Rough viscosity solutions and applications to SPDEs P.K. Friz (joint - PowerPoint PPT Presentation

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 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

Motivation T. Lyons (’98): Let ( z n ) � C 1 � [ 0 , T ] , R d � be Cauchy in rough path metric , with limit z . Assume y n = V ( y n ) ˙ z n , y n ( 0 ) = y 0 2 R n , ( ODE ) : ˙ where V = ( V 1 , . . . , V d ) are suitable vector …elds. Then y n converges uniformly to y = y ( z ) 2 C ([ 0 , T ] , R n ) , independent of the approximating sequence. P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 3 / 21

Motivation T. Lyons (’98): Let ( z n ) � C 1 � [ 0 , T ] , R d � be Cauchy in rough path metric , with limit z . Assume y n = V ( y n ) ˙ z n , y n ( 0 ) = y 0 2 R n , ( ODE ) : ˙ where V = ( V 1 , . . . , V d ) are suitable vector …elds. Then y n converges uniformly to y = y ( z ) 2 C ([ 0 , T ] , R n ) , independent of the approximating sequence. Lions–Souganidis (’03): Let ( z n ) � C 1 � [ 0 , T ] , R d � be uniformly � [ 0 , T ] , R d � convergent to some z 2 C . Assume ( visc . PDE ) : ( ∂ t � F ) u n = H ( Du n ) ˙ z n , u n ( 0 , � ) = u 0 , � � Du , D 2 u where F = F , H = ( H 1 , . . . , H d ) are suitable. Then u n converges uniformly to u = u ( z ) 2 BUC ([ 0 , T ] , R n ) , independent of the approximating sequence. P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 3 / 21

Rough di¤erential equations T. Lyons (’98): Let ( z n ) � C 1 � [ 0 , T ] , R d � be Cauchy in rough path metric , with limit z . Assume ( ODE ) : dy n = V ( y n ) dz n , y n ( 0 ) = y 0 2 R n , where V = ( V 1 , . . . , V d ) are su¢ciently smooth vector …elds. Then y n converges uniformly to y = y ( z ) 2 C ([ 0 , T ] , R n ) , independent of the approximating sequence. P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

Rough di¤erential equations T. Lyons (’98): Let ( z n ) � C 1 � [ 0 , T ] , R d � be Cauchy in rough path metric , with limit z . Assume ( ODE ) : dy n = V ( y n ) dz n , y n ( 0 ) = y 0 2 R n , where V = ( V 1 , . . . , V d ) are su¢ciently smooth vector …elds. Then y n converges uniformly to y = y ( z ) 2 C ([ 0 , T ] , R n ) , 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 ) d z , y ( 0 ) = y 0 2 R n P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

Rough di¤erential equations T. Lyons (’98): Let ( z n ) � C 1 � [ 0 , T ] , R d � be Cauchy in rough path metric , with limit z . Assume ( ODE ) : dy n = V ( y n ) dz n , y n ( 0 ) = y 0 2 R n , where V = ( V 1 , . . . , V d ) are su¢ciently smooth vector …elds. Then y n converges uniformly to y = y ( z ) 2 C ([ 0 , T ] , R n ) , 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 ) d z , y ( 0 ) = y 0 2 R n What are rough path metrics and what are rough paths ? P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

Rough di¤erential equations T. Lyons (’98): Let ( z n ) � C 1 � [ 0 , T ] , R d � be Cauchy in rough path metric , with limit z . Assume ( ODE ) : dy n = V ( y n ) dz n , y n ( 0 ) = y 0 2 R n , where V = ( V 1 , . . . , V d ) are su¢ciently smooth vector …elds. Then y n converges uniformly to y = y ( z ) 2 C ([ 0 , T ] , R n ) , 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 ) d z , y ( 0 ) = y 0 2 R n What are rough path metrics and what are rough paths ? First example, not applicable to Brownian motion: take j z s , t � ˜ z s , t j ρ α -Höl ( z , ˜ z ) : = sup for α 2 ( 1 / 2 , 1 ] ; j t � s j α s , t 2 [ 0 , T ] rough paths are just α -Hölder paths; RDEs are "Young" ODEs. P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 4 / 21

