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Rough viscosity solutions and applications to SPDEs P.K. Friz (joint work with M. Caruana, H. Oberhauser, J. Diehl) TU and WIAS Berlin July 2010 P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 1 / 21 Outline I Rough dierential equations

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Rough viscosity solutions and applications to SPDEs

P.K. Friz (joint work with M. Caruana, H. Oberhauser, J. Diehl)

TU and WIAS Berlin

July 2010

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 1 / 21

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Outline

I Rough di¤erential equations (RDEs) II Viscosity theory III A (rough-) pathwise approach to SPDEs IV BSDEs driven by rough paths

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 2 / 21

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Motivation

  • T. Lyons (’98): Let (zn) C 1 [0, T] , Rd

be Cauchy in rough path metric, with limit z. Assume (ODE) : ˙ yn = V (yn) ˙ zn, yn (0) = y0 2 Rn, where V = (V1, . . . , Vd) are suitable vector …elds. Then yn converges uniformly to y = y (z) 2 C ([0, T] , Rn), independent of the approximating sequence.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 3 / 21

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Motivation

  • T. Lyons (’98): Let (zn) C 1 [0, T] , Rd

be Cauchy in rough path metric, with limit z. Assume (ODE) : ˙ yn = V (yn) ˙ zn, yn (0) = y0 2 Rn, where V = (V1, . . . , Vd) are suitable vector …elds. Then yn converges uniformly to y = y (z) 2 C ([0, T] , Rn), independent of the approximating sequence. Lions–Souganidis (’03): Let (zn) C 1 [0, T] , Rd be uniformly convergent to some z 2 C [0, T] , Rd . Assume (visc.PDE) : (∂t F) un = H (Dun) ˙ zn, un (0, ) = u0, where F = F

  • Du, D2u
  • , H = (H1, . . . , Hd) are suitable.

Then un converges uniformly to u = u (z) 2 BUC ([0, T] , Rn), independent of the approximating sequence.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 3 / 21

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Rough di¤erential equations

  • T. Lyons (’98): Let (zn) C 1 [0, T] , Rd

be Cauchy in rough path metric, with limit z. Assume (ODE) : dyn = V (yn) dzn, yn (0) = y0 2 Rn, where V = (V1, . . . , Vd) are su¢ciently smooth vector …elds. Then yn converges uniformly to y = y (z) 2 C ([0, T] , Rn), independent of the approximating sequence.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

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Rough di¤erential equations

  • T. Lyons (’98): Let (zn) C 1 [0, T] , Rd

be Cauchy in rough path metric, with limit z. Assume (ODE) : dyn = V (yn) dzn, yn (0) = y0 2 Rn, where V = (V1, . . . , Vd) are su¢ciently smooth vector …elds. Then yn converges uniformly to y = y (z) 2 C ([0, T] , Rn), independent of the approximating sequence. Interpretation: y is the solution to a rough di¤erential equation, driven by the rough path z write (RDE) : dy = V (y) dz, y (0) = y0 2 Rn

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

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Rough di¤erential equations

  • T. Lyons (’98): Let (zn) C 1 [0, T] , Rd

be Cauchy in rough path metric, with limit z. Assume (ODE) : dyn = V (yn) dzn, yn (0) = y0 2 Rn, where V = (V1, . . . , Vd) are su¢ciently smooth vector …elds. Then yn converges uniformly to y = y (z) 2 C ([0, T] , Rn), independent of the approximating sequence. Interpretation: y is the solution to a rough di¤erential equation, driven by the rough path z write (RDE) : dy = V (y) dz, y (0) = y0 2 Rn What are rough path metrics and what are rough paths ?

