The Risk-Sensitive Switching Problem Under Knightian Uncertainty - PDF document

The Risk-Sensitive Switching Problem Under Knightian Uncertainty S.Hamad` ene & H.Wang University of Le Mans, Fr. New Advances in BSDEs for financial engineering applications, Tamerza, Oct.25-28, 2010

Let B := ( B t ) t ≤ T a BM on a probability space (Ω , F , P ) ; ( F t ) t ≤ T the completed natural fil- tration of B . A switching problem is a stochastic control where the decision maker moves among m states or modes when she decides according to the best profitability. There are several works on the switching prob- lem (H.-Jeanblanc, Djehiche-H.-Popier, H.-Zhang, Hu-Tang, Zervos (s.p.), Ly Vath-Pham, Ly Vath-Pham-XYZ , Zervos, Porchet-Touzi, Carmona-Ludkowski,...). Examples of a switching problems • In financial markets when a trader invests his/her money between several assets (economies) according their profitability

• In the energy market when a manager of a power plant puts it in the mode which occurs the best profitability. in assets investor puts his money in in the case A strategy of switching has two components (when m ≥ 3): - ( τ n ) n ≥ 0 an increasing sequence of stopping times: they are the times when the decision maker decides to switch. - a sequence ( ξ n ) n ≥ 0 ) of r.v. with values in J := { 1 , ..., m } (the different states) such that ξ n is F τ n -measurable which stands for the state to which the system is switched at τ n from its current one. Remark: When m = 2, a strategy has only one component, i.e., stopping times.

With a strategy ( δ, ξ ) = (( τ n ) n ≥ 0 , ( ξ n ) n ≥ 0 ) is associated an indicator of the state of the sys- tem which is ( u t ) t ≤ T given by: u 0 = 1 and u t = ξ n if t ∈ ] τ n , τ n +1 ] ( n ≥ 0) . When a strategy ( δ, ξ ) is implemented usually the yield is given by: J ( δ, ξ ) := ∫ T ∑ E [ 0 ψ u s ( s ) ds − ℓ u τn − 1 ,u τn ( τ n )1 1 [ τ n <T ] ] n ≥ 1 where • ψ k ( t, ω ) is the instantaneous profit in the state k • ℓ kl ( t, ω ) ≥ c > 0 is the switching cost from state k to state l at t .

The problem is to focus on J ∗ := sup J ( δ, ξ ) . ( δ,ξ ) This problem is linked to systems of Reflected BSDEs with inter-connected obstacles or oblique reflection of the following type: for i ∈ J := { 1 , ...., m } , ∫ T ∫ T  Z i u dB u + K i T − K i Y i t = ψ i ( u ) du −   t   t t   t ≥ max j ∈J − i {− ℓ ij ( t ) + Y j Y i  t } , ∫ T   0 ( Y i j ∈J − i {− ℓ ij ( u ) + Y j u } ) dK i  u − max u = 0 .     (1) where K i are continuous and non-decreasing and J − i := J − { i } .

The solution of (6) provides the optimal strat- egy ( δ ∗ , ξ ∗ ) and J ∗ 1 = Y 1 0 (Djehiche, H., Popier, 07). Knightian uncertainty: means that the proba- bility of the future is not fixed and a family of probabilities P u are likewise. Risk-sensitiveness: means that the criterion is of type E [ e θζ ] where θ is related to risk attitude of the con- troller. So let us set: J ( δ, ξ ; u ) := ∫ T 0 ( ψ u s ( s, X s ) + h ( s, X s , u s )) ds − A δ E u [exp { T } ] where

• X verifies dX t = ϱ ( t, X t ) dt + σ ( t, X t ) dB t , t ≤ T are factors which determine prices in the mar- ket and ∀ λ > 0 , E [ e λ sup t ≤ T | X t | ] < ∞ . • u := ( u t ) t ≤ T is a stochastic process valued in U (not bounded) • P u is a probability such that: ∫ . dP u dP = E T ( 0 b ( t, X t ) dB t ) • A δ T := ∑ n ≥ 1 ℓ u τn − 1 ,u τn ( τ n , X τ n )1 1 [ τ n <T ] • h is a premium which satisfies: l ( u ) ≤ h ( t, x, u ) ≤ C (1 + | x | + l ( u )) with l ( u ) → ∞ as | u | → ∞ .

