ITN-Marie Curie ”Deterministic and Stochastic Controlled Systems and Application” Some Applications of SDEs (BSDEs) with Oblique Reflection PHD Student : GASSOUS M. A. SUPERVISOR : RASCANU A. UNIVERSITY ”ALEXANDRU IOAN CUZA” IAS ¸I FACULTY OF MATHEMATICS Spring School ”Stochastic Control In Finance” Roscoff 2010.
Introduction � � Applications • Reflected SDE in time-dependent domains • Two examples in Economics (Backward case) • Switching Games (Backward case) � References
Introduction Forward case (solved): - Lions and Sznitman, 1984. - Dupuis and Ishii, 1993.
- Gassous and Rascanu: � dX t + R ( X t ) ∂ϕ ( X t ) ( dt ) ∋ f ( t, X t ) dt + g ( t, X t ) dB t , t > 0 , X 0 = ξ , (1) where ϕ : R d → ] −∞ , + ∞ ] is a proper convex lower-semicontinuous � � function, ∂ϕ is the subdifferential of ϕ and R = d × d ∈ r i,j � R d ; R 2 d � C 2 is a symmetric matrix such that for all x ∈ R d , b 1 c | u | 2 ≤ � R ( x ) u, u � ≤ c | u | 2 , ∀ u ∈ R d (for some c ≥ 1) .
When ϕ = I O , we have X t ∈ C ([0 , ∞ [ , O ) , k t ∈ C ([0 , ∞ [ , R d ) ∩ BV loc ( R + , R d ) , (1) � t � t (2) X t + k t = x 0 + 0 f ( X s ) ds + 0 g ( X s ) dBs, for t ≥ 0 , � t � t (3) � k � t = 0 1 bd ( O ) ( x ( s )) d � k � s , k ( t ) = 0 γ ( x ( s )) d � k � s . (2)
Backward case (in work): - Ramasubramanian, 2002 (special domain) � dY t − R ( Y t ) ∂ϕ ( Y t ) ( dt ) ∋ − f ( t, Y t , Z t ) dt + Z t dB t , t ≥ 0 , (3) Y T = ξ .
Application a k − dimensional standard BM on a com- We consider ( B t ) t ≥ 0 plete probability space (Ω , F , P ), and {F t : t ≥ 0 } the natural fil- tration. 1.Reflected Stochastic differential equations in time-dependent domains Let K be a subset of R + × R n such that the projection of K onto time axis is [0 , T [, and for each 0 ≤ t < T, K ( t ) = { x ∈ R n : ( t, x ) ∈ K } is a bounded connected open set in R n . Let n ( t, x ) be the unit inward normal of K ( t ) and → γ be the unit inward normal vector field on ∂K.
Theorem Suppose that → γ · n ≥ c 0 on ∂K for some c 0 > 0 . Then for each ( s, x ) ∈ K with s < T, there is a unique pair of adapted continuous processes ( X s,x , L s,x ) s.t. t, X s,x ∈ K for t ∈ [ s, T [ , with X s,x � � ( i ) = x, s t L s,x is a nondecreasing process with L s,x � � , t ∈ [ s, T [ ( ii ) = 0 s.t. s t � t L s,x s 1 ∂K ( r, X r ) dL s,x = r , t � t � t � t X s,x s b ( r, X s,x s σ ( r, X s,x s n ( r, X s,x ) dL s,x ( iii ) = x + ) ds + ) dB r + r . r r r t Proof We remark that the last equation is equivalent to an equation with an oblique reflection vector field n verified by the time-space diffusion process (t, X s,x ) in K. t
2. Two examples in Economics (BSDE) : We consider the RBSDE in an d − dimensional positive orthant G x ∈ R d : x i > 0 , 1 ≤ i ≤ d � � with oblique reflection G = : � T � T � T Y ( t ) = ξ + b ( s, Y ( s )) ds + R ( s, Y ( s )) dK ( s ) − t � Z ( s ) , dB ( s ) t t with Y ( · ) ∈ G for all 0 ≤ t ≤ T ; and K i (0) = 0 , K i ( · ) continuous, nondecreasing with � t K i ( t ) = (3) 0 I { 0 } ( Y i ( s )) dK i ( s ) . This equation has a unique solution (see [1]).
⋆ Backward stochastic analogue of subsidy-surplus model considered in Ramasubramanian [1] We consider an economy with d interdependent sectors, with the following interpretation (a) Y i ( t ) = current surplus in Sector i at time t ; (b) K i ( t ) = cumulative subsidy given to Sector i over [0 , t ]; (c) ξ i = desired surplus in Sector i at time T ; � t (d) s b i ( u, Y ( u )) du = net production of Sector i over [ s, t ] due to evolution of the system; this being negative indicates there is net consumption;
� t s r − (e) ij ( u, Y ( u )) dK j = amount of subsidy for Sector j mobi- lized from Sector i over [ s, t ]; � t s r + (f) ij ( u, Y ( u )) dK j = amount of subsidy mobilized for Sector j which is actually used in Sector i (but not as subsidy in Sector i ) over [ s, t ]. The condition (3) in RBSDE ( ξ, b, R ) means that subsidy for Sector i can be mobilized only when Sector i has no surplus. (The uniform spectral radius condition would mean that the sub- sidy mobilized from external sources is nonzero; so this would be an ‘open’ system in the jargon of economics).
