Infinite-dimensional methods for path-dependent equations

Infinite-dimensional methods for path-dependent equations (Università di Pisa) 7th General AMaMeF and Swissquote Conference EPFL, Lausanne, 8 September 215

Based on Flandoli F., Zanco G. - An infinite-dimensional approach to path-dependent Kolmogorov equations, to be published on AoP Flandoli F., Russo F., Zanco G. - Infinite-dimensional calculus under weak spatial regularity of the process, in preparation

Motivation Models with general (adapted) dependance on the past Delay in state and/or in control variable Pricing and hedging of path-dependent options Pricing and hedging of options written on stocks described by equations that depend on (at least part of) the previous history of the stocks Analysis of incentives to innovation in firms with an established market, optimal investment strategies depending on knowledge and R&D investment

{ dx(t) = bt (X t ) dt + σ(x t ) dw(t) for t [, T], X() = x

{ dx(t) = bt (X t ) dt + σ(x t ) dw(t) for t [, T], X() = x X t = { X(s) } s [,t],

{ dx(t) = bt (X t ) dt + σ(x t ) dw(t) for t [, T], X() = x X t = { X(s) } s [,t], b = {b t } t [,T], b t : D([, t]; R d ) R d

{ dx(t) = bt (X t ) dt + σ(x t ) dw(t) for t [t, T], X(t) = γ(t) for t [, t ] X t = { X(s) } s [,t], b = {b t } t [,T], b t : D([, t]; R d ) R d

{ dx(t) = bt (X t ) dt + σ(x t ) dw(t) for t [t, T], X(t) = γ(t) for t [, t ] X t = { X(s) } s [,t], b = {b t } t [,T], b t : D([, t]; R d ) R d SDEs: classical. Tsirelson s example. Calculus&PDEs: very recent, Dupire 9, Cont-Fournié 1& 13, Peng 11, Cosso 12, Ekren-Keller-Touzi-Zhang 13& 14, Di Girolami-Russo 1& 14, Cosso-Russo 15, Cosso-Federico-Gozzi-Rosestolato-Touzi 15...

Example b t (γ t ) = g (γ(t), γ(s)) ds

Example b t (γ t ) = g (γ(t), γ(s)) ds Example b t (γ t ) = h (γ(t), γ(t )) 1 [t,t](t)

Example b t (γ t ) = g (γ(t), γ(s)) ds Example b t (γ t ) = h (γ(t), γ(t )) 1 [t,t](t) Example ( b t (γ t ) = w γ(t), ) p(s) dγ(s)

Example b t (γ t ) = g (γ(t), γ(s)) ds Example b t (γ t ) = h (γ(t), γ(t )) 1 [t,t](t) Example ( b t (γ t ) = w γ(t), ) p(s) dγ(s) Example b t (γ t ) = q (γ(t), γ(t τ))

SDEs and Kolmogorov equation Delay equations framework

SDEs and Kolmogorov equation Delay equations framework L p : = R d L p (( T, ); R d ), p 2, D : = R d D b ([ T, ); R d ), { } C : = R d ϕ C b ([ T, ); R d ) : lim ϕ(s), s { } C : = y = ( ϕ x ) C s.t. x = lim ϕ(s). s

backward extension operator L t : D([, t); R d ) D([ T, ); R d ) L t (γ)(s) = γ()1 [ T, t) (s) + γ(t + s)1 [ t,) (s), s [ T, )

backward extension operator L t : D([, t); R d ) D([ T, ); R d ) L t (γ)(s) = γ()1 [ T, t) (s) + γ(t + s)1 [ t,) (s), s [ T, ) restriction operator M t : D([ T, ); R d ) D([, t); R d ) M t (ϕ)(s) = ϕ(s t), s [, t) { M t ( ϕ x Mt ϕ(s) s [, t) ) (s) = x s = t

( ) ˆb (t, ( ϕ x )) = ˆb(t, x, ϕ) := b t M t ( ϕ x ) ; b t (γ) := ˆb(t, γ(t), L t γ)

( ) ˆb (t, ( ϕ x )) = ˆb(t, x, ϕ) := b t M t ( ϕ x ) ; b t (γ) := ˆb(t, γ(t), L t γ) ( ) ( ) X(t) X(t) Y(t) = = {X(t + s)} s [ T,] X [t T,t]