Problem: Characterization, properties and com- putation of J ∗ = sup inf u J ( δ, ξ ; u ) . δ Does an optimal strategy ( δ ∗ , u ∗ ) exist? So let H be the hamiltonian of the problem, H ( t, x, z, u ) := zb ( t, x, u ) + h ( t, x, u ) and H ∗ ( t, x, z ) := inf u ∈ U H ( t, x, z, u ) . Assume hereafter m = 2. The system of reflected BSDEs associated with the problem is:

∫ T  t [ ψ 1 ( s, X s ) + H ∗ ( s, X s , Z 1 • Y 1 t = s )+      ∫ T 1  2 | Z 1 s | 2 ] ds − t Z 1 s dB s + K 1 T − K 1  t ;        ∫ T    t [ ψ 2 ( s, X s ) + H ∗ ( s, X s , Z 2  • Y 2 t = s )+      ∫ T  1 2 | Z 2 s | 2 ] ds − t Z 2 s dB s + K 2 T − K 2  t ;     • Y 1 t ≥ Y 2  t − ℓ 12 ( t, X t );     [ Y 1 t − Y 2 t + ℓ 12 ( t, X t )] dK 1 t = 0;          • Y 2 t ≥ Y 1  t − ℓ 21 ( t, X t );     [ Y 2 t − Y 1 t + ℓ 21 ( t, X t )] dK 2  t = 0 .  (2) Verification theorem: If there exist two triplets of processes ( Y i , Z i , K i ), i = 1 , 2 which satisfy (2) then we have: exp { Y 1 0 } = sup u ∈U J ( δ, u ) inf δ ∈D and the optimal strategy ( δ ∗ , u ∗ ) is given by

τ ∗ 0 := 0 and for n = 0 , · · · , τ ∗ inf { t ≥ τ ∗ 2 n : Y 1 t = Y 2 := t − ℓ 12 ( t, X t ) } 2 n +1 τ ∗ inf { t ≥ τ ∗ 2 n +1 : Y 2 t = Y 1 := t − ℓ 21 ( t, X t ) } . 2 n +2 and u ∗ [ u ∗ ( t, X t , Z 1 ∑ t := t )1 [ τ ∗ 2 n +1 ) ( t ) + 2 n ,τ ∗ n ≥ 0 u ∗ ( t, X t , Z 2 t )1 [ τ ∗ 2 n +2 ) ( t )] . 2 n +1 ,τ ∗ Sketch of the proof: the problems are related to the lack of integrability and of regularity of the data of the problem. Step 1: Expression of the payoffs via BSDEs Let ( δ, u ) admissible. Then there exists a unique pair of P -measurable processes ( Y δ,u , Z δ,u ) such ∫ T 0 | Z δ,u | 2 ds < ∞ , the process that P -a.s, s + ∫ t t e Y δ,u 0 h ( s,X s ,u s ) ds ) t ≤ T is of class [D] and ( L u t

for any t ≤ T , ∫ T Y δ,u t ( ψ δ ( s, X s ) + H ( s, X s , u s , Z δ,u = − A δ T + ) s t ∫ T 2 | Z δ,u t Z δ,u + 1 | 2 ) ds − dB s . s s (3) Moreover, we have: ∫ T exp { Y δ,u = E u [exp { 0 ( ψ δ ( s, X s ) } 0 + h ( s, X s , u s )) ds − A δ (4) T } ] = J ( δ, u ) . Step 2: Let δ ∈ D , then there exists a unique pair of P -measurable processes ( Y δ, ∗ , Z δ, ∗ ) such that ( e Y δ, ∗ p ≥ 1 S p , ) t ≤ T ∈ E := ∩ t ( e Y δ, ∗ Z δ, ∗ ) t ≤ T ∈ H 2 ,d and for any t ≤ T , t t ∫ T Y δ, ∗ t ( ψ δ ( s, X s ) + H ∗ ( s, X s , Z δ, ∗ = − A δ T + s ) t ∫ T 2 | Z δ, ∗ t Z δ, ∗ + 1 s | 2 ) ds − s dB s . (5)