Rough di¤erential equations (cont’d) A better example, applicable to Brownian motion, for α 2 ( 1 / 3 , 1 / 2 ] take � � � � � z 1 � � z 2 � z 1 z 2 s , t � ~ s , t � ~ s , t s , t ρ α -Höl ( z , ˜ z ) : = sup + j t � s j α j t � s j 2 α s , t 2 [ 0 , T ] where we introduced generalized increments of z 2 C 1 , � Z t � Z t Z r � � 2 R d � R d � d . z 1 s , t , z 2 z s , t : = : = s dz , s dz � dz s , t s The (abstract) completion of C 1 -paths with respect to ρ α -Höl leads to rough path space which can be identi…ed as a subset of ( ) � � � � � [ 0 , T ] , R d � R d � d � � z 1 � � z 2 � s , t s , t j t � s j 2 α < ∞ z 2 C : sup j t � s j α + . s , t 2 [ 0 , T ] P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 5 / 21

Rough di¤erential equations (cont’d) A better example, applicable to Brownian motion, for α 2 ( 1 / 3 , 1 / 2 ] take � � � � � z 1 � � z 2 � z 1 z 2 s , t � ~ s , t � ~ s , t s , t ρ α -Höl ( z , ˜ z ) : = sup + j t � s j α j t � s j 2 α s , t 2 [ 0 , T ] where we introduced generalized increments of z 2 C 1 , � Z t � Z t Z r � � 2 R d � R d � d . z 1 s , t , z 2 z s , t : = : = s dz , s dz � dz s , t s The (abstract) completion of C 1 -paths with respect to ρ α -Höl leads to rough path space which can be identi…ed as a subset of ( ) � � � � � [ 0 , T ] , R d � R d � d � � z 1 � � z 2 � s , t s , t j t � s j 2 α < ∞ z 2 C : sup j t � s j α + . s , t 2 [ 0 , T ] � z i z j � = z i dz j + z j dz i = � z 2 � = 1 2 z 1 � z 1 = From d ) Sym ) � � $ � � � � z 1 s , t , z 2 z 1 z 2 s , t , A s , t with "area" A s , t : = Anti s , t s , t P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 5 / 21

Example: Homogenization of highly oscillatory ODEs �� 0 � � �� 0 t t 7! z t � , � t 0 0 is the rough path limit, any α 2 ( 1 / 3 , 1 / 2 ) , of � � 2 C � z n ( t ) = n � 1 exp 2 π in 2 t = R 2 . Given two vector …elds V = ( V 1 , V 2 ) the RDE solution dy = V ( y ) d z (1) models the e¤ective behaviour of the highly oscillatory ODE dy n = V ( y n ) dz n as n ! ∞ . In fact, the RDE solution of (1) solves the ODE y = [ V 1 , V 2 ] ( y ) ˙ where [ V 1 , V 2 ] is the Lie bracket of V 1 and V 2 . P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 6 / 21

Stochastic di¤erential equations . Let B be d -dimensional 2 C 1 careful interpretation of the Brownian motion. Since B ( ω ) / stochastic di¤erential equation dy = V ( y ) ∂ B is necessary (Itô-theory). On the other hand we can set � � Z t B t ( ω ) = 0 B s � ∂ B s B t , 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

Stochastic di¤erential equations . Let B be d -dimensional 2 C 1 careful interpretation of the Brownian motion. Since B ( ω ) / stochastic di¤erential equation dy = V ( y ) ∂ B is necessary (Itô-theory). On the other hand we can set � � Z t B t ( ω ) = 0 B s � ∂ B s B t , 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 �� 0 � � �� 0 t B t + , 0 � t 0 P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 7 / 21

Advantages for the probabilist ? RDE solution to dy = V ( y ) d B 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

Advantages for the probabilist ? RDE solution to dy = V ( y ) d B 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

Advantages for the probabilist ? RDE solution to dy = V ( y ) d B 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 ) d z ; For e.g. V 2 Lip 3 + ε , can see that ψ , D ψ , D 2 ψ exist and depend continuously on z ; also for ψ � 1 , D ψ � 1 , D 2 ψ � 1 . Limit theorems for stochastic ‡ows as trivial consequence. P.K. Friz (TU and WIAS Berlin) RPDEs July 2010 8 / 21

Advantages for the probabilist ? RDE solution to dy = V ( y ) d B 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 ) d z ; For e.g. V 2 Lip 3 + ε , can see that ψ , D ψ , D 2 ψ exist and depend continuously on z ; also for ψ � 1 , D ψ � 1 , D 2 ψ � 1 . 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