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

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Rough di¤erential equations

  • T. Lyons (’98): Let (zn) C 1 [0, T] , Rd

be Cauchy in rough path metric, with limit z. Assume (ODE) : dyn = V (yn) dzn, yn (0) = y0 2 Rn, where V = (V1, . . . , Vd) are su¢ciently smooth vector …elds. Then yn converges uniformly to y = y (z) 2 C ([0, T] , Rn), independent of the approximating sequence. Interpretation: y is the solution to a rough di¤erential equation, driven by the rough path z write (RDE) : dy = V (y) dz, y (0) = y0 2 Rn What are rough path metrics and what are rough paths ? First example, not applicable to Brownian motion: take ρα-Höl (z, ˜ z) := sup

s,t2[0,T ]

jzs,t ˜ zs,tj jt sjα for α 2 (1/2, 1]; rough paths are just α-Hölder paths; RDEs are "Young" ODEs.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

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Rough di¤erential equations (cont’d)

A better example, applicable to Brownian motion, for α 2 (1/3, 1/2] take ρα-Höl (z, ˜ z) := sup

s,t2[0,T ]

  • z1

s,t ~

z1

s,t

  • jt sjα

+

  • z2

s,t ~

z2

s,t

  • jt sj2α

where we introduced generalized increments of z 2 C 1, zs,t :=

  • z1

s,t, z2 s,t

  • :=

Z t

s dz,

Z t

s

Z r

s dz dz

  • 2 Rd Rdd.

The (abstract) completion of C 1-paths with respect to ρα-Höl leads to rough path space which can be identi…ed as a subset of ( z 2 C

  • [0, T] , Rd Rdd

: sup

s,t2[0,T ]

  • z1

s,t

  • jt sjα +
  • z2

s,t

  • jt sj2α < ∞

) .

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 5 / 21

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Rough di¤erential equations (cont’d)

A better example, applicable to Brownian motion, for α 2 (1/3, 1/2] take ρα-Höl (z, ˜ z) := sup

s,t2[0,T ]

  • z1

s,t ~

z1

s,t

  • jt sjα

+

  • z2

s,t ~

z2

s,t

  • jt sj2α

where we introduced generalized increments of z 2 C 1, zs,t :=

  • z1

s,t, z2 s,t

  • :=

Z t

s dz,

Z t

s

Z r

s dz dz

  • 2 Rd Rdd.

The (abstract) completion of C 1-paths with respect to ρα-Höl leads to rough path space which can be identi…ed as a subset of ( z 2 C

  • [0, T] , Rd Rdd

: sup

s,t2[0,T ]

  • z1

s,t

  • jt sjα +
  • z2

s,t

  • jt sj2α < ∞

) . From d

  • zizj = zidzj + zjdzi =

) Sym

  • z2 = 1

2z1 z1 =

)

  • z1

s,t, z2 s,t

$

  • z1

s,t, As,t

  • with "area" As,t := Anti
  • z2

s,t

  • P.K. Friz (TU and WIAS Berlin)

RPDEs July 2010 5 / 21

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Example: Homogenization of highly oscillatory ODEs t 7! zt

  • ,
  • t

t

  • is the rough path limit, any α 2 (1/3, 1/2), of

zn (t) = n1 exp

  • 2πin2t

2 C = R2. Given two vector …elds V = (V1, V2) the RDE solution dy = V (y) dz (1) models the e¤ective behaviour of the highly oscillatory ODE dyn = V (yn) dzn as n ! ∞. In fact, the RDE solution of (1) solves the ODE ˙ y = [V1, V2] (y) where [V1, V2] is the Lie bracket of V1 and V2.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 6 / 21

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Stochastic di¤erential equations. Let B be d-dimensional Brownian motion. Since B (ω) / 2 C 1 careful interpretation of the stochastic di¤erential equation dy = V (y) ∂B is necessary (Itô-theory). On the other hand we can set Bt (ω) =

  • Bt,

Z t

0 Bs ∂Bs

  • where ∂ indicates (Stratonovich) integration. For any α 2 (1/3, 1/2)

P [B is a α-Hölder rough path] = 1. Any reasonable (smooth) approximation to Brownian motion converges to B in rough path metric.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 7 / 21

