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The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications A white noise approach to optimal insider control of systems with delay Olfa Draouil Joint work with Bernt Øksendal Department of Mathematics - University of Oslo November, 2018 Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications The Donsker delta functional 1 Optimal insider Control problem for SDDE 2 Transforming the insider control problem to a related 3 parameterized non-insider problem Maximum principle theorems 4 A sufficient-type maximum principle Necessary maximum principle Applications 5 Optimal inside harvesting in a population modelled by a delay equation Optimal insider portfolio in a financial market with delay Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications ◮ Let (Ω , F , F , P ) a filtered probability space, where ◮ Ω = S ′ ( R ) the dual of Schwartz space ◮ F = {F t } t ≥ 0 is the sigma algebra generated by a Brownian motion B ( t ) and an independent compensated Poisson random measure ˜ N ( dt , d ζ ) ◮ P is the gaussian measure on S ′ ( R ) characterized by � e i � ω,φ � P ( d ω ) = e − 1 2 � φ � 2 , φ ∈ S ( R ) Ω � ◮ � φ � 2 = � φ � 2 R | φ ( x ) | 2 dx L 2 ( R ) = Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications List of notation : ◮ ( S ) ∗ = the Hida stochastic distribution space. ◮ F ⋄ G = the Wick product of random variables F and G . n ! F ⋄ n (The Wick exponential of F .) ◮ exp ⋄ ( F ) = Σ ∞ 1 n = 0 ◮ D t F = the Hida-Malliavin derivative of F at t with respect to B ( · ) . G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance , Universitext, Springer, 2009. Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications We assume that the inside information is of initial enlargement type. Specifically, we assume that the inside filtration H has the form H = {H t } 0 ≤ t ≤ T , where H t = F t ∨ σ ( Z ) (0.1) for all t , where Z is a given F T 0 -measurable random variable, for some T 0 > 0 (constant). ◮ Z has a Donsker delta functional. Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications The Donsker delta functional Definition Let Z : Ω → R be a random variable which also belongs to ( S ) ∗ . Then a continuous functional δ Z ( . ) : R → ( S ) ∗ (1.1) is called a Donsker delta functional of Z if it has the property that � g ( z ) δ Z ( z ) dz = g ( Z ) a . s . (1.2) R for all (measurable) g : R → R such that the integral converges. Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications The Donsker delta functional for a class of Itô - Lévy processes Consider the special case when Z is a first order chaos random variable of the form Z = Z ( T 0 ); where � t � t � ψ ( s , ζ )˜ Z ( t ) = β ( s ) dB ( s ) + N ( ds , d ζ ); t ∈ [ 0 , T 0 ] (1.3) 0 0 R for some deterministic functions β � = 0 , ψ satisfying � T 0 � { β 2 ( t ) + ψ 2 ( t , ζ ) ν ( d ζ ) } dt < ∞ a.s. (1.4) 0 R Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications In this case it is well known that the Donsker delta functional exists in ( S ) ∗ and is given by � exp ⋄ � � T 0 � � T 0 δ Z ( z ) = 1 ( e ix ψ ( s ,ζ ) − 1 )˜ N ( ds , d ζ ) + ix β ( s ) dB ( s ) 2 π R 0 R 0 � T 0 � � ( e ix ψ ( s ,ζ ) − 1 − ix ψ ( s , ζ )) ν ( d ζ ) − 1 2 x 2 β 2 ( s ) } ds − ixz + { dx . 0 R (1.5) Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications The Donsker delta functional for a Gaussian process Consider the special case when Z is a Gaussian random variable of the form � t Z = Z ( T 0 ); where Z ( t ) = β ( s ) dB ( s ) , for t ∈ [ 0 , T 0 ] (1.6) 0 for some deterministic function β ∈ L 2 [ 0 , T 0 ] . In this case it is well known that : 2 exp ⋄ [ − ( Z − z ) ⋄ 2 δ Z ( z ) = ( 2 π v ) − 1 ] (1.7) 2 v where ◮ v := � β � 2 [ 0 , T 0 ] . Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications We consider an insider’s optimal control problem for a stochastic process X ( t ) = X ( t , Z ) = X u ( t , Z ) defined as the solution of a stochastic differential delay equation of the form   dX ( t ) = dX ( t , Z ) = b ( t , X ( t , Z ) , Y ( t , Z ) , u ( t , Z ) , Z ) dt     + σ ( t , X ( t , Z ) , Y ( t , Z ) , u ( t , Z ) , Z ) dB ( t ) � R γ ( t , X ( t , Z ) , Y ( t , Z ) , u ( t , Z ) , Z , ζ )˜  + N ( dt , d ζ ) , 0 ≤ t ≤ T     X ( t ) = ξ ( t ) , − δ ≤ t ≤ 0 (2.1) where Y ( t , Z ) = X ( t − δ, Z ) , (2.2) δ > 0 being a fixed constant (the delay). Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications ◮ u ( t , Z ) = u ( t , x , z ) z = Z is our insider control process, which is allowed to depend on both Z and F t . ◮ In other words, u ( . ) is assumed to be H -adapted, ◮ such that u ( ., z ) is F -adapted for each z ∈ R . Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications ◮ Let U denote the set of admissible control values. ◮ We assume that the functions b ( t , x , y , u , z ) = b ( t , x , y , u , z , ω ) : [ 0 , T ] × R × R × U × R × Ω �→ R σ ( t , x , y , u , z ) = σ ( t , x , y , u , z , ω ) : [ 0 , T ] × R × R × U × R × Ω �→ R γ ( t , x , y , u , z , ζ ) = γ ( t , x , y , u , z , ζ, ω ) : [ 0 , T ] × R × R × U × R × R × Ω �→ ◮ are given C 1 functions with respect to x , y and u ◮ adapted processes in ( t , ω ) for each given x , y , u , z , ζ . ◮ Jacod condition holds. Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications ◮ Let A be a given family of admissible H − adapted controls u . ◮ The performance functional J ( u ) of a control process u ∈ A is defined by � T J ( u ) = E [ f ( t , X ( t , Z ) , u ( t , Z ) , Z )) dt + g ( X ( T , Z ) , Z )] , (2.3) 0 ◮ where f ( t , x , u , z ) : [ 0 , T ] × R × U × R �→ R g ( x , z ) : R × R �→ R (2.4) ◮ C 1 with respect to x and u . Olfa Draouil A white noise approach to optimal insider control of systems with delay

The Donsker delta functional Optimal insider Control problem for SDDE Transforming the insider control problem to a related parameterized non-insider problem Maximum principle theorems Applications The problem we consider is the following : Problem Find u ⋆ ∈ A such that J ( u ) = J ( u ⋆ ) . sup (2.5) u ∈A Olfa Draouil A white noise approach to optimal insider control of systems with delay