Moreover, ∀ t ≤ T , ∀ δ ∈ D , Y δ, ∗ = essinf u ∈U Y δ,u . t t Step 3: Reduction of the problem sup u ∈U J ( δ, u ) = sup inf u ∈U J ( δ, u ) . inf δ ∈D δ ∈B where B := { δ := ( τ n ) n ≥ 0 ∈ D , ∃ K δ , such that τ n = T, for any n ≥ K δ } . Step 4: end of the proof by induction. Let δ ∈ B then by a backward induction we have: 0 ≥ Y δ, ∗ Y 1 . 0 As (in using the system of reflected BSDEs) we have: 0 = Y δ ∗ , ∗ Y 1 0

therefore Y δ, ∗ u ∈U Y δ,u Y 1 0 = sup = sup inf 0 0 δ ∈D δ ∈D which implies that exp( Y 1 u ∈U J ( δ, u ) = J ( δ ∗ , u ∗ ) . 0 ) = sup inf δ ∈D Therefore the problem turns into solving the system (2). Theorem: The system of reflected BSDEs with inter-connected obstacles (2) has a unique so- lution. Sketch of the proof: Step 1: Let us consider the following system:

For i = 1 , ..., m , ∫ T  f i ( u, Y 1 u , ..., Y m u , Z i Y i t = ξ i + u ) du     t ∫ T    Z i u dB u + K i T − K i  −   t  (6) t t ≥ max j ∈J − i h ij ( ω, t, Y j Y i t )    ∫ T    0 ( Y i j ∈J − i h ij ( ω, u, Y j u )) dK i  u − max u = 0 .     We first extend the result by H.-Zhang (07) to the case of continuous coefficients f j with lin- ear growth in using inf-convolution techniques. Step 2: We use an exponential transform for (2) and we obtain:

∫ T Y 1 Y 1 s ) + [ ψ 1 ( s, X s )+ • ¯ t (¯ t = 1 + Z 1 ∫ T ¯ H ∗ ( s, X s , Z 1 K 1 K 1 ¯ s dB s + ¯ T − ¯ s s ) + )] ds − t ; t ¯ ( Y 1 ∫ T Y 2 Y 2 s ) + [ ψ 2 ( s, X s )+ • ¯ t (¯ t = 1 + Z 2 ∫ T ¯ H ∗ ( s, X s , Z 2 K 2 K 2 ¯ s dB s + ¯ T − ¯ s ) + )] ds − s t ; t ¯ ( Y 2 Y 1 t ≥ e − g 12 ( t,X t ) ¯ Y 2 Y 2 t ≥ e − g 21 ( t,X t ) ¯ Y 1 • ¯ t ; ¯ t Y 1 t − e − g 12 ( t,X t ) ¯ Y 2 K 1 • (¯ t ) d ¯ t = 0 and Y 2 t − e − g 21 ( t,X t ) ¯ Y 1 K 2 (¯ t ) d ¯ t = 0 (7) Finally we show that this system has a solution and we go back to (2). Dynamic Programming Principle: Y 1 and Y 2 satisty the following DPP:

∫ τ n Φ u s ( s, X s , Z u s Y 1 t = esssup δ =( τ n ) n ≥ 0 ∈D 1 t E [ s ) ds t k =1 ,n ℓ u τk − 1 ,uτk 1 [ τ k <T ] + Y u τn − ∑ τ n 1 [ τ n <T ] | F t ] where • D 1 is the set of admissible strategies such t that τ 1 ≥ t and u 0 = 1 • Φ i ( t, x, z ) = ψ i ( t, x ) + H ∗ ( t, x, z ) + 1 2 | z | 2 . The same is true for Y 2 . With the help of this DPP we show that: Theorem: Assume that: ( i ) U is compact and h is bounded