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Stochastic di¤erential equations. Let B be d-dimensional Brownian motion. Since B (ω) / 2 C 1 careful interpretation of the stochastic di¤erential equation dy = V (y) ∂B is necessary (Itô-theory). On the other hand we can set Bt (ω) =

  • Bt,

Z t

0 Bs ∂Bs

  • where ∂ indicates (Stratonovich) integration. For any α 2 (1/3, 1/2)

P [B is a α-Hölder rough path] = 1. Any reasonable (smooth) approximation to Brownian motion converges to B in rough path metric. But: there are "unreasonable" approximations, e.g. those of [McShane ’72], which converge to Bt +

  • ,
  • t

t

  • P.K. Friz (TU and WIAS Berlin)

RPDEs July 2010 7 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution. +) construction of stochastic ‡ows trivial (no null-set trouble)

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution. +) construction of stochastic ‡ows trivial (no null-set trouble) +) consider solution ‡ow ψ to (RDE) : dy = V (y) dz; For e.g. V 2 Lip3+ε, can see that ψ, Dψ, D2ψ exist and depend continuously on z; also for ψ1, Dψ1, D2ψ1. Limit theorems for stochastic ‡ows as trivial consequence.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution. +) construction of stochastic ‡ows trivial (no null-set trouble) +) consider solution ‡ow ψ to (RDE) : dy = V (y) dz; For e.g. V 2 Lip3+ε, can see that ψ, Dψ, D2ψ exist and depend continuously on z; also for ψ1, Dψ1, D2ψ1. Limit theorems for stochastic ‡ows as trivial consequence. +) no trouble if V = V (; ω) anticipating

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution. +) construction of stochastic ‡ows trivial (no null-set trouble) +) consider solution ‡ow ψ to (RDE) : dy = V (y) dz; For e.g. V 2 Lip3+ε, can see that ψ, Dψ, D2ψ exist and depend continuously on z; also for ψ1, Dψ1, D2ψ1. Limit theorems for stochastic ‡ows as trivial consequence. +) no trouble if V = V (; ω) anticipating +) no restriction to semi-martingale driving process

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution. +) construction of stochastic ‡ows trivial (no null-set trouble) +) consider solution ‡ow ψ to (RDE) : dy = V (y) dz; For e.g. V 2 Lip3+ε, can see that ψ, Dψ, D2ψ exist and depend continuously on z; also for ψ1, Dψ1, D2ψ1. Limit theorems for stochastic ‡ows as trivial consequence. +) no trouble if V = V (; ω) anticipating +) no restriction to semi-martingale driving process +) various applications of continuity in B; e.g. support theorems

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Advantages for the probabilist? RDE solution to dy = V (y) dB is solved for …xed ω, depends continuously on B and yields a (classical) Stratonovich SDE solution. +) construction of stochastic ‡ows trivial (no null-set trouble) +) consider solution ‡ow ψ to (RDE) : dy = V (y) dz; For e.g. V 2 Lip3+ε, can see that ψ, Dψ, D2ψ exist and depend continuously on z; also for ψ1, Dψ1, D2ψ1. Limit theorems for stochastic ‡ows as trivial consequence. +) no trouble if V = V (; ω) anticipating +) no restriction to semi-martingale driving process +) various applications of continuity in B; e.g. support theorems

  • ) one additional degree of smoothness compared to Itô-theory.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

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Part II: Second order parabolic viscosity theory

Crandall, Evans, Ishii, Lions (’80ties) ... : Let u = u (t, x) and Fu = F

  • t, x, u, Du, D2u
  • be continuous, weakly elliptic. Subject to

(TC), satis…ed in many examples (in particular from stochastic control theory) (∂t F) u = 0, u (0, ) = u0 2 BUC (Rn) has a unique solution, in viscosity sense, say u 2 BUC ([0, T] , Rn). In fact, one has comparison in the sense that (∂t F) u

