Some remarks on large deviation estimates for controlled - PowerPoint PPT Presentation

Some remarks on large deviation estimates for controlled semi-martingales March 9, 2012 at NCTS Hideo Nagai Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531, Japan, E-mail: nagai@sigmath.es.osaka-u.ac.jp

Large deviation estimates for controlled semi-martingale X 0 = x ∈ R N , (1 . 1) dX t = λ ( X t ) dW t + β ( X t ) dt, λ ( x ): R N �→ N ⊗ M , β ( x ): R N �→ R N W t : M - dim. F t B.M., 1 ( 1 ) (1 . 2) J ( κ ) := lim T inf log P T F T ( X . , h . ) ≤ κ . h . T →∞ ∫ T ∫ T 0 ϕ ( X s , h s ) ∗ dW s F T ( X . , h . ) = F 0 + 0 f ( X s , h s ) ds + h s : F t - prog. m’ble, R m -valued , m, N ≤ M F 0 : F 0 - m’ble r.v., f ( x, h ) := − 1 2 h ∗ S ( x ) h + h ∗ g ( x ) + U ( x ) , ϕ ( x, h ) = δ ( x ) h, S ( x ) : R N �→ R m ⊗ R m , g ( x ) : R N �→ R m , δ ( x ) : R N �→ R M ⊗ R m ,

Robust version of large deviation estimates dP ζ ∫ T ∫ T � s dW s − 1 0 ζ ∗ 0 | ζ s | 2 ds � := e 2 � � dP � F T ∫ t W ζ 0 ζ s ds : B.M. under P ζ t := W t − dX t = λ ( X t ) dW ζ t + { β ( X t ) + λ ( X t ) ζ t } dt. (1 . 3) ∫ T ( ) 1 1 T { F T ( X . , h . ) + µ 0 | ζ s | 2 ds } ≤ κ log P ζ J 1 ( κ ) := lim T inf sup . 2 h . T →∞ ζ .

”Market model” Motivated examples Riskless asset: dS 0 ( t ) = r ( X t ) S 0 ( t ) dt, S 0 (0) = s 0 . (1 . 4) Risky assets:  dS i ( t ) = S i ( t ) { α i ( X t ) dt + ∑ n + m k =1 σ i k ( X t ) dW k t } ,  (1 . 5) S i (0) = s i , i = 1 , ..., m  Factors: dX t = ˜ β ( X t ) dt + ˜ λ ( X t ) dW t , (1 . 6) X (0) = x,

Total wealth: m N i t S i ∑ V t = t i =0 N i t : Number of the shares t = N i t S i h i t : Portfolio proportion i = 0 , 1 , 2 , . . . , m. V t h t = ( h 1 t , . . . , h m t ) dV t = r ( X t ) dt + h ( t ) ∗ ( α ( X t ) − r ( X t ) 1 ) dt + h ( t ) ∗ σ ( X t ) dW t , V t log V T = log V 0 ∫ T ∫ T 0 {− 1 2 h ∗ s σσ ∗ ( x s ) h s + h ∗ 0 h ∗ + s ˆ α ( X s ) + r ( X s ) } dt + s σ ( X s ) dW s , α ( x ) = α ( x ) − r ( x ) 1 . ˆ

Asymptotics of the minimizing probability : 1 h ∈H ( T ) log P (1 (1 . 7) J 0 ( κ ) := lim inf T log V T ( h ) ≤ κ ) . T T →∞ concerns the problem of down-side risk minimization for the given target growth rate κ . Setting n + m = M , n = N , and f ( x, h ) = − 1 2 h ∗ σσ ∗ ( x ) h + h ∗ ˆ α ( x ) + r ( x ) , ϕ ( x, h ) = σ ∗ ( x ) h, we arrive at the above problem (1.2) with S ( x ) = σσ ∗ ( x ) = δ ∗ δ ( x ). X i t = log S i Complete market case : n = 0, m = N = M , t m t = { α i ( X t ) − 1 j ( X t ) dW j 2( σσ ∗ ( X t )) ii } + dX i σ i X i 0 = x i ∑ t , j =1 t = e x i e X i t satisfies (1.5) with s i = e x i . regarded as factors and S i

Related contributions • Upside chance maximization Pham ’03; Hata-Sekine ’05,’10; Hata-Iida ’06; Sekine ’06 Knispel ’12 • Downside risk minimization for specified models Hata - N. - Sheu ’10, AAP; Hata ’11 APFM, • Under partial information ( for hidden Markov models) Y. Watanabe, ’11, Ph.D. thesis, Osaka Univ. • Under partial information (for Linear Gaussian Models) N. ’11 May, QF • Full information (General factor models) N. : to appear in AAP; Hata and Sheu: on going • With constraints Sekine: to appear in FS • Robust downside risk minimization for one factor models T. Kagawa ’12 master thesis, Osaka Univ.

