An algebraic framework for It os formula David Kelly Martin Hairer - PowerPoint PPT Presentation

An algebraic framework for Itˆ o’s formula David Kelly Martin Hairer Mathematics Institute University of Warwick Coventry UK CV4 7AL dtbkelly@gmail.com April 26, 2013 Algebra and Combinatorics Seminar 2013, ICMAT Madrid. David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 1 / 27

Outline 1. A little bit about rough path theory 2. The geometric assumption 3. Two approaches to non-geometric rough paths 3.1 Branched 3.2 Quasi geometric 4. Geometric vs non-geometric 5. Itˆ o’s formula David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 2 / 27

Outline 1. A little bit about rough path theory 2. The geometric assumption 3. Two approaches to non-geometric rough paths 3.1 Branched 3.2 Quasi geometric 4. Geometric vs non-geometric 5. Itˆ o’s formula David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 2 / 27

Outline 1. A little bit about rough path theory 2. The geometric assumption 3. Two approaches to non-geometric rough paths 3.1 Branched 3.2 Quasi geometric 4. Geometric vs non-geometric 5. Itˆ o’s formula David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 2 / 27

Outline 1. A little bit about rough path theory 2. The geometric assumption 3. Two approaches to non-geometric rough paths 3.1 Branched 3.2 Quasi geometric 4. Geometric vs non-geometric 5. Itˆ o’s formula David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 2 / 27

Outline 1. A little bit about rough path theory 2. The geometric assumption 3. Two approaches to non-geometric rough paths 3.1 Branched 3.2 Quasi geometric 4. Geometric vs non-geometric 5. Itˆ o’s formula David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 2 / 27

Outline 1. A little bit about rough path theory 2. The geometric assumption 3. Two approaches to non-geometric rough paths 3.1 Branched 3.2 Quasi geometric 4. Geometric vs non-geometric 5. Itˆ o’s formula David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 2 / 27

The problem We are interested in equations of the form � V i ( Y t ) dX i dY t = t , i where X : [0 , T ] → V is path with some H¨ older exponent γ ∈ (0 , 1), Y : [0 , T ] → U and V i : U → U are smooth vector fields. The theory of rough paths (Lyons) tells us that we should think of the equation as � dY t = V i ( Y t ) d X t , ( † ) i where X is an object containing X as well as information about the iterated integrals of X . We call X a rough path above X . David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 3 / 27

The problem We are interested in equations of the form � V i ( Y t ) dX i dY t = t , i where X : [0 , T ] → V is path with some H¨ older exponent γ ∈ (0 , 1), Y : [0 , T ] → U and V i : U → U are smooth vector fields. The theory of rough paths (Lyons) tells us that we should think of the equation as � dY t = V i ( Y t ) d X t , ( † ) i where X is an object containing X as well as information about the iterated integrals of X . We call X a rough path above X . David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 3 / 27

Illustrating the idea Consider the formal calculation (where everything is one dimensional) and X has γ ∈ (1 / 4 , 1 / 3]. � t Y t = Y 0 + V ( Y s ) dX s 0 David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 4 / 27

Illustrating the idea Consider the formal calculation (where everything is one dimensional) and X has γ ∈ (1 / 4 , 1 / 3]. � t Y t = Y 0 + V ( Y s ) dX s 0 � � � t V ( Y 0 ) + V ′ ( Y 0 ) δ Y 0 , s + 1 2 V ′′ ( Y 0 ) δ Y 2 = Y 0 + 0 , s + . . . dX s 0 David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 4 / 27

Illustrating the idea Consider the formal calculation (where everything is one dimensional) and X has γ ∈ (1 / 4 , 1 / 3]. � t Y t = Y 0 + V ( Y s ) dX s 0 � � � t V ( Y 0 ) + V ′ ( Y 0 ) δ Y 0 , s + 1 2 V ′′ ( Y 0 ) δ Y 2 = Y 0 + 0 , s + . . . dX s 0 � t � t � s 2 dX s + V ′ ( Y 0 ) V ( Y 0 ) = Y 0 + V ( Y 0 ) dX s 1 dX s 2 0 0 0 � t � s 3 � s 2 + V ′ ( Y 0 ) V ′ ( Y 0 ) V ( Y 0 ) dX s 1 dX s 2 dX s 3 0 0 0 � t + 1 2 V ′′ ( Y 0 ) V ( Y 0 ) V ( Y 0 ) X s 3 X s 3 dX s 3 + . . . 0 David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 4 / 27

