Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.

Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs Andrea Cosso LPMA, Université Paris Diderot joint work with Francesco Russo ENSTA, ParisTech 7th General AMaMeF and Swissquote Conference September 8th, 2015

Outline 1 Functional vs Banach space stochastic calculus Functional Itô calculus via regularization Comparing the two approaches 2 Path-dependent PDE Path-dependent SDE Strict solutions 3 Strong-viscosity solutions Towards a weaker notion of solution Definition, existence and uniqueness

Functional Itô & Banach space stochastic calculus Functional Itô calculus is an extension of classical Itô calculus designed ad hoc for functionals F (t, X +t, X t ) depending on time t, past and present values of the process X. B. Dupire (2009) Functional Itô calculus. Portfolio Research Paper, Bloomberg. R. Cont & D.-A. Fournié (2013) Functional Itô calculus and stochastic integral representation of martingales. Annals of Probability, 41 (1), 109-133. Banach space stochastic calculus also gives an expansion of F (t, X +t, X t ), but considering the path X +t = (X s+t ) s [ T,0] as an element of the Banach space B = C([ T, 0]) (B can be a generic separable Banach space). C. Di Girolami & F. Russo (2010) Infinite dimensional stochastic calculus via regularization and applications. Ph.D. Thesis, preprint inria-00473947. C. Di Girolami & F. Russo (2014) Generalized covariation for Banach space valued processes, Itô formula and applications. Osaka J. Math., 51 (3), 729-783.

First step: functional Itô calculus via regularization Using regularization techniques, instead of discretization techniques of Föllmer type, is not the only issue. We also investigate other possible improvements of functional Itô calculus: To define functional derivatives, we do not need to extend a functional from C([ T, 0]) to D([ T, 0]), but to a space C ([ T, 0]) which gets stuck as much as possible to the natural space C([ T, 0]). Time and path plays two distinct roles in our setting. = we define the horizontal derivative independently of the time derivative.

The space C ([ T, 0]): motivation In the classical literature on path-dependent SDEs, it is usual to consider the state space L 2 ([ T, 0]) R( past present Here we take where C b ([ T, 0[) R( past present C b ([ T, 0[) = { f : [ T, 0[ R: f is bounded and continuous}.

Definition Definition C ([ T, 0]) : set of bounded functions η : [ T, 0] R continuous on [ T, 0[, equipped with an inductive topology which induces the following convergence η n n in C ([ T,0]) η if: (i) η n C. (ii) sup t K η n (t) η(t) 0, compact set K [ T, 0[. (iii) η n (0) η(0).

Remarks C([ T, 0]) is dense in C ([ T, 0]), when endowed with the topology of C ([ T, 0]). Examples of continuous functionals: (a) U(η) = g(η(t 1 ),..., η(t n )), with T t 1 < < t n 0 and g : R n R continuous. (b) U(η) = [ T,0] ϕ(t)d η(t), with ϕ: [ T, 0] R a càdlàg bounded variation function. The functional U(η) = sup t [ T,0] η(t) is not continuous.

Functional derivatives Definition Let u: C ([ T, 0]) R and η C ([ T, 0]). (i) Horizontal derivative at η: D H u(η) u(η( )1 [ T,0[ + η(0)1 {0} ) u(η( ε)1 [ T,0[ + η(0)1 {0} ) := lim. ε 0 + ε (ii) First-order vertical derivative at η: D V u(η) := a ũ(η [ T,0[, η(0)). (iii) Second-order vertical derivative at η: D V V u(η) := 2 aaũ(η [ T,0[, η(0)). ũ(γ, a) := u(γ1 [ T,0[ + a1 {0} ), (γ, a) C b ([ T, 0[) R.

