Uniqueness, existence and regularity of stochastic Volterra integral - PowerPoint PPT Presentation

Uniqueness, existence and regularity of stochastic Volterra integral equations Alexander Kalinin 2 nd Imperial-CUHK Workshop on Quantitative Finance 22 May 2019

1 Kernel functions and path-dependency 2 Deterministic Volterra integral equations 3 Stochastic Volterra integral equations

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Kernel functions and path-dependency Let us regard the following stochastic Volterra integral equation with an initial value condition: � t � t X t = X 0 + K ( t , s ) b ( s , X s ) ds + K ( t , s ) σ ( s , X s ) dW s (1) 0 0 for t ∈ [0 , T ] a.s. In this one-dimensional setting, K : [0 , T ] 2 → R denotes the kernel function and b , σ : [0 , T ] × R → R are measurable. For instance, one can take K ( t , s ) = ( t − s ) β − 1 for all s , t ∈ [0 , T ] with s < t and some β > 0. 1 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations The case of regular kernels: (i) Such equations were first analyzed by Berger and Mizel (1980), even in a multidimensional setting. (ii) Protter (1985) showed the existence and uniqueness of strong solutions, allowing for path-dependency and a semimartingale as integrator. The case of irregular kernels: (iii) Zhang (2010) proved the existence and uniqueness of strong solutions with continuous paths, even in a Banach space. (iv) Abi Jaber, Larsson and Pulido (2017) considered weak solutions if the coefficients are affine. 2 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations (i) In the sequel, let C α r ([0 , T ] , E ) denote the Banach space of all x ∈ C ([0 , T ] , E ) that are α -Hölder-continuous on [ r , T ], endowed with the ‘delayed α -Hölder norm’ | x ( s ) − x ( t ) | � x � α, r := � x r � ∞ + sup . | s − t | α s , t ∈ [ r , T ]: s � = t (ii) Let L α, q ([ r , T ] 2 , E ) be the linear space of all measurable maps K : [ r , T ] 2 → E for which there is c ≥ 0 such that � � t � � s � 1 / q � 1 / q | K ( t , s ′ ) | q ds ′ | K ( t , r ′ ) − K ( s , r ′ ) | q dr ′ + s r is bounded by c ( t − s ) α for every s , t ∈ [ r , T ] with s < t . 3 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Example Let E be a Banach algebra, k ∈ C α ([ r , T ] , E ) and l : [ r , T ] → E be measurable. Then K : [ r , T ] 2 → E given by K ( t , s ) := k ( t ) l ( s ) belongs to L α, q ([ r , T ] 2 , E ) as soon as there is ˜ q > q such that � T q dt < ∞ | l ( t ) | ˜ and α ≤ 1 / q − 1 / ˜ q . r 4 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Deterministic Volterra integral equations First, we consider the following path-dependent Volterra integral equation coupled with a running value condition: � t K ( t , s ) ϕ ( s , x s ) ds x ( t ) = ˆ x ( t ) + (v) r for t ∈ [ r , T ] and x ( q ) = ˆ x ( q ) for q ∈ [0 , r ]. In this framework, D is an open set in the separable Banach space E and K : [ r , T ] 2 → R and ϕ : [ r , T ] × C ([0 , T ] , D ) → E are two measurable maps. 5 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations We call an interval I in [0 , T ] admissible if it agrees with either [0 , t ] or [0 , t ) for some t ∈ ( r , T ]. Definition A solution to (v) on an admissible interval I is a map x ∈ C ( I , D ) � t such that x r = ˆ x r , as well as r | K ( t , s ) ϕ ( s , x s ) | ds < ∞ and � t K ( t , s ) ϕ ( s , x s ) ds x ( t ) = ˆ x ( t ) + r for all t ∈ I with t ≥ r . 6 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations We let K satisfy a Hölder continuity condition and ϕ a boundedness and Lipschitz condition on a class of bounded sets. (v.1) There are α ∈ (0 , 1] and q > 1 so that K ∈ L α, q ([ r , T ] 2 , R ). (v.2) For any closed and bounded set B in D that is bounded away from ∂ D there are k , λ ∈ L p ([ r , T ] , R + ) such that | ϕ ( s , x ) | ≤ k ( s ) , | ϕ ( s , x ) − ϕ ( s , y ) | ≤ λ ( s ) � x − y � ∞ for any s ∈ [ r , T ] and x , y ∈ C ([0 , T ] , B ), where p is the dual exponent of q . 