  • 0 (∂t F) v

and u0

  • v0

implies u v.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 9 / 21

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Part II: Second order parabolic viscosity theory

Crandall, Evans, Ishii, Lions (’80ties) ... : Let u = u (t, x) and Fu = F

  • t, x, u, Du, D2u
  • be continuous, weakly elliptic. Subject to

(TC), satis…ed in many examples (in particular from stochastic control theory) (∂t F) u = 0, u (0, ) = u0 2 BUC (Rn) has a unique solution, in viscosity sense, say u 2 BUC ([0, T] , Rn). In fact, one has comparison in the sense that (∂t F) u

  • 0 (∂t F) v

and u0

  • v0

implies u v. Here (∂t F) u 0 means (∂t F) ϕj¯

t,¯ x 0 for any smooth

test-function which touches u from above at (¯ t, ¯ x).

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 9 / 21

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Part II: Second order parabolic viscosity theory

Crandall, Evans, Ishii, Lions (’80ties) ... : Let u = u (t, x) and Fu = F

  • t, x, u, Du, D2u
  • be continuous, weakly elliptic. Subject to

(TC), satis…ed in many examples (in particular from stochastic control theory) (∂t F) u = 0, u (0, ) = u0 2 BUC (Rn) has a unique solution, in viscosity sense, say u 2 BUC ([0, T] , Rn). In fact, one has comparison in the sense that (∂t F) u

  • 0 (∂t F) v

and u0

  • v0

implies u v. Here (∂t F) u 0 means (∂t F) ϕj¯

t,¯ x 0 for any smooth

test-function which touches u from above at (¯ t, ¯ x). Motivation for this de…nition: if u 2 C 1,2 is a classical subsolution, ∂t ϕ F

  • t, x, ϕ, Dϕ, D2ϕ

∂tu F

  • t, x, Du, D2u

0.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 9 / 21

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Stability properties of viscosity theory

Stability: If (∂t Fε) uε 0 and Fε ! F, uε ! u locally uniformly then (∂t F) u 0.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 10 / 21

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Stability properties of viscosity theory

Stability: If (∂t Fε) uε 0 and Fε ! F, uε ! u locally uniformly then (∂t F) u 0. Nice feature: no need to show convergence of the derivatives

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 10 / 21

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Stability properties of viscosity theory

Stability: If (∂t Fε) uε 0 and Fε ! F, uε ! u locally uniformly then (∂t F) u 0. Nice feature: no need to show convergence of the derivatives Stability [Barles–Perthame ’88] If (∂t Fε) uε 0 with Fε ! F locally uniformly and (ε, t, x) 7! uε (t, x) locally bounded; then (∂t F) ¯ u 0 where ¯ u (t, x) := lim

ε!0 uε (t, x) =

lim

ε!0,t0!t,x0!t

uε

  • t0, x0

. Note: ¯ u (upper) semi-continuous. Similar for supersolutions.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 10 / 21

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Stability properties of viscosity theory

Stability: If (∂t Fε) uε 0 and Fε ! F, uε ! u locally uniformly then (∂t F) u 0. Nice feature: no need to show convergence of the derivatives Stability [Barles–Perthame ’88] If (∂t Fε) uε 0 with Fε ! F locally uniformly and (ε, t, x) 7! uε (t, x) locally bounded; then (∂t F) ¯ u 0 where ¯ u (t, x) := lim

ε!0 uε (t, x) =

lim

ε!0,t0!t,x0!t

uε

  • t0, x0

. Note: ¯ u (upper) semi-continuous. Similar for supersolutions. Typical application: (∂t Fε) uε = 0 with Fε ! F locally uniformly; fuεg locally bounded and comparison holds for ∂t F. Then uε converges local uniformly to some u = u (t, x) and (∂t F) u = 0.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 10 / 21

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Part III: Stochastic viscosity solutions