( ) ˆb (t, ( ϕ x )) = ˆb(t, x, ϕ) := b t M t ( ϕ x ) ; b t (γ) := ˆb(t, γ(t), L t γ) dy(t) dt ( ) ( ) X(t) X(t) Y(t) = = {X(t + s)} s [ T,] ( ) ( ) Ẋ(t) = = {Ẋ(t } + + s) Ẋ [t T,t] s X [t T,t] ( ) bt (X t ) + ( ) σt (X t )Ẇ(t)

A ( ) ˆb (t, ( ϕ x )) = ˆb(t, x, ϕ) := b t M t ( ϕ x ) ; b t (γ) := ˆb(t, γ(t), L t γ) ( ) ( ) X(t) X(t) Y(t) = = {X(t + s)} s [ T,] ( ) ( ) dy(t) Ẋ(t) = = {Ẋ(t } + dt Ẋ [t T,t] + s) s ) ( ) ( ( )) x (ˆb (t, ( ϕ x )) :=, B t, := ϕ ϕ ( x ϕ Dom(A) = X [t T,t] ( ) bt (X t ) + ), Σ ( t, ( ) σt (X t )Ẇ(t) ( )) x (ˆσ (t, ( x ) := ϕ )) ϕ { ( x ϕ ) C s.t. ϕ C 1 ( [ T, ); R d)} Y(t) / Dom(A)

Assumption ( ) B L, T; C 2,α b (E, E), Σ constant

Assumption ( ) B L, T; C 2,α b (E, E), Σ constant { dy(t) = AY(t) dt + B(t, Y(t)) dt + Σ dβ(t), t [t, T], Y(t ) = y

Assumption ( ) B L, T; C 2,α b (E, E), Σ constant Y t,y (ω, t) = e (t t)a y + e (t s)a B(s, Y t,y (ω, s)) ds + t e (t s)a Σ dβ(ω, s) t

Assumption ( ) B L, T; C 2,α b (E, E), Σ constant Y t,y (ω, t) = e (t t)a y + e (t s)a B(s, Y t,y (ω, s)) ds + t e (t s)a Σ dβ(ω, s) t Existence and uniqueness of solutions

Assumption ( ) B L, T; C 2,α b (E, E), Σ constant Y t,y (ω, t) = e (t t)a y + e (t s)a B(s, Y t,y (ω, s)) ds + t e (t s)a Σ dβ(ω, s) t Existence and uniqueness of solutions Continuous if E = L p, L if E = D Two times Fréchet differentiable w.r.t. initial data with α-hölder second differential

u t (t, y)+ Du(t, y), Ay + B (t, y) +1 2 Tr ( ΣΣ D 2 u(t, y) ) =, u(t, ) = Φ,

u t (t, y)+ Du(t, y), Ay + B (t, y) +1 2 Tr ( ΣΣ D 2 u(t, y) ) =, u(t, ) = Φ, u(t, y) = E [Φ (Y t,y (T))]

u t (t, y)+ Du(t, y), Ay + B (t, y) +1 2 Tr ( ΣΣ D 2 u(t, y) ) =, u(t, ) = Φ, Tr ( ΣΣ D 2 u(t, y) ) = σj 2 D 2 u(t, y)(e j, e j ) j=1

u t (t, y)+ Du(t, y), Ay + B (t, y) +1 2 Tr ( ΣΣ D 2 u(t, y) ) =, u(t, ) = Φ, Tr ( ΣΣ D 2 u(t, y) ) = d σj 2 D 2 u(t, y)(e j, e j ) j=1

u (t, y) Φ (y) = T t Du (s, y), Ay + B (s, y) ds+ + 1 2 T d t j=1 σ 2 j D 2 u(s, y)(e j, e j ) ds

u (t, y) Φ (y) = T t Du (s, y), Ay + B (s, y) ds+ + 1 2 T d t j=1 σ 2 j D 2 u(s, y)(e j, e j ) ds Definition Given Φ C 2,α b (D, R), we say that u : [, T] D R is a classical solution of the Kolmogorov equation with final condition Φ if ( ) u L, T; C 2,α b (D, R) C ([, T] D, R) for some α (, 1), and satisfies the above identity for every t [, T] and y Dom (A), with the duality terms understood with respect to the topology of D.