7 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Unique non-extendible solutions (K., 2019) Under (v.1) and (v.2), there is a unique non-extendible solution x r , ˆ x to (v) on a maximal interval of existence I r , ˆ x that is open in [0 , T ]. For t + x := sup I r , ˆ x we either have I r , ˆ x = [0 , T ] or r , ˆ � 1 � lim min dist ( x r , ˆ x ( t ) , ∂ D ) , = 0 . (2) 1 + | x r , ˆ x ( t ) | t ↑ t + r , ˆ x 8 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Example Assume that (i) there is β ∈ (0 , 1] such that K ( t , s ) = ( t − s ) β − 1 for all s , t ∈ [ r , T ] with s < t , and (ii) there is a map ψ : D → E that is Lipschitz continuous on any bounded set in D that is bounded away from ∂ D with ϕ ( s , x ) = ψ ( x ( s )) for any s ∈ [ r , T ] and x ∈ C ([0 , T ] , D ). Then (v.1) and (v.2) are valid, and the proposition applies. 9 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Next, we require that � ˆ x � α, r < ∞ and add an affine boundedness condition on ϕ . (v.3) There are k , l ∈ L p ([ r , T ] , R + ) so that | ϕ ( s , x ) | ≤ k ( s ) + l ( s ) � x � ∞ for all s ∈ [ r , T ] and x ∈ C ([0 , T ] , D ). 10 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations From (v.3) it follows that any global solution to (v) lies in the set R α, p x of all x ∈ C α r ([0 , T ] , E ) satisfying r , ˆ � � t � x t − ˆ x t � p � t � r l ( s ) p ds k ( s ) p + l ( s ) p � ˆ α, r ≤ e r p x s � p ∞ ds r p r for each t ∈ [ r , T ] and some constant r p ≥ 0. 11 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Global delayed Hölder continuous solutions Let (v.1)-(v.3) hold and D = E . Then I r , ˆ x = [0 , T ] and for any x 0 ∈ R α, p x the sequence ( x n ) n ∈ N in R α, p x , recursively given by r , ˆ r , ˆ � r ∨ t K ( t , s ) ϕ ( s , x s x n +1 ( t ) := ˆ x ( t ) + n ) ds r for all n ∈ N 0 , converges in the delayed α -Hölder norm � · � α, r to x r , ˆ x , the unique global solution to (v). 12 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Estimates in the delayed Hölder norm Moreover, the Picard sequence ( x n ) n ∈ N satisfies n − 1 � � t L p ( t ) i � k ( s ) p + l ( s ) p � ˆ � � x t x t � p x s � p n − ˆ α, r ≤ r p ∞ ds i ! r i =0 + L p ( t ) n � x t x t � p 0 − ˆ ∞ , n ! � t r l ( s ) p ds , and where L p ( t ) := r p � 1 � t ∞ � 1 / p � � i / p λ ( s ) p ds � � x t n − x t x � α, r ≤ � x t 1 − x t 0 � ∞ r p r , ˆ i ! r i = n for all n ∈ N , every t ∈ [ r , T ] and some λ ∈ L p ([ r , T ] , R + ). 13 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Example Suppose that D = E , x ∈ C β − (i) there exists β ∈ (0 , 1] with ˆ ([0 , T ] , E ) and K ( t , s ) r = ( t − s ) β − 1 for any s , t ∈ [ r , T ] with s < t , and (ii) there is a map ψ : E → E that is Lipschitz continuous on bounded sets and of linear growth such that ϕ ( s , x ) = ψ ( x ( s )) for each s ∈ [ r , T ] and x ∈ C ([0 , T ] , E ). x ∈ C β − Then x r , ˆ ([0 , T ] , E ) and the Picard sequence converges in r the delayed α -Hölder norm � · � α, r to x r , ˆ x for each α ∈ [0 , β ). 14 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Continuity of the solution map Let D = E and (v.1)-(v.3) be satisfied. Then the map C α r ([0 , T ] , E ) → C α r ([0 , T ] , E ) , x �→ x r , ˆ ˆ x is Lipschitz continuous on bounded sets. That is, for any n ∈ N there is λ n ≥ 0 such that � x r , ˆ x − x r , ˆ y � α, r ≤ λ n � ˆ x − ˆ y � α, r y ∈ C α for each ˆ x , ˆ r ([0 , T ] , E ) with � ˆ x � α, r ∨ � ˆ y � α, r ≤ n . 15 / 26

Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations Stochastic Volterra integral equations We turn to the following path-dependent stochastic Volterra integral equation combined with a running value condition: � t � t K ( t , s ) b ( s , X s ) ds + K ( t , s ) σ ( s , X s ) dW s X t = ξ t + (V) r r for t ∈ [ r , T ] and X r = ξ r a.s. Here, H and L are two separable Hilbert spaces, D ⊂ H is open, W is an id L -cylindrical Wiener process and b : [ r , T ] × C ([0 , T ] , D ) → H , σ : [ r , T ] × C ([0 , T ] , D ) → L ( L , H ) are two measurable maps. 16 / 26