Aim: a pathwise theory of fully nonlinear SPDE of the form du F

  • t, x, u, Du, D2u
  • dt

= H (x, u, Du) ∂B, u (0, ) = u0 2 BUC (Rn) for H = (H1, . . . , Hd ) and a d-dimensional Brownian motion B.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 11 / 21

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Part III: Stochastic viscosity solutions

Aim: a pathwise theory of fully nonlinear SPDE of the form du F

  • t, x, u, Du, D2u
  • dt

= H (x, u, Du) ∂B, u (0, ) = u0 2 BUC (Rn) for H = (H1, . . . , Hd ) and a d-dimensional Brownian motion B. Theorem [Lions–Souganidis ’03] Fix u0 2 BUC (Rn) and let zn 2 C 1 [0, T] , Rd uniformly convergent to some z 2 C [0, T] , Rd . Assume (∂t F) un = H (Dun) ˙ zn, un (0, ) = u0. Then un converges locally uniformly to a limit u = u (z) 2 BUC ([0, T] , Rn), independent of the approximating

  • sequence. Interpretation:

du F

  • t, x, Du, D2u
  • dt = H (Du) ∂z;

applicable to a.e. (continuous) sample path of Brownian motion.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 11 / 21

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Remarks: Lions-Sougandis give intrinsic de…nition by using test-functions propagated by the Hamiltonian ‡ow of du = H (Du) ∂z.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 12 / 21

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Remarks: Lions-Sougandis give intrinsic de…nition by using test-functions propagated by the Hamiltonian ‡ow of du = H (Du) ∂z. The case when H = H (x, u) driven by a multi-dimensional Brownian motion is di¢cult.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 12 / 21

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Remarks: Lions-Sougandis give intrinsic de…nition by using test-functions propagated by the Hamiltonian ‡ow of du = H (Du) ∂z. The case when H = H (x, u) driven by a multi-dimensional Brownian motion is di¢cult. They also have a note [CRAS ’00] on du F

  • Du, D2u
  • dt = H (u) ∂z.

Stability of solutions as functions of z only in rough path sense (unless [Hi, Hj] = 0); we shall return to this later. In this case, x-dependence does not add real di¢culty.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 12 / 21

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Theorem [Caruana-F-Oberhauser AIHP ’10] Fix u0 2 BUC (Rn) and let zn 2 C 1 [0, T] , Rd Cauchy in rough path metric with (geometric) rough path limit z. Assume (∂t F) un = hDun, σ (x)i ˙ zn, un (0, ) = u0. Then un ! u = u (z) 2 BUC ([0, T] , Rn) locally uniformly, independent of the approximating sequence. Interpretation: du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz;

applicable to a.e. sample path of B i.e. Brownian motion and Lévy area.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 13 / 21

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Theorem [Caruana-F-Oberhauser AIHP ’10] Fix u0 2 BUC (Rn) and let zn 2 C 1 [0, T] , Rd Cauchy in rough path metric with (geometric) rough path limit z. Assume (∂t F) un = hDun, σ (x)i ˙ zn, un (0, ) = u0. Then un ! u = u (z) 2 BUC ([0, T] , Rn) locally uniformly, independent of the approximating sequence. Interpretation: du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz;

applicable to a.e. sample path of B i.e. Brownian motion and Lévy area. Remark: with x-dependence a pathwise SPDE theory must be rough

  • pathwise. Indeed, take F = 0, σ = (V1, ..., Vd). The resulting

(linear, …rst-order) SPDE is solved globally by the method of characteristics and involves the ‡ow of the SDE dy = V (y) ∂B. Now think of McShane type approximations to B.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 13 / 21

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Idea of proof: we have H (x, Du) = hDu, σ (x)i , linear in the

  • gradient. Then

"dyn = σ (yn) dzn ! dy = σ (y) dz " as ‡ow (of di¤eomorphisms), say ψn ! ψ. This induces a new coordinate chart in which, setting vn (t, x) = un (t, ψn (t, x)) , (∂t F) un = hDun, σ (x)i ˙ zn , (∂t F n) vn = 0. and where F n F ψn ! F z F ψ locally uniformly. Assuming local boundedness of vn then implies, using stability of viscosity solutions, (∂t F z) h lim

n!∞ vn (t, x)

i 0 (∂t F z)