Theorem ( ) Let Φ C 2,α (D, R) be given and let B L, T; C 2,α b (D, D). The function u : [, T] D R given by u(t, y) = E [Φ (Y t,y (T))], is a solution of the Kolmogorov equation with final condition Φ under the assumption that for any s [ T, ], any r s, any y C and for almost every a [ T, ] the following hold: ( ( 1 1 DΦ(y), J n 1 [a,] ) DΦ(y), 1 [a,] ) ; and idem for B. ( ( ) D 2 1 Φ(y) J n 1 [a,] ( ( D 2 1 Φ(y) ( ( ) D 2 1 Φ(y) J n 1 [a,] 1 [a,] ) ( ) 1 1 [a,], J n ( 1 1 [a,] ) ( ) 1 1 [a,] ( 1, 1 [a,] ) ) ; ( 1, J n ( 1 1 [a,] ) 1 [a,] ) ) ; ( 1 1 [a,] ) ) ;

Theorem ( ) If Φ : L p R is C 2,α b and B L, T; C 2,α b (L p, L p ) u (t, y) := E [Φ (Y t,y (T))], (t, y) [, T] L p, then the function is a solution of the Kolmogorov equation in L p with final condition Φ.

Change of variables formula What about uniqueness?

Change of variables formula What about uniqueness? Need Ito formula

Change of variables formula What about uniqueness? Need Ito formula dy(t) = AY(t) dt+b(t, Y(t)) dt+σ(t, Y(t)) dβ(t) C,

Change of variables formula What about uniqueness? Need Ito formula dy(t) = AY(t) dt+b(t, Y(t)) dt+σ(t, Y(t)) dβ(t) C, F : [, T] C R

Change of variables formula What about uniqueness? Need Ito formula dy(t) = AY(t) dt+b(t, Y(t)) dt+σ(t, Y(t)) dβ(t) C, F : [, T] C R + F(t, Y(t)) F(, Y()) = t F(s, Y(s)) ds AY(s) + B(s, Y(s), DF(s, Y(s)) ds + + DF(s, Y(s)), Σ(s, Y(s)) dβ(s) 1 2 Tr R d [ Σ(s, Y(s))Σ(s, Y(s)) D 2 F(s, Y(s)) ] ds

Change of variables formula What about uniqueness? Need Ito formula dy(t) = AY(t) dt+b(t, Y(t)) dt+σ(t, Y(t)) dβ(t) C, F : [, T] C R F(t, Y(t)) F(, Y()) = t F(s, Y(s)) ds + AY(s) + B(s, Y(s), DF(s, Y(s)) ds+ DF(s, Y(s)), Σ(s, Y(s)) dβ(s) 1 + 2 Tr [ R d Σ(s, Y(s))Σ(s, Y(s)) D 2 F(s, Y(s)) ] ds Problems: Y(t) / Dom(A) and usually also t F(t, ) makes sense only on Dom(A)

Theorem Let F C ([, T] C) be such that DF C ([, T] C; C ), D 2 F C ([, T] C; L (C; C )), t F C([, T] Dom(A); R). Assume that there exists a continuous function G : [, T] C R such that G(t, y) = t F(t, y) + Ay, DF(t, y) on [, T] Dom(A). Then F(t, Y(t)) = F(, Y()) + + G(s, Y(s)) ds B(s, Y(s)), DF(s, Y(s)) ds + + DF(s, Y(s)), Σ(s, Y(s)) dβ(s) 1 2 Tr R d [ Σ(s, Y(s))Σ(s, Y(s)) D 2 F(s, Y(s)) ] ds

Theorem Let F C ([, T] C) be such that DF C ([, T] C; C ), D 2 F C ([, T] C; L (C; C )), t F exists on T Dom(A), λ(t ) = T and T does not depend on y. Assume that there exists a continuous function G : [, T] C R such that Then G(t, y) = t F(t, y) + Ay, DF(t, y) on T Dom(A). F(t, Y(t)) = F(, Y()) + + G(s, Y(s)) ds B(s, Y(s)), DF(s, Y(s)) ds + + DF(s, Y(s)), Σ(s, Y(s)) dβ(s) 1 2 Tr R d [ Σ(s, Y(s))Σ(s, Y(s)) D 2 F(s, Y(s)) ] ds

Theorem If F, DF, D 2 F are piecewise continuous, i.e. t 1,..., t n s.t. F C ([t j, t j+1 ) E), y t F(t, y) is càdlàg, t y F(t, y) is continuous, t F exists on T Dom(A), then F(t, Y(t)) = F(, Y()) + + B(s, Y(s)), DF(s, Y(s)) ds + + Ĝ(s, Y(s)) ds + F(Y; t 1,..., t n ) DF(s, Y(s)), Σ(s, Y(s)) dβ(s) 1 2 Tr R d [ Σ(s, Y(s))Σ(s, Y(s)) D 2 F(s, Y(s)) ] ds

