Martingale Problem under Nonlinear Expectations Chen Pan USTC, - - PowerPoint PPT Presentation

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Martingale Problem under Nonlinear Expectations

Chen Pan

USTC, China and UC Berkeley

Sixth WCMF 2014, Santa Barbara joint work with Xin Guo (UC Berkeley) & Shige Peng (Shandong University, China)

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Outline

1

Background

2

Motivation for Martingale Problem

3

Martingale Problem under Nonlinear Expectations Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

4

Related Result – Weak Solution to G-SDE Weak solution

5

Discussions

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Linearity in probability theory

A random variable X is uniquely determined by its probability distribution P, or equivalently its expectation, P(X ∈ A) = E[1A] 1-1 correspondence between linear expectation and additive probability measure

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Sublinear Expectation ˜ E

Given χ (e.g. all bounded measurable random variables), ˜ E : χ → R is sublinear iff (a) Monotonicity: If X ≤ Y , then ˜ E[X] ≤ ˜ E[Y ] (b) Constant preserving: ˜ E[X + c] = ˜ E[X] + c (c) Sublinearity: ˜ E[X + Y ] ≤ ˜ E[X] + ˜ E[Y ]. (d) Positive homogeneity: ˜ E[λX] = λ˜ E[X] for all λ > 0

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

No more 1-1 correspondence between ˜ E and ˜ P

Clearly ˜ P(A) = ˜ E[1A] = ˜ Ef [1A] for all f continuous and strictly increasing, f (x) = x for x ∈ [0, 1], where ˜ Ef [X] = f −1(˜ E[f (X)]).

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Connecting linear and sublinear expectations

Denis, Hu and Peng (2011) There exists a weakly compact family of probability measures P on (Ω, B(Ω)) such that ˜ E[X] = max

P∈P E P[X],

where E P is the linear expectation with respect to P, for a proper class

• f random process X.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Sublinear expectation and model uncertainty

Sublinear expectation “measures” the model uncertainty, the bigger the expectation ˜ E, the more the uncertainty. ˜ E1[X] ≤ ˜ E2[X] iff P1 ⊂ P2

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Sublinear expectation and risk measure

Let ρ(X) = ˜ E[−X] Then we get a coherent risk measure ρ : χ → R (a) Monotonicity: If X ≥ Y , then ρ(X) ≤ ρ(Y ). (b) Constant translatability: ρ(X + c) = ρ(X) − c (c) Convexity: ρ(αX + (1 − α)Y ) ≤ αρ(X) + (1 − α)ρ(Y ), α ∈ [0, 1]. (d) Positive homogeneity: ρ(λX) = λρ(X) for all λ > 0.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

G− normal distribution and G-heat equation

Starting point (Peng 2005) G-normal distribution X, N(0 × [σ2, σ2]) is characterized by the G-heat equation ∂tu − G(D2u) = 0, u|t=0 = φ. Here G(·) : R → R is a monotonic, sublinear function, with G(γ) = 1 2 sup

α∈[σ2,σ2]

γα, where σ2 ≤ σ2 are constants.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

An Alternative Way

Many difficulties in dealing with the optimization problem in G-expectation framework, other ways are needed to explore to avoid the inconvenience. Peng’s way(sublinear case): G-normal distribution + “independence” ⇒ G-Brownian motion ⇒ general G-(semi)-martingales Our idea: General stochastic processes in a nonlinear expectation space

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Martingale Problem under Nonlinear Expectations

Definition of martingale problem Find a family of operators { Et}t≥0 on a nonlinear expectation space (Ω, H) such that ϕ(Xt) − t

• G(Xθ, ϕx(Xθ), ϕxx(Xθ)) dθ, t ≥ 0

is an { Et}-martingale for all ϕ ∈ C ∞

0 (Rd).

• G : Rd × Rd × Sd → R continuous with desirable properties

Ω = Cx0([0, ∞); Rd) and Xt(ω) = ω(t), ω ∈ Ω.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Comparison with classical martingale problems

Classical M.P.’s Our M.P.’s to find a probability measure P

• n (Ω, F)

to find a nonlinear expectation ˜ E on (Ω, H) X0 = x P-a.s. X0 = x in L1

• E

ϕ(Xt) − t

0 Lθϕ(Xθ) dθ is a P-

martingale for ∀ϕ ∈ C ∞ ϕ(Xt) − t G(θ, Xθ, ϕx(Xθ), ϕxx(Xθ))dθ is an E-martingale for ∀ϕ ∈ C ∞ Lθ =

1 2

aij(θ, ·)

∂2 ∂xi∂xj

+ bi(θ, ·) ∂

∂xi is a linear differen-

tial operator Nonlinear PDE associated with

• G

: [0, ∞) × Rd × Rd × Sd → R is a con- tinuous function with some properties

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

The critical step

To identify appropriate classes of G

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Recipe and Ingredients

PDEs associated with G For a given G, the associated state-dependent parabolic PDE

• ∂tu −

G(x, Du, D2u) = 0, (t, x) ∈ (0, T] × Rd, u(0, x) = ϕ(x), x ∈ Rd. ( P) Comparison and existence theorems for the PDEs Constructing conditional expectations from solutions of PDEs

Finite dimensional distribution + Kolmogorov’s time consistency theorem

Properties of the conditional expectations

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Class D for G

A continuous function G : Rd × Rd × Sd → R is of class D, if

• G(x, 0, 0) ≡ 0 for all x ∈ Rd
• G is positive homogeneous,

(DOM)