Large deviation estimates for controlled semi-martingale X 0 = x ∈ R N , (1 . 1) dX t = λ ( X t ) dW t + β ( X t ) dt, 1 ( 1 ) (1 . 2) J ( κ ) := lim T inf log P T F T ( X . , h . ) ≤ κ . h . T →∞ ∫ T ∫ T 0 ϕ ( X s , h s ) ∗ dW s F T ( X . , h . ) = F 0 + 0 f ( X s , h s ) ds + f ( x, h ) := − 1 2 h ∗ S ( x ) h + h ∗ g ( x ) + U ( x ) , ϕ ( x, h ) = δ ( x ) h, S ( x ) : R N �→ R m ⊗ R m , g ( x ) : R N �→ R m , δ ( x ) : R N �→ R M ⊗ R m ,

Risk-sensitive control and its H-J-B equation Assume that F 0 = 0 Consider 1 (2 . 1) χ ( θ ) = lim ˆ h ∈A ( T ) J ( x ; h ; T ) , inf θ < 0 , T T →∞ where J ( x ; h ; T ) = log E [ e θ { ∫ T 0 f ( X s ,h s ) ds + ∫ T 0 ϕ ( X s ,h s ) ∗ dW s } ] , (2 . 2) and h ranges over the set A ( T ) of all admissible investment strate- gies defined by A ( T ) = { h ∈ H ( T ); E [ e θ ∫ T ∫ T s δ ∗ ( X s ) dW s − θ 2 0 h ∗ 0 h ∗ s δ ∗ δ ( X s ) h s ds ] = 1 } . 2 Then, we shall see that (2.1) could be considered the dual problem to (1.2).

Assumptions (2 . 3) λ, β, S, g, δ are smooth and globally Lipschitz , U is smooth c 4 | x | 2 − c 5 ≤ U ( x ); U, | DU | ≤ c 6 | x | 2 + c 7 c 2 | ξ | 2 ≤ ξ ∗ λλ ∗ ( x ) ξ ≤ c 3 | ξ | 2 , c 2 , c 3 > 0 , ξ ∈ R n , (2 . 4) δ ∗ δ ( x ) ≥ c δ I, (2 . 5) c δ > 0 c 0 δ ∗ δ ( x ) ≤ S ( x ) ≤ c 1 δ ∗ δ ( x ) , x ∈ R N , c 0 , c 1 > 0 (2 . 6)

Note that, when setting Q θ := S ( x ) − θδ ∗ δ ( x ) , θ ≤ 0 , Q θ satisfies ( c 0 − θ ) δ ∗ δ ( x ) ≤ Q θ ( x ) ≤ ( c 1 − θ ) δ ∗ δ ( x ) (2 . 7) and θ θ c 0 − θ ( δ ∗ δ ( x )) − 1 ≤ θQ − 1 θQ − 1 c 1 − θ ( δ ∗ δ ( x )) − 1 , (2 . 8) ( x ) ≤ ( x ) θ θ Moreover, c 0 δ ∗ ≤ I c 0 − θI ≤ I + θδQ − 1 (2 . 9) θ holds. Indeed, (2.7) follows directly from (2.6) and thus (2.8) is obtained from (2.7). The lefthand side of (2.9) is seen since θ θ c 0 − θδ ( δ ∗ δ ) − 1 δ ∗ ≤ θδQ − 1 δ ∗ , c 0 − θI ≤ θ which follows from (2.8). The right hand side of (2.9) is obvious.

Value function (2 . 13) h . ∈A ( T − t ) log E [ e θ { ∫ T − t f ( X s ,h s ) ds + ∫ T − t ϕ ( X s ,h s ) ∗ dW s } ] . v ( t, x ) = inf 0 0 Under P h ( A ) = E [ e θ ∫ T ∫ T s δ ∗ ( X s ) dW s − θ 2 0 h ∗ 0 h ∗ s δ ∗ δ ( X s ) h s ds : A ] , 2 X t satisfies dX t = { β ( X t ) + θλδ ( X t ) h t } dt + λ ( X t ) dW h X 0 = x t , with B. M. W h t defined by ∫ t W h t := W t − γ 0 δ ( X s ) h s ds h . ∈A ( T ) log E h [ e θ ∫ T − t { f ( X s ,h s )+ θ 2 h ∗ s δ ∗ δ ( X s ) h s } ds ] (2 . 14) v ( t, x ) = inf 0

The H-J-B equation : ∂t + 1 2 tr[ λλ ∗ D 2 v ] + 1  ∂v 2 ( Dv ) ∗ λλ ∗ Dv     + inf h { [ β + θλδh ] ∗ Dv + θf ( x, h ) + θ 2  2 | ϕ ( x, h ) | 2 ) } = 0 ,     v ( T, x ) = 0  which is written as (2 . 15) 2 ( Dv ) ∗ λ ( I + θδQ − 1 ∂t + 1 θ Dv + 1 ∂v 2 tr[ λλ ∗ D 2 v ] + β ∗ δ ∗ ) λ ∗ Dv  θ      2 g ∗ Q − 1 + θ g + θU = 0 , θ     v ( T, x ) = 0 ,  where β θ = β + θλδQ − 1 Q θ = S − θδδ ∗ . g, θ