⋆ Backward stochastic (oblique) analogue of projected dy- namical system Suppose the system represents d traders each specializing in a different commodity. For this model we assume: r ij ( · , · ) ≤ 0 , i � = j ; Y i ( t ) = current price of Commodity i at time t ; there is a price floor viz. prices cannot be negative; K i ( t ) = cumulative adjustment involved in the price of Com- modity i over [0 , t ]; b i ( t, Y ( t )) dt = infinitesimal change in price of Commodity i due to evolution of the system;
ξ i = desired price level of Commodity i at time T . Condition (3) then means that adjustment dK i ( t ) can take place only if the price of Commodity i is zero. � t s r − ij ( u, Y ( u )) dK j ( u ) = adjustment from Trader i when price of Commodity j is zero. Note that dK j ( · ) can be viewed upon as a sort of artificial/forced infinitesimal consumption when the price of Commodity j is zero to boost up the price;
hence r − ij ( t, Y ( t )) dK j ( t ) is the contribution of Trader i towards this forced consumption. (As before, the uniform spectral radius condition) implies that there is nonzero ‘external adjustment’, like perhaps governmental intervention/consumption to boost prices when prices crash).
3. Switching Games(Ying-Hu and Shanjian Tang) Consider two players I and II, who use their respective switching control processes a ( · ) and b ( · ) to control the following BSDE : A ( a ) ( T ) − A ( a ) ( t ) B ( b ) ( T ) − B ( b ) ( t ) � � � � U ( t ) = ξ + − � T � T f ( s, U ( s ) , V ( s ) , a ( s ) , b ( s )) ds − + V ( s ) dB ( s ) , t t A a ( . ) ( · ) and B b ( · ) ( · ) are the cost processes associated where with the switching control processes a ( · ) and b ( · ). Under suitable conditions, the above BSDE has a unique adapted solution, denoted by ( U a ( · ) ,b ( · ) , V a ( · ) ,b ( · ) ).
Player I chooses the switching control a ( · ) from a given finite set to minimize the cost min −− > J ( a ( · ) , b ( · )) = U a ( · ) ,b ( · ) (0) and each of his instantaneous switching from one scheme i ∈ Λ ′ ∈ Λ incurs a positive cost which to another different scheme i � i, i ′ � . will be specified by the function k While Player II chooses the switching control b ( · ) from a given finite set Π to maximize the cost max −− > J ( a ( · ) , b ( · )) and each of his instantaneous switching from one scheme j ∈ Π to another different scheme j ′ ∈ Π incurs a positive cost which will be specified by the function l ( j, j ′ ),
� ∞ � Let j =0 increasing sequence of stopping time, α j F θ j − measurable θ j r.v with value in Λ , then a admissible switching strategy for player I: N � a ( s ) = α 0 χ { θ 0 } ( s ) + α j − 1 χ ( θ j − 1 ,θ j ] ( s ) , j =1 therefore N − 1 A a ( · ) ( s ) = � � � χ [ θ j ,T ] ( s ) . k α j − 1 , α j j =1
We are interested in the existence and the construction of the value process as well as the saddle point. The solution of the above-stated switching game will appeal the reflected backward stochastic differential equation with oblique reflection:
� T � � Y i,j ( t ) = ξ i,j + f s, Y ij ( s ) , Z ij ( s ) , i, j ds t � T � T � T − dK ij ( s ) + dL ij ( s ) − Z ij ( s ) dB ( s ) t t t � � i, i ′ �� Y i,j ( t ) ≤ min Y i ′ ,j ( t ) + k , i ′ � = i � � j, j ′ �� Y i,j ( t ) ≥ max Y i,j ′ ( t ) − l , (4) i ′ � = i T i, i ′ �� � � � dK ij ( s ) = 0 , Y i,j ( s ) − min Y i ′ ,j ( s ) + k i ′ � = i 0 T j, j ′ �� � � � dL ij ( s ) = 0 . Y i,j ( s ) − max Y i,j ′ ( t ) − l i ′ � = i 0
We define ( a ∗ ( · ) , b ∗ ( · )) as follows: θ ∗ 0 := 0 , τ ∗ 0 := 0; α ∗ 0 := i, β ∗ 0 := j. We define stopping times θ ∗ p , τ ∗ p ; α ∗ p , β ∗ p in the following inductive manner: θ ∗ p := inf { s ≥ θ ∗ p − 1 ∧ τ ∗ p − 1 : Y α ∗ p − 1 ( s ) = min { Y i ′ ,β ∗ p − 1 ( s ) p − 1 ,β ∗ i ′ � = i ′ ) }} ∧ T, + k ( α ∗ p − 1 , i τ ∗ p := inf { s ≥ θ ∗ p − 1 ∧ τ ∗ p − 1 : Y α ∗ p − 1 ( s ) = max { Y α ∗ p − 1 ,j ′ ( s ) p − 1 ,β ∗ j ′ � = j ′ ) }} ∧ T. − l ( β ∗ p − 1 , j
Theorem Under the usual hypothesis. Let ( Y, Z, K, L ) solution in the space S 2 × M 2 × N 2 × N 2 to RBSDE (4 ). Then we have the represen- tation : U a ( · ) Y ij ( t ) = ess inf ( t ) . j a ( · ) ∈A i t
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