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Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Report Title Stochastic Volterra Equations in Banach Spaces and SPDEs

Xicheng Zhang(HUST, Wuhan)

Workshop on Stochastic Analysis & Finance City University of Hong Kong June 29-July 3

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Contents

1 ASVE 2 Motivations 2 Local Maximal Solutions 3 Strong Solutions of Semilinear SPDEs 4 Large Deviations for ASVE 5 Reference

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Abstract Stochastic Volterra Equations (ASVE)

Let X be a 2-smooth and real separable Banach space, i.e., there exists a constant CX 2 such that for all x, y ∈ X x + y2

X + x − y2 X 2x2 X + CXy2 X.

Example: Any Hilbert space and any Lp-space over measure space (E, E , µ) with p 2 are 2-smooth. In fact, it follows from the following two elementary inequalities: for α ∈ [0, 1] and any a, b 0, 2α−1(aα + bα) (a + b)α aα + bα and for p 2 and any a, b ∈ R, 1 2(|a + b|p + |a − b|p) |a|p + (2p−1 − 1)|b|p. In this case, CLp = 2(2p−1 − 1)2/p.

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Abstract Stochastic Volterra Equations

Let l2 be the usual sequence Hilbert space and (ek)k∈N the canonical basis of l2. Let W(t) := (W k(t))k∈N be an l2- valued cylindrical Brownian motion on a complete filtered probability space (Ω, F, P; (Ft)t0).

Definition (See Ondrej´ at(2004))

A bounded linear operator B : l2 → X is called radonifying if the series Bek · W k(1) converges in L2(Ω; X). We shall denote by L2(l2; X) the space of all radonifying op- erators, and write for B ∈ L2(l2; X) BL2(l2;X) :=

• E
• Bek · W k(1)
• 2

X

1/2 . L2(l2; X) together with ·L2(l2;X) is a separable Banach space.

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Abstract Stochastic Volterra Equations

We shall consider the following stochastic Volterra integral equation in a 2-smooth Banach space X (Berger-Mizel(1980)): Xt = g(t) + t

0 A(t, s, Xs)ds +

t

0 B(t, s, Xs)dW(s) ,

(1) where g(t) is an X-valued measurable (Ft)-adapted process, and A : △ × Ω × X → X ∈ M△ × B(X)/B(X) B : △ × Ω × X → L2(l2; X) ∈ M△ × B(X)/B(L2(l2; X)). Here and below, △ := {(t, s) ∈ R2

+ : s t}, and M△ denotes

the progressively measurable σ-field on △ × Ω generated by the sets E ∈ B(△) × F with properties: 1E(t, s, ·) ∈ Fs for all (t, s) ∈ △, and s → 1E(t, s, ω) is right continuous for any t ∈ R+ and ω ∈ Ω.

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 1. SPDE driven by space-time white noise

Let W(t, x) be a Brownian sheet on R+ ×[0, 1]. Consider the following SPDE (Walsh [1986]):    ∂ut(x) ∂t = ∂2ut(x) ∂x2 + f(x, ut(x)) + σ(x, ut(x))∂2W ∂t∂x, u(t, 0) = u(t, 1) = 0, u(t, x)|t=0 = u0(x), where f(x, r), σ(x, r) : [0, 1] × R → R are two Borel measur- able functions. This equation is understood as ut(x) = 1 Gt(x, y)u0(y)dy + t 1 Gt−s(x, y)f(y, us(y))dyds + t 1 Gt−s(x, y)σ(y, us(y))W(ds, dy), where the stochastic integral is the usual Itˆ

• ’s integral and

Gt(x, y) is the fundamental solutions of homogeneous heat equation.

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 1. SPDE driven by space-time white noise

Let (φk)k∈N be an orthogonal basis of L2(0, 1). Define Ttf(x) := 1 Gt(x, y)f(y)dy, W k(t) := 1 φk(y)W(t, dy). Then we can write ut(x) = Ttu0(x) + t Tt−sf(·, us(·))(x)ds + t Tt−s(σ(·, us(·))φk(·))(x)W k(ds), which takes form (1). We can take X = Lp(0, 1), p 2.