Illustrating the idea In more than one dimension, we similarly have � t � t � t dX i dX j s 1 dX i Y t = Y 0 + V i ( Y 0 ) s + DV i · V j ( Y 0 ) s 2 0 0 0 � t � t � t dX k s 1 dX j s 2 dX i + DV i · ( DV j · V k )( Y 0 ) s 3 0 0 0 � t + 1 2 D 2 V i : ( V j , V k )( Y 0 ) X j s 3 X k s 3 dX i s 3 + . . . 0 The blue integrals are the components of X . We always have � Y t = Y 0 + V w ( Y 0 ) X t ( e w ) w The only thing that distinguishes geo and non-geo is which algebra w comes from. David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 5 / 27

Illustrating the idea In more than one dimension, we similarly have � t � t � t dX i dX j s 1 dX i Y t = Y 0 + V i ( Y 0 ) s + DV i · V j ( Y 0 ) s 2 0 0 0 � t � t � t dX k s 1 dX j s 2 dX i + DV i · ( DV j · V k )( Y 0 ) s 3 0 0 0 � t + 1 2 D 2 V i : ( V j , V k )( Y 0 ) X j s 3 X k s 3 dX i s 3 + . . . 0 The blue integrals are the components of X . We always have � Y t = Y 0 + V w ( Y 0 ) X t ( e w ) w The only thing that distinguishes geo and non-geo is which algebra w comes from. David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 5 / 27

The geometric assumption Roughly speaking, a geometric rough path X above X is a path indexed by tensors .The tensor components are “iterated integrals” of X . � t � s 2 � X t , e i � = X i dX i s 1 dX j � X t , e ij � “ = ” t s 2 0 0 � t � s 3 � s 2 dX i s 1 dX j s 2 dX k and � X t , e ijk � “ = ” s 3 0 0 0 They must be “classical integrals”, in that they satisfy the classical laws of calculus. For example, integration by parts holds ... � t � s 2 � t � s 2 t X j X i dX i s 1 dX j dX i s 1 dX j t = s 2 + s 2 . 0 0 0 0 Hence, this is an assumption on the types of integrals appearing in the equation ( † ). David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 6 / 27

The geometric assumption Roughly speaking, a geometric rough path X above X is a path indexed by tensors .The tensor components are “iterated integrals” of X . � t � s 2 � X t , e i � = X i dX i s 1 dX j � X t , e ij � “ = ” t s 2 0 0 � t � s 3 � s 2 dX i s 1 dX j s 2 dX k and � X t , e ijk � “ = ” s 3 0 0 0 They must be “classical integrals”, in that they satisfy the classical laws of calculus. For example, integration by parts holds ... � t � s 2 � t � s 2 t X j X i dX i s 1 dX j dX i s 1 dX j t = s 2 + s 2 . 0 0 0 0 Hence, this is an assumption on the types of integrals appearing in the equation ( † ). David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 6 / 27

The geometric assumption Roughly speaking, a geometric rough path X above X is a path indexed by tensors .The tensor components are “iterated integrals” of X . � t � s 2 � X t , e i � = X i dX i s 1 dX j � X t , e ij � “ = ” t s 2 0 0 � t � s 3 � s 2 dX i s 1 dX j s 2 dX k and � X t , e ijk � “ = ” s 3 0 0 0 They must be “classical integrals”, in that they satisfy the classical laws of calculus. For example, integration by parts holds ... � t � s 2 � t � s 2 t X j X i dX i s 1 dX j dX i s 1 dX j t = s 2 + s 2 . 0 0 0 0 Hence, this is an assumption on the types of integrals appearing in the equation ( † ). David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 6 / 27

The geometric rough path approach (For a more rigorous definition ...) Let T ( A ) be the tensor product space generated by the alphabet A . (If V = R d then A = { 1 , . . . d } ). A geometric rough path of regularity γ is a path X : [0 , T ] → T ( A ) ∗ , such that 1. � X t , e w �� X t , e v � = � X t , e w ✁ e v � , 2. |� X s , t , e w �| ≤ C | t − s | | w | γ for every word w ∈ T ( A ) where ✁ is the shuffle product and where X s , t = X − 1 ⊗ X t . s And Chen’s relation follows from the definition X s , t = X s , u ⊗ X u , t . David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 7 / 27

The geometric rough path approach (For a more rigorous definition ...) Let T ( A ) be the tensor product space generated by the alphabet A . (If V = R d then A = { 1 , . . . d } ). A geometric rough path of regularity γ is a path X : [0 , T ] → T ( A ) ∗ , such that 1. � X t , e w �� X t , e v � = � X t , e w ✁ e v � , 2. |� X s , t , e w �| ≤ C | t − s | | w | γ for every word w ∈ T ( A ) where ✁ is the shuffle product and where X s , t = X − 1 ⊗ X t . s And Chen’s relation follows from the definition X s , t = X s , u ⊗ X u , t . David Kelly (Warwick) An algebraic framework for Itˆ o’s formula April 26, 2013 7 / 27