Robust Maximization of Asymptotic Growth under Covariance Uncertainty Erhan Bayraktar and Yu-Jui Huang Department of Mathematics, University of Michigan

Our Goal

The Question


Choose an π ∈ V s.t. V

How to maximize the growth rate of one’s wealth when precise covariance structure of the underlying assets is not known?

Take η ∈ Hλ∗(E)(E). Define

attains the rate

∗ πt

sup inf g(π; P) π∈V P∈Π

uniformly over all P in Π (or at least in a large enough subset Π∗ of Π).

The Set-up d

When c ∈ C is fixed...


Let E ⊂ R be an open connected set, and S be the set of d × d symmetric matrices. • X: price process of d assets taking values in E. 0,α • θ, Θ : E 7→ (0, ∞) are functions in Cloc (E), and satisfy θ < Θ in E. • C: set of functions c : E 7→ Sd s.t. for any x ∈ E,

For any D ⊂ E and λ ∈ R, we consider c Hλ(D)



:= {η ∈ C (D) | L η + λη = 0, η > 0},

:= e

and set


∇η (Xt) ∀ t ≥ 0,

∗ log η (Xt) ∗ Π := P ∈ Π P- lim inf ≥ 0 P-a.s. . t→∞ t Then, we have • Π∗ is large enough to include all the probabilities in Π under which X is stable. • λ∗(E) = sup inf∗ g(π; P) = inf∗ sup g(π; P).

• π∗


 




∈ V and g(π ∗; P) ≥ λ∗(E) for all P ∈ Π∗.

and define the principal eigenvalue for Lc(·) on D as ∗,c

λ (D) := sup{λ ∈ R |

c Hλ(D)

6= ∅}.

In [3], the authors take η ∗,c


 

:= P ∈

c Π


P- lim inf

c ∈ Hλ∗,c(E)(E) log η ∗,c(Xt)

and define  

≥ 0 P-a.s. .




Proving “λ (E) = inf c∈C λ (E)”

θ(x)Id ≤ c(x) ≤ Θ(x)Id. Remark. Each c ∈ C represents a possible covariance structure that might materialize. The (Knightian) uncertainty is captured by θ and Θ.

Conclusions and Outlook

Main Result

Assume: there exist {En}n∈N of bounded open 2,α ¯ convex subsets of E s.t. ∂En is of C , En ⊂ En+1 S∞ ∀ n ∈ N, and E = n=1 En.

They show that • Π∗,c is large enough to include all the probabilities in Πc under which X is stable. • λ∗,c(E) = sup inf∗,c g(π; P) = inf∗,c sup g(π; P). ∗,c • πt

π∈V ∗,c

:= eλ





• (L

Selected References

• Qc:

Recall Pucci’s operator: given 0 < λ ≤ Λ, + Mλ,Λ(M )

the solution to the (generalized) martingale problem on E for the operator Lc(·). • Πc:= {P | P loc Qc, X doesn’t explode P-a.s.} S c • Π:= c∈C Π . • π ∈ V (admissible trading strategy): predictable process s.t. the following holds for all c ∈ C: (i) π is X-integrable under Q ; Rt 0 π (ii) Vt := 1 + 0 πsdXs > 0 Qc-a.s., for all t ≥ 0. c

• Asymptotic

growth rate of V π under P:

:= sup γ ∈ R n

(≈ sup γ ∈ R |


sup A∈A(λ,Λ)

Tr(AM ), ∀ M ∈ S ,


Define the operator F : E × S 7→ R by 1 F (x, M ) := sup Tr(AM ). 2 A∈A(θ(x),Θ(x)) For any D ⊂ E and λ ∈ R, we consider 2

Hλ(D) := {η ∈ C (D) | F (x, D η)+λη ≤ 0, η > 0}, 

π P- lim inf (log Vt /t) ≥ γ P-a.s. t→∞ o π γt Vt ≥ e as t large P-a.s. )


Sketch of proof: 1. On each En, show the existence of a positive viscosity solution ηn (by using [5]) to

and define the principal eigenvalue for F on D as ∗

λ (D) := sup{λ ∈ R | Hλ(D) 6= ∅}. Now, by using the arguments in [3] and the relation λ∗(E) = inf c∈C λ∗,c(E), we obtain


F (x, D η) + λ (En)η ≤ 0.


where A(λ, Λ) denotes the set of matrices in Sd with eigenvalues lying in [λ, Λ].


g(π; P)

For Further Information

∇η ∗,c(Xt) ∈ V satisfies

When c ∈ C is NOT fixed... ∂ 2f 1 Pd f )(x):= 2 i,j=1 cij (x) ∂xi∂xj (x) 1 2 = 2 Tr[c(x)D f (x)].

The covariance uncertainty we consider is similar to the “Knightian uncertainty” formulated in [2], in the sense that the constraint on covariance is Markovian. The latter, however, is more general as it allows the covariance itself to be non-Markovian. It is of interest to generalize our results to the case with non-Markovian covariances, which would lead to eigenvalue problems for path-dependent PDEs.

• Yu-Jui Huang is available at [email protected] • Preprint of our paper can be downloaded from • This poster can be downloaded from∼jayhuang

g(π ∗,c; P) ≥ λ∗,c(E), ∀ P ∈ Π∗,c.


Among an appropriate class C of covariance structures, we characterize the largest possible robust asymptotic growth rate as the principle eigenvalue ∗ λ (E) of the fully nonlinear elliptic operator F , and identify the optimal trading strategy in terms of ∗ λ (E) and the associated eigenfunction.

2. Show that ηn is actually smooth (using [6]). 3. Show λ∗(En) = inf c∈C λ∗,c(En).

(i) ≤: Use a maximum principle related to F . (ii) ≥: Use the theory of continuous selection in [1] to construct {cm}m∈N ⊂ C s.t. ∗


λ (En) ≥ lim inf λ m→∞


4. Show λ∗(E) = λ0 := limn→∞ λ∗(En).

(i) ≤: obvious from definitions. (ii) ≥: Prove a Harnack inequality for F , and use it to show ηn converges uniformly on E to some η ∗ ∈ Hλ0 (E).

5. Since λ∗,c(E) = inf n∈N λ∗,c(En) for each c ∈ C (by [4]), we have ∗,c



inf λ (E) = inf inf λ (En) = inf inf λ (En)



c∈C n∈N ∗ inf λ (En) n∈N

n∈N c∈C

= λ (E).

[1] A. L. Brown, Set valued mappings, continuous selections, and metric projections, J. Approx. Theory 57 (1997), pp. 48-68. [2] D. Fernholz and I. Karatzas, Optimal arbitrage under model uncertainty, Ann. Appl. Probab. 21 (2011), pp. 2191-2225. [3] C. Kardaras and S. Robertson, Robust maximization of asymptotic growth, to appear in Ann. Appl. Probab, (2012). [4] R. G. Pinsky, Positive harmonic functions and diffusion, Cambridge University Press, Cambridge, 1995. [5] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), pp. 105-135. [6] M. V. Safonov, Classical solution of second-order nonlinear elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), pp. 1272-1287.

Robust Maximization of Asymptotic Growth under ... - CiteSeerX

Robust Maximization of Asymptotic Growth under Covariance Uncertainty. Erhan Bayraktar and Yu-Jui Huang. Department of Mathematics, University of Michigan. The Question. How to maximize the growth rate of one's wealth when precise covariance structure of the underlying assets is not known? The Set-up. Let E ⊂ R.

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