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 2. Stochastic Reaction-Diffusion Equation

Consider the following stochastic reaction diffusion equation in Rd:

• dut(x) = [∆ut(x) − u3

t (x)]dt + σk(x, ut(x))dW k(t),

u(0, x) = u0(x). (2) Define Ttf(x) := 1

• (2πt)d
• Rd e− |x−y|2

2t

f(y)dy. The mild solution of (2) is given by ut(x) = Ttu0(x) − t Tt−su3

s(x)ds

+ t Tt−s(σk(·, us(·)))(x)W k(ds), which takes form (1).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 3. Stochastic Evolutionary Integral Equation

Consider the following stochastic evolutionary integral equa- tion (α ∈ [0, 1)): ut(x) = u0(x) + t ∆us(x) (t − s)α ds + t f(x, us(x))ds + t σk(x, us(x))dW k(s). (3) Let Stf be the solution of (See Pr¨ uss [1993]) ut(x) = f(x) + t ∆us(x) (t − s)α ds. By a solution of (3), we mean that ut = Stu0 + t St−sf(us)ds + t St−sσk(us)W k(ds), which still takes form (1).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 4. Stochastic Navier-Stokes Equation

Consider the following d-dimensional stochastic Navier-Stokes equation in Rd:      dut =

• ∆ut + (ut · ∇)ut + ∇π(t)
• dt

+ f(t, ut)dt + σk(t, ut)dW k(t) divut = 0, u|t=0 = u0, (4) Let (Tt)t0 be the Gaussian heat semigroup and P the or- thogonal projection from L2(Rd; Rd) to the divergence free subspace L2

σ(Rd; Rd). The solution of (4) can be written as

ut = Ttu0 + t Tt−sP[(us · ∇)us]ds + t Tt−sPf(s, us)ds + t Tt−sPσk(s, us)W k(ds), which also takes form (1).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 5. SPDEs driven by additive fractional Brownian motions

For H ∈ (0, 1), let KH(t, s) :=

• cH(t − s)H− 1

2 + sH− 1 2 F(t/s)

• 1{s<t}, s, t ∈ [0, 1],

where cH :=

• 2HΓ(3/2−H)

Γ(H+1/2)Γ(2−2H)

1/2 , Γ denotes the usual Gamma function, and F(u) := cH(1 2 − H) u

1

(r − 1)H− 3

2 (1 − rH− 1 2 )dr.

The sequence of independent fractional Brownian motions with Hurst parameter H ∈ (0, 1) may be defined by (See Decreusefond and ¨ Ust¨ unel [1999]) W k

H(t) :=

t KH(t, s)dW k(s), k = 1, 2, · · · , which has the covariance function RH(t, s) = E(W k

H(t)W k H(s)) = 1

2(s2H + t2H − |t − s|2H).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Motivations: 5. SPDEs driven by additive fractional Brownian motions

Consider the following SPDE driven by {W k

H, k ∈ N}

• dut(x) = [∆ut(x) + f(t, ω, x, ut(x))]dt + σk(t, x)dW k

H(t),

u(t, x)|t=0 = u0(x), where σk(t, x) is a deterministic function. As above, we con- sider the mild solution: ut = Ttu0 + t Tt−sf(s, us)ds + t Tt−sσk(s)dW k

H(s).

By the integration by parts formula, the stochastic integral can be formally written as t Tt−sσk(s)dW k

H(s) =

t

• σk(t)KH(t, s)

+ t

s

KH(u, s)

• ∆Tt−uσk(u) − Tt−s ˙

σk(s)

• du
• dW k(s).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Existence and Uniqueness of Local Maximal Solutions for ASVE

Definition (Local Maximal Solutions)

Let τ be an (Ft)-stopping time, and {Xt; t ∈ [0, τ)} an X- valued continuous (Ft)-adapted process. The pair of (X, τ) is called a local solution of Eq.(1) if P-a.s., for all t ∈ [0, τ) Xt = g(t) + t A(t, s, Xs)ds + t B(t, s, Xs)dW(s); (X, τ) is called a maximal solution of Eq.(1) if lim

t↑τ(ω) Xt(ω)X = +∞ on

{ω : τ(ω) < +∞}, P − a.s. We call (X, τ) a non-explosion solution of Eq.(1) if P{ω : τ(ω) < +∞} = 0.