Since ( c 0 − θ ) δ ∗ δ ( x ) ≤ Q θ ( x ) ≤ ( c 1 − θ ) δ ∗ δ ( x ) (2 . 7) N θ := I + θδQ − 1 δ ∗ θ satisfies c 0 (2 . 9) c 0 − θI ≤ N θ ≤ I Further, = ( δ ∗ δ ) − 1 δ ∗ N θ δ ( δ ∗ δ ) − 1 − ( δ ∗ δ ) − 1 θQ − 1 θ

Rewritten ∂t + 1 ∂v 2 tr[ λλ ∗ D 2 v ] + { β − λδ ( δ ∗ δ ) − 1 g } ∗ Dv       + 1 2 [ λ ∗ Dv + δ ( δ ∗ δ ) − 1 g ] ∗ N θ [ λ ∗ Dv + δ ( δ ∗ δ ) − 1 g ]       − 1 2 g ∗ ( δ ∗ δ ) − 1 g + θU = 0 ,       v ( T, x ) = 0      is the H-J-B equation of the stochastic control problem: ∫ T v ∗ (0 , x ; T ; θ ) = sup z . E [ 0 Φ( Y s , z s ) ds ] subject to dY t = λ ( Y t ) dB t + { G ( Y t ) + λ ( Y t ) z t } dt, Y 0 = x, G ( y ) = β ( y ) − λδ ( δ ∗ δ ) − 1 g Φ( y, z ; θ ) = − 1 z + g ∗ ( δ ∗ δ ) − 1 δ ∗ ( y ) z − 1 2 z ∗ N − 1 2 g ∗ ( δ ∗ δ ) − 1 g ( y ) + θU ( y ) . θ

Convexity We can see that Φ( y, z ; θ ) is nothing but a linear function of θ and thus the value function of this stochastic control problem is a convex function of θ . Further, under suitable conditions we have a solution to H-J-B equation of ergodic type: χ ( θ ) = 1 θ Dv + 1 2 tr[ λλ ∗ D 2 v ] + β ∗ 2 ( Dv ) ∗ λN θ λ ∗ Dv (2 . 16) 2 g ∗ Q − 1 + θ g + θU, θ and 1 (2 . 17) χ ( θ ) = lim T v (0 , x ; T ; θ ) , T →∞ Then, we see the convexity of χ ( θ ).

Analytical results Under the assumptions (2.3) - (2.6) H-J-B equation (2.15) has a unique solution such that v ( t, x ) ≤ K 0 ∂ 2 v ∂x i ∂x j , ∈ L p (0 , T ; L p v, ∂v ∂v loc ( R n )) ∂t , ∂x i , ∂v ∂t ≥ − C ∂ 2 v ∂ 2 v ∂ 3 v ∂ 3 v ∂x i ∂x j ∂t ∈ L p (0 , T ; L p loc ( R n )) ∂ 2 t , ∂x i ∂t , ∂x i ∂x j ∂x k , | Dv | 2 + c 0 ν 1 ( ∂v ∂t + C ) ≤ c ( | DN θ | 2 2 r + | N θ | 2 2 r + | D ( λλ ∗ ) | 2 2 r + | Dβ θ | 2 2 r + | β θ | 2 2 r + | U | 2 r + | DU | 2 r + | g | 2 2 r + | Dg | 2 2 r + 1) x ∈ B r , t ∈ [0 , T ) cf. Bensoussan-Frehse-N ’98 AMO, N. ’03 SICON

Verification h ( t, x ) := Q − 1 ( δ ∗ λ ∗ v ( t, x ) + g ( x )) ˆ θ h ( T ) ˆ := ˆ h ( t, X t ) is the optimal strategy: t log E [ e θ { ∫ T ) ds + ∫ T h ( T ) h ( T ) ) ∗ dW s } ] 0 f ( X s , ˆ 0 ϕ ( X s , ˆ v (0 , x ; T ) = s s inf h . ∈A ( T ) log E [ e θ { ∫ T 0 f ( X s ,h s ) ds + ∫ T 0 ϕ ( X s ,h s ) ∗ dW s ] = Moreover, z ( t, y ) := N θ { λ ∗ Dv ( t, x ) + δ ( δ ∗ δ ) − 1 g ( x ) } ˆ dY t = λ ( Y t ) dB t + { G ( Y t ) + λ ( Y t )ˆ z ( t, Y t ) } dt ∫ T ∫ T v (0 , x ; T ) = E [ 0 Φ( Y s , ˆ z s ) ds ] = sup z . E [ 0 Φ( Y s , z s ) ds ]