Example but f t (γ t ) = g (γ(t), γ(s)) ds F ( ) t (t, y) = g y (1), y (2) ( t) F ( t (t, y) + Ay, DF(t, y) = g y (1), y (1))

Example but f t (γ t ) = Example g (γ(t), γ(s)) ds F ( ) t (t, y) = g y (1), y (2) ( t) F ( t (t, y) + Ay, DF(t, y) = g y (1), y (1)) f t (γ t ) = h (γ(t), γ(t )) F ( ) t (t, y) = 2h y (1), y (2) (t t) but F (t, y) + Ay, DF(t, y) = t y (2) (t t)

Example ( F t (t, y) = 2w y (1), ( f t (γ t ) = w γ(t), = Ay, DF(t, y) ) p(s) dγ(s) ) p(s) dy (2) (s t) ṗ(s) dẏ (2) (s t)

Example ( F t (t, y) = 2w y (1), ( f t (γ t ) = w γ(t), = Ay, DF(t, y) ) p(s) dγ(s) ) p(s) dy (2) (s t) ṗ(s) dẏ (2) (s t) For the Kolmogorov equation the equation itself provides the extension: G(t, y) = B(t, y), DF(t, y) 1 2 Tr R d [ ΣΣ D 2 F(t, y) ] therefore uniqueness follows by standard arguments.

Example For f t (γ t ) = q (γ(t), γ(t τ)) we have f t (W t ) q(, ) = + 1 2 + τ 1 q (W(s), W(s τ)) 2 11 q (W(s), W(s τ)) ds + 1 2 2 q (W(s + τ), W(s)) δw(s) + τ 2 22 q (W(s), W(s τ)) ds 2 12 q (W(s), W(s τ)) ds

Comparison with functional Ito calculus ν t i ν t (γ t ) = lim h ( γ he i t ) ν t (γ t ) h D t ν (γ t ) = lim h + ν t+h (γ t,h ) ν t (γ t ) h

Comparison with functional Ito calculus ν t i ν t (γ t ) = lim h ( γ he i t ) ν t (γ t ) h D t ν (γ t ) = lim h + ν t+h (γ t,h ) ν t (γ t ) h { D t ν(γ t ) + b t (γ t ) ν t (γ t ) + 1 2 Tr [ σσ 2 ν t (γ t ) ] =, ν T (γ T ) = f (γ T )

Comparison with functional Ito calculus ν t i ν t (γ t ) = lim h ( γ he i t ) ν t (γ t ) h D t ν (γ t ) = lim h + ν t+h (γ t,h ) ν t (γ t ) h { D t ν(γ t ) + b t (γ t ) ν t (γ t ) + 1 2 Tr [ σσ 2 ν t (γ t ) ] =, ν T (γ T ) = f (γ T ) Theorem (Dupire - 29; Cont, Fournié - 213) (Under suitable assumptions) ν t (X t ) ν (X ) = D s ν s (X s ) ds + + 1 2 ν s (X s ) dx(s) + Tr 2 ν s (X s ) d [X] (s)

Theorem ( ) If ν t (γ): = u(t, γ(t), L t γ) or u(t, x, ϕ): = ν t M t ( ϕ x ) then D i ν t (γ) = x i u(t, x, L t γ), i = 1,..., d. If ν t (γ): = u(t, γ(t), L t γ) and Du and t u exists on Dom(A), then D t ν exists on Dom(A) and D t ν(γ t ) = u t (t, γ(t), L tγ t ) + Du(t, γ(t), L t γ t ), A(L t γ t ). ( ) If u(t, x, ϕ): = ν t M t ( ϕ x ), D t ν exists everywhere and Du, t u exist on Dom(A) then D t ν is the extension.

Theorem If the infinite dimensional liftings B and Φ of b t and f satisfy the assumptions of the previous theorems, then, for almost every t, the function ν t (γ t ) = E [f (X γt (T))] is a solution of the path dependent Kolmogorov equation. Proof: set y = (γ(t), L t γ t ) C ; ν t (γ t ) : = u(t, γ(t), L t γ t ) = u(t, y) = E [Φ (Y t,y (T))] [ ( = E f M (Y (T)))] t,y = E [f (X γt (T))]

Thank you for your kind attention