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Existence and Uniqueness of Local Maximal Solutions for ASVE

Notations: Let K>1 be the set of all positive measurable functions κ on △ with the property that for any T > 0 and some β = β(T) > 1 t → t κβ(t, s)ds ∈ L∞(0, T). We introduce the following conditions on g and A, B: (H1) The process t → g(t) is continuous and (Ft)-adapted, and for any p 2 and T > 0 E

• sup

t∈[0,T]

g(t)p

X

• < +∞.

(H2) For any R > 0, there exists κ1,R ∈ K>1 such that for all (t, s) ∈ △, ω ∈ Ω and x ∈ X with xX R A(t, s, ω, x)X + B(t, s, ω, x)2

L2(l2;X) κ1,R(t, s).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Existence and Uniqueness of Local Maximal Solutions for ASVE

(H3) For any R > 0, there exists κ2,R ∈ K>1 such that for all (t, s) ∈ △, ω ∈ Ω and x, y ∈ X with xX, yX R A(t, s, ω, x) − A(t, s, ω, y)X κ2,R(t, s) · x − yX, B(t, s, ω, x) − B(t, s, ω, y)2

L2(l2;X) κ2,R(t, s) · x − y2 X.

(H4) For any R > 0, there exists a measurable function λR satisfying that for any T > 0 and some γ, C > 0 t λR(t′, t, s)ds C|t′ − t|γ, 0 t < t′ T, (5) such that for all s < t < t′, ω ∈ Ω and xX R, A(t′, s, ω, x) − A(t, s, ω, x)X +B(t′, s, ω, x) − B(t, s, ω, x)2

L2(l2;X) λR(t′, t, s).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Existence and Uniqueness of Local Maximal Solutions for ASVE

(H5) For some κ1 ∈ K>1 and all (t, s) ∈ △, ω ∈ Ω, x ∈ X A(t, s, ω, x)X κ1(t, s) · (xX + 1) B(t, s, ω, x)2

L2(l2;X) κ1(t, s) · (x2 X + 1).

(H6) For all s < t < t′, ω ∈ Ω and x ∈ X A(t′, s, ω, x) − A(t, s, ω, x)X λ(t′, t, s) · (xX + 1) B(t′, s, ω, x) − B(t, s, ω, x)2

L2(l2;X) λ(t′, t, s) · (x2 X + 1),

where λ satisfies (5).

Theorem (Existence and Uniqueness)

Under (H1)-(H4), there exists a unique maximal solution (X, τ) for Eq.(1) in the sense of Definition 2. If (H5) and (H6) also hold, then there is no explosion for Eq.(1).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Strong Solutions for Semilinear SPDEs

Let L be a positive sectorial operator on X which generates an analytic semigroup Tt = e−Lt, t 0. Assume that 0 belongs to the resolvent set of L. Then, for any α ∈ R, the fractional power Lα is well defined (See Pazy [1978]). Thus, we define the fractional Sobolev space Xα by Xα := Domain of Lα with the norm xXα := LαxX. It is easy to see that Xα is still 2-smooth, and B ∈ L2(l2; Xα) if and only if LαB ∈ L2(l2; X), i.e., BL2(l2;Xα) = LαBL2(l2;X).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Strong Solutions for Semilinear SPDEs

Consider the following semilinear SPDE: dXt = [−LXt + Φ(t, Xt)]dt + Ψ(t, Xt)dW(t), X0 = x0 ∈ X. (6) where the coefficients satisfy that for some α ∈ (0, 1) (M1) Φ : R+ × Ω × Xα → X ∈ M × B(Xα)/B(X) and Ψ : R+ × Ω × Xα → L2(l2; X α

2 ) ∈ M × B(Xα)/B(L2(l2; X α 2 )).