• G(x, p, A) −

G(x, p′, A′) ≤ G(x, p − p′, A − A′) for each x ∈ Rd, (p, A), (p′, A′) ∈ Rd × Sd, and the continuous function G : Rd × Rd × Sd → R satisfies

• A. Subadditivity G(x, p + ¯

p, A + ¯ A) ≤ G(x, p, A) + G(x, ¯ p, ¯ A);

• B. Positive Homogeneity G(x, λp, λA) = λG(x, p, A);
• C. Monotonicity G(x, p, A) ≤ G(x, p, A + ˜

A);

• D. G is uniformly Liptschitz continuous with respect to x.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Example of G and G

G is sublinear with the form G(x, p, A) = sup

γ∈Γ

1 2tr[a(x, γ)A] + (b(x, γ), p)

• ,

and G has the form

• G(x, p, A) = sup

γ∈Γ

inf

λ∈Λ

1 2tr[σ(x, γ, λ)σ′(x, γ, λ)A] + (b(x, γ, λ), p)

• r
• G(x, p, A) = inf

γ∈Γ sup λ∈Λ

1 2tr[σ(x, γ, λ)σ′(x, γ, λ)A] + (b(x, γ, λ), p)

• ,

where Γ, Λ are index sets, and the coefficients satisfy some proper conditions

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Constructing E

Now, define Et according to the operator: Tt[ϕ(·)](x) := u(t, x), where u is the unique viscosity solution of the PDE. Ω = Cx0([0, ∞); Rd), Xt(ω) = ωt, ω ∈ Ω is canonical, Take ϕ0 in a proper function space on (Rd)N denoted by C((Rd)N) Set ξ(ω) = ϕ0(Xt1, · · · , XtN), 0 = t0 ≤ t1 ≤ · · · ≤ tN ≤ T, Define

• Et[ξ] = ϕN−j(ωt1, · · · , ωtj), if t = tj, 0 ≤ j ≤ N,

and denote E := E0.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Here for 0 ≤ j ≤ N, ϕ1(x1, · · · , xN−1) = TtN−tN−1[ϕ0(x1, · · · , xN−1, ·)](xN−1), . . . ϕN−j(x1, · · · , xj) = Ttj+1−tj[ϕN−j−1(x1, · · · , xj, ·)](xj), . . . ϕN−1(x1) = Tt2−t1[ϕN−2(x1, ·)](x1), ϕN = Tt1[ϕN−1(·)](x0), where ϕk ∈ C((Rd)N−k), 0 ≤ k ≤ N − 1 and ϕN ∈ R.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Properties of E and E

(Monotonicity)

• Et[ξ] ≤

Et[η] if ξ ≤ η (Constant preserving) For c ∈ R constant,

• E[ξ + c] =

E[ξ] + c (Tower property)

• Es ◦

Es+h = Es, h > 0. (Domination)

• Et[ξ] −

Et[η] ≤ Et[ξ − η]

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Specially, for the nonlinear expectation E, we have (Subadditivity) Et[ξ + η] ≤ Et[ξ] + Et[η]. (Positive homogeneity) Es[ξη] = ξ+Es[η] + ξ−Es[−η]. In particular, Et[λξ] = λEt[ξ] for any constant λ ≥ 0.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem

Existence to Martingale Problem Assuming unique solutions to PDEs w.r.p. G and

• G. For any

ϕ ∈ C ∞

0 (R), α0 > 0.

• Es[ϕ(Xt) − ϕ(Xs) −

t

s

• G (Xθ, ϕx(Xθ), ϕxx(Xθ)) dθ] = 0, 0 ≤ s ≤ t < ∞.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Weak solution

Weak solution of G-SDE

Definition A weak solution of G-SDE is a triple ((Ω, H, E), X, B), where (Ω, H, E) is a nonlinear expectation space, X is a continuous process on the nonlinear expectation space (Ω, H, E), and B is a G-Brownian motion, the identity Xt = X0 + t b(Xθ) dθ + t r(Xθ) dBθ + t σ(Xθ) dBθ holds in the nonlinear expectation space.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Weak solution

Theorem of Existence (1-dim. case) There exists at least one weak solution of the G-SDE. Especially, take ˆ G(x, p, A) = 1 2 sup

γ∈Γ

{[2r(x)p + σ2(x)A]γ} + b(x)p, where Γ = [σ2, σ2], σ2 > 0, then the nonlinear expectation ˆ E derived from the PDE ∂tu − ˆ G(x, u′, u′′) = 0, (t, x) ∈ (0, T] × R, together with the canonical space ˆ Ω = Cx([0, ∞)) and the process Bt = t dXθ σ(Xθ) − t b(Xθ) σ(Xθ) dθ − t r(Xθ) σ3(Xθ) dXθ, t ≥ 0 is a weak solution of the G-SDE.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions Weak solution

Remark

For d-dimensional case, one needs to establish a generalized Girsanov type theorem: the existence of a dominated sublinear expectation replaces the classical “absolute continuity of probability measures”. The weak solution is not unique. Actually, one can choose different pairs (σ2

• , σ2
• ) to derive different ˆ

G◦, as long as σ2

• > 0.

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

Summary

A martingale problem under non-linear expectations is studied Notion of weak solutions of SDE under nonlinear expectation space is proposed The PDE is state dependent in this version. One can generalize to a path-dependent version

Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G-SDE Discussions

THANK YOU!

Reference:

• X. Guo, C. Pan and S. Peng, Martingale Problem under Nonlinear

Expectations, under review (http://arxiv.org/abs/1211.2869)

Pan Martingale Problem under Nonlinear Expectations