  • lim

n!∞ vn (t, x)

  • Comparison implies vn ! v locally uniformly and (∂t F z) v = 0.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 14 / 21

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Unwrapping the change-of-coordinates then yields a solution to the fully non-linear PDE with "rough path" noise du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 15 / 21

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Unwrapping the change-of-coordinates then yields a solution to the fully non-linear PDE with "rough path" noise du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz.

Let us emphasize that u is the locally uniform limit of un obtained by replacing z by zn

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 15 / 21

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Unwrapping the change-of-coordinates then yields a solution to the fully non-linear PDE with "rough path" noise du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz.

Let us emphasize that u is the locally uniform limit of un obtained by replacing z by zn ... and now u = uz, independent of the approximating sequence, as

  • desired. Moreover, (u0, z) 7! uz is continuous.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 15 / 21

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Unwrapping the change-of-coordinates then yields a solution to the fully non-linear PDE with "rough path" noise du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz.

Let us emphasize that u is the locally uniform limit of un obtained by replacing z by zn ... and now u = uz, independent of the approximating sequence, as

  • desired. Moreover, (u0, z) 7! uz is continuous.

Moral: stability properties of viscosity - and rough path theory work perfectly together.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 15 / 21

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Unwrapping the change-of-coordinates then yields a solution to the fully non-linear PDE with "rough path" noise du F

  • t, x, Du, D2u
  • dt = hDu, σ (x)i dz.

Let us emphasize that u is the locally uniform limit of un obtained by replacing z by zn ... and now u = uz, independent of the approximating sequence, as

  • desired. Moreover, (u0, z) 7! uz is continuous.

Moral: stability properties of viscosity - and rough path theory work perfectly together. Applications to stochastic PDEs are roughpathwise i.e. by taking zt = Bt (ω) =

  • Bt,

Z t

0 Bs ∂Bs

  • btain Stratonovich SPDE solution (no need for stochastic

parabolicity assumption!)

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 15 / 21

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SLIDE 41

Theorem [F-Oberhauser ArXiv ’10] Let L = L

  • t, x, u, Du, D2u
  • and Mi (t, x, u, Du) be linear di¤erential operators. Let u0 2 BUC

and z a rough path. Then there exists a unique u = uz 2 BUC ([0, T] Rn), write du L

  • t, x, u, Du, D2u
  • dt = M (t, x, u, Du) dz, u (0, ) u0,

such that for any smooth sequence zn ! z in rough path metric, un, BUC viscosity solutions of ˙ un L

  • t, x, u, Dun, D2un = M (t, x, un, Dun) ˙

zn, u (0, ) u0, converge locally uniformly to uz. Moreover, (u0, z) 7! uz is continuous.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 16 / 21

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SLIDE 42

Theorem [F-Oberhauser ArXiv ’10] Let L = L

  • t, x, u, Du, D2u
  • and Mi (t, x, u, Du) be linear di¤erential operators. Let u0 2 BUC

and z a rough path. Then there exists a unique u = uz 2 BUC ([0, T] Rn), write du L

  • t, x, u, Du, D2u
  • dt = M (t, x, u, Du) dz, u (0, ) u0,

such that for any smooth sequence zn ! z in rough path metric, un, BUC viscosity solutions of ˙ un L

  • t, x, u, Dun, D2un = M (t, x, un, Dun) ˙

zn, u (0, ) u0, converge locally uniformly to uz. Moreover, (u0, z) 7! uz is continuous. Useful? Well-known SPDE results (support theorems, approximations theorems, splitting methods, maximum principles) become immediate corollaries.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 16 / 21