(M2) For any R > 0, there exist CR > 0 and β ∈ [0, 1 − α) such that for all s > 0, ω ∈ Ω and x, y ∈ BR(Xα), Φ(s, ω, x)X + Ψ(s, ω, x)2

L2(l2;X α

2 )

CR (s ∧ 1)β , Φ(s, ω, x) − Φ(s, ω, y)X CR (s ∧ 1)β x − yXα

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Strong Solutions for Semilinear SPDEs

and Ψ(s, ω, x) − Ψ(s, ω, y)2

L2(l2;X α

2 )

CR (s ∧ 1)β x − y2

Xα.

(M3) For any R, T > 0, there exist δ > 0 and α′ > 1 such that for all s, s′ ∈ [0, T], ω ∈ Ω and x ∈ BR(Xα), Φ(s′, ω, x) − Φ(s, ω, x)X CT,R|s′ − s|δ, Ψ(s, ω, x)2

L2(l2;X α′

2

) CT,R.

(M4) There is a β ∈ [0, 1 − α) such that for all s > 0, ω ∈ Ω and x ∈ Xα, Φ(s, ω, x)X C (s ∧ 1)β (1 + xXα), Ψ(s, ω, x)2

L2(l2;X α

2 )

C (s ∧ 1)β (1 + x2

Xα).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Strong Solutions for Semilinear SPDEs

Theorem (Unique Maximal Strong Solutions for SPDEs)

Assume that (M1), (M2) and (M3) hold. For any x0 ∈ X1, there exists a unique pair of (X, τ) such that (i) t → Xt ∈ X1 is continuous on [0, τ) a.s. and Xt∧τ is (Ft)-adapted; (ii) limt↑τ XtXα = +∞ on {ω : τ(ω) < +∞}; (iii) it holds that in X Xt = x0 − t LXsds + t Φ(s, Xs)ds + t Ψ(s, Xs)dW(s), ∀t ∈ [0, τ), P − a.s.. We shall call (X, τ) the unique maximal strong solution of Eq.(6). If (M4) also holds, then τ = +∞ a.s..

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Large Deviation Principle for ASVE

Below, we assume that g and A, B are non-random. Consider the following small perturbation of ASVE (1) Xǫ,t = g(t) + t A(t, s, Xǫ,s)ds + √ǫ t B(t, s, Xǫ,s)dW(s), where ǫ ∈ (0, 1). Besides (H3)-(H6), we assume that (H1)′ There are δ, α > 0 such that for any T > 0, g(t) − g(t′)X CT |t − t′|δ, t, t′ ∈ [0, T] sup

t∈[0,T]

g(t)Xα < +∞. (H2)′ For the same α as in (H1)′ and any R > 0, there exists a kernel function κα,R ∈ K>1 such that for all (t, s) ∈ △ and x ∈ X with xX R A(t, s, x)Xα + B(t, s, x)2

L2(l2;X α

2 ) κα,R(t, s).

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

Large Deviation Principle for ASVE

Theorem (Large Deviation Principle)

Assume that (H1)′, (H2)′ and (H3)-(H6) hold and L−1 is a compact operator. Then the law µǫ of Xǫ,· in C([0, T]; X) satisfies an LDP, i.e., for any E ⊂ B(C([0, T]; X)) − inf

f∈Eo I(f) lim ǫ→0

ǫ log µǫ(E) lim

ǫ→0 ǫ log µǫ(E) − inf f∈ ¯ E I(f),

where the rate function I(f) := 1

2 inf{h∈ℓ2

T : f=Xh} h2

ℓ2

T , and

Xh

t = g(t) +

t A(t, s, Xh

s )ds +

t B(t, s, Xh

s )˙

h(s)ds and ℓ2

T :=

• h =

· ˙ h(s)ds : ˙ h ∈ L2(0, T; l2)

• .

Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

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Stochastic Volterra Equations in Banach Spaces and SPDEs Contents ASVE Motivations Local Solutions of ASVE Strong Solutions of Semilinear SPDEs Large Deviation Principle for ASVE Reference

End

Thank you!

Xicheng Zhang Huazhong University of Science and Technology Email: XichengZhang@gmail.com