The space C 1,2 (([0, T ] past) present) Definition U : [0, T ] C([ T, 0]) R is in C 1,2 (([0, T ] past) present)) if: U admits a (necessarily unique) continuous extension u: [0, T ] C ([ T, 0]) R. t u, D H u, D V u, D V V u exist and are continuous. Then we define on [0, T ] C([ T, 0]): D H U := D H u, D V U := D V u, D V V U := D V V u.

Functional Itô s formula Theorem Let U be in C 1,2 (([0, T ] past) present) and X = (X t ) t [0,T ] be a real continuous finite quadratic variation process. U(t, X t ) = U(0, X 0 ) + + t 0 t 0 ( t U(s, X s ) + D H U(s, X s ) ) ds D V U(s, X s )d X s + 1 2 t 0 D V V U(s, X s )d[x] s for all 0 t T, where X = (X t ) t denotes the window process associated with X, defined by X t := {X t+s, s [ T, 0]}.

Comparing the two approaches Identification of the functional derivatives Our aim is to prove formulae which allow to express functional derivatives in terms of differential operators arising in the Banach space stochastic calculus. Notation: we denote by D U the Fréchet derivative of U, which can be written as D U(η)ϕ = ϕ(x)d dx U(η) = ϕ(x) ( Ddx U(η)+Dδ 0 U(η)δ 0 (dx) ) [ T,0] [ T,0] for some uniquely determined finite signed Borel measure D dx U(η) on [ T, 0]. Vertical derivative Horizontal derivative D V U(η) = D δ 0 U(η) D H U(η) =?

Identification of D H U: definition of χ 0 χ 0 subspace of M([ T, 0] 2 ): µ M([ T, 0] 2 ) belongs to χ 0 if µ(dx, dy) = g 1 (x, y)dxdy + λ 1 δ 0 (dx) δ 0 (dy) + g 2 (x)dx λ 2 δ 0 (dy) + λ 3 δ 0 (dx) g 3 (y)dy + g 4 (x)δ y (dx) dy, with g 1 L 2 ([ T, 0] 2 ), g 2, g 3 L 2 ([ T, 0]), g 4 L ([ T, 0]), λ 1, λ 2, λ 3 R.

Identification of D H U Theorem Let η C([ T, 0]) be such that the quadratic variation [η] on [ T, 0] exists. Let U : C([ T, 0]) R be C 2 -Fréchet such that: (i) D 2 U : C([ T, 0]) χ 0. (ii) Dx 2,Diag U(η) ( g 4 ) has a [η]-zero set of discontinuity (e.g., if it is countable). (iii) There exist continuous extensions of U and D 2 dx dy U u: C ([ T, 0]) R, D 2 dx dy u: C ([ T, 0]) χ 0. (iv) The horizontal derivative D H U(η) exists at η. Then D H U(η) = [ T,0] D dx U(η)d+ η(x) 1 2 [ T,0] Dx 2,Diag U(η)d[η](x)

Semilinear parabolic path-dependent PDE Consider the semilinear parabolic path-dependent PDE on [0, T ] C([ T, 0]): t U + D H U + b(t, η)d V U + 1 2 σ(t, η)2 D V V U + F (t, η, U, σ(t, η)d V U) = 0, U(T, η) = H(η). Standing Assumption (A). b, σ, F, H are Borel measurable functions satisfying, for some positive constants C and m, b(t, η) b(t, η ) + σ(t, η) σ(t, η ) C η η, F (t, η, y, z) F (t, η, y, z ) C ( y y + z z ), b(t, 0) + σ(t, 0) C, F (t, η, 0, 0) + H(η) C ( 1 + η m), for all t [0, T ], η, η C([ T, 0]), y, y, z, z R.

