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

Uniqueness, existence and regularity of stochastic Volterra integral equations

Alexander Kalinin

2nd Imperial-CUHK Workshop

• n 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: Xt = X0 + t K(t, s)b(s, Xs) ds + t K(t, s)σ(s, Xs) dWs (1) 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.

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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.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

(i) In the sequel, let Cα

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’ xα,r := xr∞ + sup

s,t∈[r,T]: s=t

|x(s) − x(t)| |s − t|α . (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

|K(t, s′)|q ds′ 1/q + s

r

|K(t, r ′) − K(s, r ′)|q dr ′ 1/q is bounded by c(t − s)α for every s, t ∈ [r, T] with s < t.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Example

Let E be a Banach algebra, k ∈ Cα([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

r

|l(t)|˜

q dt < ∞

and α ≤ 1/q − 1/˜ q.

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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: x(t) = ˆ x(t) + t

r

K(t, s)ϕ(s, xs) ds (v) 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.

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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]

• r [0, t) for some t ∈ (r, T].

Definition

A solution to (v) on an admissible interval I is a map x ∈ C(I, D) such that xr = ˆ xr, as well as t

r |K(t, s)ϕ(s, xs)| ds < ∞ and

x(t) = ˆ x(t) + t

r

K(t, s)ϕ(s, xs) ds for all t ∈ I with t ≥ r.

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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.

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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 xr,ˆ

x to (v) on a maximal interval of existence Ir,ˆ x that is open in

[0, T]. For t+

r,ˆ x := sup Ir,ˆ x we either have Ir,ˆ x = [0, T] or

lim

t↑t+

r,ˆ x

min

• dist(xr,ˆ

x(t), ∂D),

1 1 + |xr,ˆ

x(t)|

• = 0.

(2)

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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.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Next, we require that ˆ xα,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).

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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 Rα,p

r,ˆ x of all x ∈ Cα r ([0, T], E) satisfying

xt − ˆ xtp

α,r

rp ≤ erp

t

r l(s)p ds

t

r

k(s)p + l(s)pˆ xsp

∞ ds

• for each t ∈ [r, T] and some constant rp ≥ 0.

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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 Ir,ˆ

x = [0, T] and for any

x0 ∈ Rα,p

r,ˆ x the sequence (xn)n∈N in Rα,p r,ˆ x , recursively given by

xn+1(t) := ˆ x(t) + r∨t

r

K(t, s)ϕ(s, xs

n) ds

for all n ∈ N0, converges in the delayed α-Hölder norm · α,r to xr,ˆ

x, the unique global solution to (v).

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Estimates in the delayed Hölder norm

Moreover, the Picard sequence (xn)n∈N satisfies xt

n − ˆ

xtp

α,r ≤ rp n−1

• i=0

Lp(t)i i! t

r

k(s)p + l(s)pˆ xsp

∞ ds

• + Lp(t)n

n! xt

0 − ˆ

xtp

∞,

where Lp(t) := rp t

r l(s)p ds, and

xt

n − xt r,ˆ xα,r ≤ xt 1 − xt 0∞ ∞

• i=n

1 i! 1/p rp t

r

λ(s)p ds i/p for all n ∈ N, every t ∈ [r, T] and some λ ∈ L p([r, T], R+).

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Example

Suppose that D = E, (i) there exists β ∈ (0, 1] with ˆ x ∈ Cβ−

r

([0, T], E) and K(t, s) = (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). Then xr,ˆ

x ∈ Cβ− r

([0, T], E) and the Picard sequence converges in the delayed α-Hölder norm · α,r to xr,ˆ

x for each α ∈ [0, β).

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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 Cα

r ([0, T], E) → Cα r ([0, T], E),

ˆ x → xr,ˆ

x

is Lipschitz continuous on bounded sets. That is, for any n ∈ N there is λn ≥ 0 such that xr,ˆ

x − xr,ˆ yα,r ≤ λnˆ

x − ˆ yα,r for each ˆ x, ˆ y ∈ Cα

r ([0, T], E) with ˆ

xα,r ∨ ˆ yα,r ≤ n.

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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: Xt = ξt + t

r

K(t, s)b(s, X s) ds + t

r

K(t, s)σ(s, X s) dWs (V) for t ∈ [r, T] and X r = ξr a.s. Here, H and L are two separable Hilbert spaces, D ⊂ H is open, W is an idL-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.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Definition

A strong solution to (V) is a pair (X, τ) consisting of a stopping time τ satisfying τ > r and some adapted continuous process X : {(t, ω) ∈ [0, T] × Ω | t < τ(ω)} → H such that t

r

|K(t, s)b(s, X s)| ds + t

r

|K(t, s)σ(s, X s)|2 ds < ∞ and (V) hold a.s. on {t < τ} for each t ∈ [0, T].

