and maximizing this is hard . . .
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Robust control
Hansen/Sargent/Turmuhambetova/Williams, Maenhout, Skiadas 2003 (Finance and Stochastics)
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Robust control
Hansen/Sargent/Turmuhambetova/Williams, Maenhout, Skiadas 2003 (Finance and Stochastics)
BSDEs
Briand/Hu, Kobylanski
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Robust control
Hansen/Sargent/Turmuhambetova/Williams, Maenhout, Skiadas 2003 (Finance and Stochastics)
BSDEs
Briand/Hu, Kobylanski
Utility maximization under uncertainty
Gundel (Finance and Stochastics), Quenez, Schied (Finance and Stochastics), Schied/Wu
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Robust control
Hansen/Sargent/Turmuhambetova/Williams, Maenhout, Skiadas 2003 (Finance and Stochastics)
BSDEs
Briand/Hu, Kobylanski
Utility maximization under uncertainty
Gundel (Finance and Stochastics), Quenez, Schied (Finance and Stochastics), Schied/Wu
Stochastic differential utility
Duffie/Epstein, Lazrak/Quenez, Schroder/Skiadas, Skiadas 2003 (Finance and Stochastics)
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
stochastic control approach to a robust utility maximization problem, in: F. E. Benth et al. (eds.), ‘Stochastic Analysis and Applications. Proceedings of the Second Abel Symposium, Oslo, 2005’, Springer, 125–152
(or google “Martin Schweizer”)
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . .
Martin Schweizer Using stochastic control in robust portfolio selection