Path-dependent SDE For every (t, η) [0, T ] C([ T, 0]), consider the path-dependent SDE: { dx s = b(s, X s )dt + σ(s, X s )dw s, s [t, T ], X s = η(s t), s [ T + t, t]. W is a real Brownian motion on (Ω, F, P). F is the completion of the natural filtration generated by W. Proposition (t, η) [0, T ] C([ T, 0]),! (up to indistinguishability) F-adapted continuous process X t,η = (Xs t,η ) s [ T +t,t ] solution to the path-dependent SDE. Moreover, for any p 1 there exists a positive constant C p such that [ E sup s [ T +t,t ] X t,η s p] ( C p 1 + η p ).

Strict solutions Definition A map U in C 1,2 (([0, T [ past) present) and C([0, T ] C([ T, 0])), satisfying the path-dependent PDE, is called a strict solution. Notation S 2 (t, T ), t T, the set of real càdlàg adapted processes Y = (Y s ) t s T such that [ Y 2 := E sup Y S 2 (t,t ) s 2] <. t s T H 2 (t, T ), t T, the set of real predictable processes Z = (Z s ) t s T such that Z 2 H 2 (t,t ) := E [ T t ] Z s 2 ds <.

Strict solutions: Feynman-Kac formula & uniqueness Theorem Let U : [0, T ] C([ T, 0]) R be a strict solution to the path-dependent PDE, satisfying the polynomial growth condition U(t, η) C ( 1 + η m ), (t, η) [0, T ] C([ T, 0]), for some positive constant m. Then, we have where (Ys t,η, Z t,η U(t, η) = Y t,η t, (t, η) [0, T ] C([ T, 0]), s ) s = (U(s, X t,η s ), σ(s, X t,η s )D V U(s, X t,η s )1 [t,t [ (s)) s with (Y t,η, Z t,η ) S 2 (t, T ) H 2 (t, T ), is the solution to the Backward Stochastic Differential Equation (BSDE) T T Ys t,η = H(X t,η T ) + F (r, X t,η r, Yr t,η, Zr t,η )dr Zr t,η dw r. s s

Strict solutions: existence (I) Theorem Suppose that b, σ, F, H are cylindrical and smooth, i.e. ( b(t, η) = b ϕ 1(x + t)d η(x),..., [ t,0] ( σ(t, η) = σ ϕ 1(x + t)d η(x),..., [ t,0] ( F (t, η, y, z) = F t, ϕ 1(x + t)d η(x),..., where [ t,0] [ t,0] [ t,0] ( H(η) = H ϕ 1(x + T )d η(x),..., [ T,0] [ t,0] ) ϕ N (x + t)d η(x) [ T,0] ) ϕ N (x + t)d η(x) ) ϕ N (x + t)d η(x), y, z ) ϕ N (x + T )d η(x) (i) b, σ, F, H are continuous and satisfy Assumption (A) with x R N in place of η. (ii) b and σ are of class C 3 with partial derivatives from order 1 up to order 3 bounded.

Strict solutions: existence (II) Theorem (cont d) (iii) For all t [0, T ], F (t,,, ) C 3 (R N ) and moreover we assume the validity of the properties below. (a) F (t,, 0, 0) belongs to C 3 and its third order partial derivatives satisfy a polynomial growth condition uniformly in t. (b) D y F, Dz F are bounded on [0, T ] R N R R, as well as their derivatives of order one and second with respect to x 1,..., x N, y, z. (iv) H C 3 (R N ) and its third order partial derivatives satisfy a polynomial growth condition. (v) ϕ 1,..., ϕ N C 2 ([0, T ]). Then, the map U given by U(t, η) = Y t,η t, (t, η) [0, T ] C([ T, 0]), is the unique strict solution to the path-dependent PDE.

Towards a weaker notion of solution Consider the lookback-type payoff: H(η) = sup η(x), η C([ T, 0]). x [ T,0] In this case, we expect that the map U(t, η) = E [ H(W t,η T )], (t, η) [0, T ] C([ T, 0]) is virtually a solution to the path-dependent PDE: t U + D H U + 1 2 DV V U = 0, U(T, η) = H(η). Unfortunately, U is not continuous with respect to the topology of C ([ T, 0]), therefore it can not be a strict solution. U is a strong-viscosity solution.