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Let K satisfy a Hölder continuity condition and (b, σ) a boundedness and Lipschitz condition on certain bounded sets. (V.1) There are α ∈ (0, 1] and q > 2 with 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+) so that |b(s, x)| ∨ |σ(s, x)| ≤ k(s), |b(s, x) − b(s, y)| ∨ |σ(s, x) − σ(s, y)| ≤ λ(s)x − y∞ for every s ∈ [r, T] and x, y ∈ C([0, T], B), where p is the dual exponent of q/2.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Unique non-extendible strong solutions (K., 2019)

Under (V.1) and (V.2), there is a unique strong solution (X r,ξ, τr,ξ) to (V) that is non-extendible in the sense that lim

t↑τr,ξ

min

• dist(X r,ξ

t

, ∂D), 1 1 + |X r,ξ

t

|

• = 0

(3) a.s. on {τr,ξ < ∞}.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

First, we see that the linear space C α,p

r

([0, T], H) of all adapted continuous processes X : [0, T] × Ω → H satisfying E[Xp

α,r] < ∞

is a complete seminormed space.

Convergence in · α,r in p-th moment

A sequence (nX)n∈N in C α,p

r

([0, T], H) converges if and only if (nXα,r)n∈N is p-fold uniformly integrable and lim

n↑∞ P(nX − Xα,r ≥ ε) = 0

for all ε > 0 and an adapted continuous process X : [0, T] × Ω → H. In the latter case, E[Xp

α,r] < ∞ and limn↑∞ E[nX − Xp α,r] = 0.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Now let E

• ξp

α,r] < ∞ and consider an affine boundedness and a

Lipschitz condition on (b, σ). (V.3) There are k, l ∈ L p([r, T], R+) such that |b(s, x)| ∨ |σ(s, x)| ≤ k(s) + l(s)x∞ for any s ∈ [r, T] and x ∈ C([0, T], D). (V.4) There is λ ∈ L p([r, T], R+) such that |b(s, x) − b(s, y)| ∨ |σ(s, x) − σ(s, y)| ≤ λ(s)x − y∞ for every s ∈ [r, T] and x, y ∈ C([0, T], D).

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Condition (V.3) implies that any global solution to (V) belongs to the set C α,p

r,ξ of all X ∈ C α,p r

([0, T], H) such that E

• X t − ξtp

α,r

• cα,p

≤ ecα,p

t

r l(s)p ds

t

r

k(s)p + l(s)pE

• ξsp

∞

• ds
• for any t ∈ [r, T] and some constant cα,p ≥ 0.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Global delayed Hölder continuous strong solutions

Let (V.1)-(V.4) be valid and D = H. Then τr,ξ = ∞ and for any

0X ∈ C α,p r,ξ the sequence (nX)n∈N in C α,p r,ξ , recursively given by n+1Xt = ξt +

r∨t

r

K(t, s)b(s, nX s) ds + r∨t

r

K(t, s)σ(s, nX s) dWs a.s. for all n ∈ N0, converges in the delayed α-Hölder norm · α,r in p-th mean to X r,ξ, the unique global solution to (V). That is, lim

n↑∞ E

• nX − X r,ξp

α,r

• = 0.

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Estimates in the delayed Hölder norm in p-th mean

Further, for any n ∈ N and each t ∈ [r, T], E

• nX t − ξtp

α,r

• is

bounded by cα,p

n−1

• i=0

Lα,p(t)i i! t

r

k(s)p + l(s)pE

• ξsp

∞

• ds
• + Lα,p(t)n

n! E

• 0X t − ξtp

∞

• ,

where Lα,p(t) := cα,p t

r l(s)p ds, and (E

• nX·∧t − X r,ξ

·∧tp α,r

• )1/p

cannot exceed

• E
• 1X t − 0X tp

∞

1/p

∞

• i=n

1 i! 1/p cα,p t

r

λ(s)p ds i/p .

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Example

Assume that D = H, (i) there is β ∈ (0, 1/2] with ξ ∈ C β−,p

r

([0, T], H) for any p ≥ 1 and K(t, s) = (t − s)β−1/2 for all s, t ∈ [r, T] with s < t, and (ii) there are two maps ϕ : H → H and ψ : H → L (L, H) that are Lipschitz continuous on bounded sets and of linear growth such that for each s ∈ [r, T] and x ∈ C([0, T], H), b(s, x) = ϕ(x(s)) and σ(s, x) = ψ(x(s)). Then X r,ξ ∈ C β−,p

r

([0, T], H) and the Picard sequence converges in the norm · α,r in p-th moment to X r,ξ for each p ≥ 1 and α ∈ [0, β).

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Kernel functions and path-dependency Deterministic Volterra integral equations Stochastic Volterra integral equations

Continuity of the strong solution map

Let D = H and (V.1)-(V.4) be valid. Then the map C α,p

r

([0, T], H) → C α,p

r

([0, T], H), ξ → X r,ξ is Lipschitz continuous. Put differently, there is λ ≥ 0 satisfying (E

• X r,ξ − X r,ηp

α,r

1/p ≤ λ(E[ξ − ηp

α,r

1/p for any ξ, η ∈ C α,p

r

([0, T], H).

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Thank you for your attention!

References

[1] Rama Cont and Alexander Kalinin. On the support of solutions of stochastic differential equations with path-dependent coefficients. arXiv preprint 1806.08988, 2018. [2] Alexander Kalinin. Deterministic and stochastic path-dependent Volterra integral equations. Preprint, 2019.