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SLIDE 43

Idea of proof: W.l.o.g. noise term writtes as

d

∑

i=1

Mi (x, u, Du) dzi =

d1

∑

j=1

hσj (x) , Dui dξj + u

d2

∑

k=1

ηk (x) dθk

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 17 / 21

slide-44
SLIDE 44

Idea of proof: W.l.o.g. noise term writtes as

d

∑

i=1

Mi (x, u, Du) dzi =

d1

∑

j=1

hσj (x) , Dui dξj + u

d2

∑

k=1

ηk (x) dθk Inner transformation: ψ (t, ) solution ‡ow to ˙ ψ = σ (ψ) dξ = ) w = u (t, ψ (t, x)) solves

  • ∂t Lψ

w = w

d1

∑

k=1

ηk (ψ (t, x)) dθk =: w η (ψ) dθ

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 17 / 21

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SLIDE 45

Idea of proof: W.l.o.g. noise term writtes as

d

∑

i=1

Mi (x, u, Du) dzi =

d1

∑

j=1

hσj (x) , Dui dξj + u

d2

∑

k=1

ηk (x) dθk Inner transformation: ψ (t, ) solution ‡ow to ˙ ψ = σ (ψ) dξ = ) w = u (t, ψ (t, x)) solves

  • ∂t Lψ

w = w

d1

∑

k=1

ηk (ψ (t, x)) dθk =: w η (ψ) dθ Outer transformation: φ (t, ; x) solution ‡ow to ˙ φ = φη (ψ) dθ = ) log φ is rough integral based on (ψ, θ). Then v = φ1 (t, w (t, x) ; x) solves

  • ∂t φ

Lψ v = 0 where φ Lψ = φ Lψ t, x, v, Dv, D2v

  • is linear if L is ...

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 17 / 21

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SLIDE 46

What about non-linear noise? [Gubinelli–Tindel, AoP ’10; Deya et al.] consider non-linear rough heat equations that include (but are not restricted to) du (∆u) dt = H (u, x) dz (2) and establish uniqueness and local existence in Sobolev-spaces.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 18 / 21

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SLIDE 47

What about non-linear noise? [Gubinelli–Tindel, AoP ’10; Deya et al.] consider non-linear rough heat equations that include (but are not restricted to) du (∆u) dt = H (u, x) dz (2) and establish uniqueness and local existence in Sobolev-spaces. In the present framework: focus on H = H (u) and ∆ for

  • simplicity. Perform outer transformation via solution ‡ow of RDE

˙ φ = H (φ) dz. Then v = φ1 (t, u (t, x)) solves ∂tv (∆v) dt = φ00(t, v) φ0(t, v) jDvj2 c (t, v) jDvj2 . Since jc (t, v)j < ε for 0 t < h (ε) we can apply the comparison results of [Kobylanski AoP ’00 or Lions-Souganidis ’00] on small

  • intervals. We then obtain, as before, solutions to (2) on [0, h], then on

[h, 2h] ... and hence on [0, T]. Moreover, (u0, z) 7! uz is continuous.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 18 / 21

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SLIDE 48

Part IV: BSDEs driven by rough paths

Rewrite non-linear rough heat equation as terminal value problem du + (Lu) dt + H (u, x) dz = 0; uT = g when z 2 C 1 this relates to the BSDE [Pardoux–Peng, ’92] dY x,s

t

= H (Y x,s

t

, X x,s

t

) dzt Z x,s

t

dWt YT = g (XT ) where L is the generator of X.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 19 / 21

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SLIDE 49

Part IV: BSDEs driven by rough paths

Rewrite non-linear rough heat equation as terminal value problem du + (Lu) dt + H (u, x) dz = 0; uT = g when z 2 C 1 this relates to the BSDE [Pardoux–Peng, ’92] dY x,s

t

= H (Y x,s

t

, X x,s

t

) dzt Z x,s

t

dWt YT = g (XT ) where L is the generator of X. In fact, a non-linear Feynman-Kac formula holds: Y x,t

t

= u (t, x) .