Strong-viscosity solutions: introduction Various definitions of viscosity-type solutions for path-dependent PDEs have been given. We recall in particular: I. Ekren, C. Keller, N. Touzi, and J. Zhang (2014) On viscosity solutions of path dependent PDEs. Annals of Probability, 42 (1), 204-236. We propose a notion of viscosity-type solution with the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. We call it strong-viscosity solution to distinguish it from the classical notion of viscosity solution and from the definition introduced by Ekren, Keller, Touzi, Zhang.

Strong-viscosity solutions: path-dependent case Path-dependent PDE on [0, T ] C([ T, 0]): t U + D H U + b(t, η)d V U + 1 2 σ(t, η)2 D V V U + F (t, η, U, σ(t, η)d V U) = 0, U(T, η) = H(η). Standing Assumption (A). b, σ, F, H are Borel measurable functions satisfying, for some positive constants C and m, b(t, η) b(t, η ) + σ(t, η) σ(t, η ) C η η, F (t, η, y, z) F (t, η, y, z ) C ( y y + z z ), b(t, 0) + σ(t, 0) C, F (t, η, 0, 0) + H(η) C ( 1 + η m), for all t [0, T ], η, η C([ T, 0]), y, y, z, z R.

Definition (I) Definition A function U : [0, T ] C([ T, 0]) R is called a strong-viscosity solution to the path-dependent PDE if there exists a sequence (U n, H n, F n, b n, σ n ) n of Borel measurable functions satisfying: (i) For some positive constants C and m, b n (t, η) b n (t, η ) + σ n (t, η) σ n (t, η ) C η η F n (t, η, y, z) F n (t, η, y, z ) C( y y + z z ) b n (t, 0) + σ n (t, 0) C U n (t, η) + H n (η) + F n (t, η, 0, 0) C ( 1 + η m ) for all t [0, T ], η, η C([ T, 0]), y, y R, z, z R. Moreover, the functions U n (t, ), H n ( ), F n (t,,, ), n N, are equicontinuous on compact sets, uniformly with respect to t [0, T ].

Definition (II) Definition (cont d) (ii) U n is a strict solution to t U n + D H U n + b n (t, η)d V U n + 1 2 σ n(t, η) 2 D V V U n + F n (t, η, U n, σ n (t, η)d V U n ) = 0, (t, η) [0, T [ C([ T, 0]), U n (T, η) = H n (η), η C([ T, 0]). (iii) (U n, H n, F n, b n, σ n ) n converges pointwise to (U, H, F, b, σ) as n.

Feynman-Kac formula & uniqueness Theorem Let U : [0, T ] C([ T, 0]) R be a strong-viscosity solution to the path-dependent PDE. Then, we have where (Y t,η s Y t,η s U(t, η) = Y t,η t, (t, η) [0, T ] C([ T, 0]), = U(s, X t,η s the BSDE Y t,η s, Zs t,η ) s [t,t ] S 2 (t, T ) H 2 (t, T ), with ), is the unique solution in S 2 (t, T ) H 2 (t, T ) to = H(X t,η T ) + T s F (r, X t,η r, Y t,η r T, Zr t,η )dr Zr t,η dw r, s for all t s T. In particular, there exists at most one strong-viscosity solution to the path-dependent PDE.

Existence Consider the path-dependent heat equation { t U + D H U + 1 2 DV V U = 0, (t, η) [0, T [ C([ T, 0]), U(T, η) = H(η), η C([ T, 0]). Theorem Suppose that H is continuous. Then, the map U(t, η) = E [ H(W t,η T )], for all (t, η) [0, T ] C([ T, 0]), is the unique strong-viscosity solution to the path-dependent heat equation. H can be in particular the lookback-type payoff H(η) = sup η(x), η C([ T, 0]). x [ T,0]

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