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 19 / 21

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SLIDE 50

Part IV: BSDEs driven by rough paths

Rewrite non-linear rough heat equation as terminal value problem du + (Lu) dt + H (u, x) dz = 0; uT = g when z 2 C 1 this relates to the BSDE [Pardoux–Peng, ’92] dY x,s

t

= H (Y x,s

t

, X x,s

t

) dzt Z x,s

t

dWt YT = g (XT ) where L is the generator of X. In fact, a non-linear Feynman-Kac formula holds: Y x,t

t

= u (t, x) . Related: dz = ∂B (SPDEs-BDSDEs; Pardoux–Peng ’94, Buckdahn-Ma ’01)

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 19 / 21

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SLIDE 51

Work in progress (Diehl-F): Consider dXt = α (t, ω) dWt + β (t, ω) dt (Itô-di¤usion) dYt = f (t, Yt, Zt; ω) dt + H (Yt, Xt) dzt Zt (ω) dWt YT = ξ 2 L∞(FT ); and write (Y n, Z n) for the BSDE solution pair when z = zn.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 20 / 21

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SLIDE 52

Work in progress (Diehl-F): Consider dXt = α (t, ω) dWt + β (t, ω) dt (Itô-di¤usion) dYt = f (t, Yt, Zt; ω) dt + H (Yt, Xt) dzt Zt (ω) dWt YT = ξ 2 L∞(FT ); and write (Y n, Z n) for the BSDE solution pair when z = zn. Using stability results for "quadratic BSDEs" (Kobylanski) we can show that zn ! z entails convergence of (Y n, Z n); giving meaning to (rough BSDE): dYt = H (Yt, Xt) dzt Z (ω) dWt,

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 20 / 21

slide-53
SLIDE 53

Work in progress (Diehl-F): Consider dXt = α (t, ω) dWt + β (t, ω) dt (Itô-di¤usion) dYt = f (t, Yt, Zt; ω) dt + H (Yt, Xt) dzt Zt (ω) dWt YT = ξ 2 L∞(FT ); and write (Y n, Z n) for the BSDE solution pair when z = zn. Using stability results for "quadratic BSDEs" (Kobylanski) we can show that zn ! z entails convergence of (Y n, Z n); giving meaning to (rough BSDE): dYt = H (Yt, Xt) dzt Z (ω) dWt, Existence, uniqueness, stability!

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 20 / 21

slide-54
SLIDE 54

Work in progress (Diehl-F): Consider dXt = α (t, ω) dWt + β (t, ω) dt (Itô-di¤usion) dYt = f (t, Yt, Zt; ω) dt + H (Yt, Xt) dzt Zt (ω) dWt YT = ξ 2 L∞(FT ); and write (Y n, Z n) for the BSDE solution pair when z = zn. Using stability results for "quadratic BSDEs" (Kobylanski) we can show that zn ! z entails convergence of (Y n, Z n); giving meaning to (rough BSDE): dYt = H (Yt, Xt) dzt Z (ω) dWt, Existence, uniqueness, stability! Idee of proof: Outer transformation via solution ‡ow of RDE ˙ φ = H (φ, x) dz. Then φ1 (t, Yt, Xt) solves a BSDE with quadratic driver ( term c (t, v) jDvj2) in PDE world); as studied in Koblylanski. Exploit stability and comparison principles for such BSDEs.

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 20 / 21

slide-55
SLIDE 55

References

Lyons–Qian: System Control and Rough Paths, Oxford University Press (2002). F–Victoir: Stochastic Processes as Rough Paths, Cambridge University Press (2010) Caruana-F: PDEs driven by rough paths; J. Di¤. Equ. (2009) Caruana-F-Oberhauser: A (rough-)pathwise approach to a class of nonlinear SPDEs; Ann. IHP Nonlinear Analysis various preprints on the arxiv

Thank you for your attention!

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 21 / 21

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SLIDE 56

P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 21 / 21