Supermartingales as Radon-Nikodym densities, Novikov's and ... · Overview Supermartingales as RN...

Overview Supermartingales as RN derivatives Novikov-Kazamaki-type conditions Distribution of explosion time

Supermartingales as Radon-Nikodym densities,Novikov’s and Kazamaki’s criteria, and the

distribution of explosion times

Johannes Ruf

Oxford-Man Institute of Quantitative Finance

This presentation is based on joint papers withNicolas Perkowski, Martin Larsson, and Ioannis Karatzas

Financial/Actuarial Mathematics SeminarUniversity of Michigan

October 15, 2013

This presentation has three parts1. Supermartingales as Radon-Nikodym densities.2. Canonical proof that certain conditions of the form

supσ∈T

E[Hσ] <∞

are sufficient for the uniform integrability / martingaleproperty of a nonnegative local martingale Z .

3. Computation of the distribution of the explosion time S of thediffusion

dX (t) = s(X (t)) (dW (t) + b(X (t))dt) , X (0) = ξ.

Unifying theme: A one-to-one correspondence of• the lack of martingale property of a nonnegative local

martingale;• a positive probability of explosions of a related process.

(due to McKean and Follmer)

supσ∈T

E[Hσ] <∞

supσ∈T

E[Hσ] <∞

supσ∈T

E[Hσ] <∞

supσ∈T

E[Hσ] <∞

Supermartingales as Radon-Nikodym densities

1. Given:• Filtered probability space (Ω,F , (Ft)t≥0,P);• nonnegative, right-continuous P–supermartingale Z = (Zt)t≥0

with EP [Z0] = 1.

2. “Can we somehow assign a measure Q to Z , in the sense of achange of measure?”

3. Possible applications:• Deriving certain properties of Z (martingale property,

decompositions, ...);• Duality;• Filtration shrinkage;• Financial mathematics: study of arbitrage opportunities.

with EP [Z0] = 1.

Constructions of measures associated to Z

• Construction of finitely additive measure on (Ω× [0,∞],A ),where A ⊂P is a suitable algebra, and where P denotes thepredictable sigma algebra. (Doleans; Metivier & Pellaumail).

• Under certain topological assumptions on the filtered space(Ω,F , (Ft)t≥0), construction of countably additive measureon (Ω× [0,∞],P) (3 different constructions: Follmer; Meyer;Stricker). If Z is local martingale then construction on (Ω,F).

• If Z is the pointwise limit of a family of uniformly integrablemartingales, then existence of a finitely additive measure on(Ω,F). (Cvitanic & Schachermayer & Wang; Karatzas &Zitkovic).

Definitions: countably additive caseIf Q and τ are a probability measure and a stopping time on(Ω,F , (Ft)t≥0), then (Q, τ) is called a Follmer pair for Z if

P[τ =∞] = 1 and

Q[A ∩ ρ < τ] = EP [Zρ1A] for all A ∈ Fρ and finite s.t. ρ.(1)

We also call Q a Follmer (countably additive) measure for (Z , τ),or, slightly abusing notation, a Follmer (countably additive)measure for Z .

• Inspired by Kunita-Yoeurp decomposition.• Follmer pair for Z is unique if given Q, Q and τ , τ such that

(Q, τ) and (Q, τ) both satisfy (1), we have Q = Q andQ[τ = τ ] = 1.

• If τ is a stopping time, then Follmer (c.a.) measure for (Z , τ)is unique if, given Q, Q such that (Q, τ) and (Q, τ) bothsatisfy (1), we have Q = Q.

P[τ =∞] = 1 and

Definitions: finitely additive case

• ba1(Ω,F ,P): space of finitely additive set functions Q on F ,weakly absolutely continuous with respect to P, withQ[Ω] = 1.

• Q can be uniquely decomposed as Q = Qr + Qs . Here, Qr issigma-additive and Qs is purely finitely additive.

• ba(Ω,F ,P) can be identified with the dual spaceL∞(Ω,F ,P)∗ of L∞ = L∞(Ω,F ,P).

A weakly absolutely continuous, finitely additive probabilitymeasure Q ∈ ba1(Ω,F ,P), such that

(Q|Fρ)r [A] = EP [Zρ1A] for all A ∈ Fρ and finite s.t. ρ,

is called Follmer finitely additive measure for Z .

Comparison of Follmer f.a. and c.a. measures

• If Z is a uniformly integrable P–martingale and ifF =

∨t≥0Ft , then each Follmer countably additive measure

for Z is a Follmer finitely additive measure for Z . Moreover,the class of Follmer finitely additive measures is strictly largerthan the class of Follmer countably additive measures (inmost cases).

• If Z is not a uniformly integrable P–martingale, then the setsof Follmer countably additive measures for Z and of Follmerfinitely additive measures for Z are disjoint.

• In general, the existence of a Follmer countably additivemeasure does not imply the existence of a Follmer finitelyadditive measure (e.g., finite probability space), nor does theopposite implication hold (e.g., under “usual assumptions”).

Existence and uniqueness: c.a. case (preparation)

Assumption P: Let E be a state space, and let ∆ /∈ E be a

cemetery state. For all ω ∈ (E ∪ ∆)[0,∞) define

ζ(ω) = inft ≥ 0 : ω(t) = ∆.

Let Ω ⊂ (E ∪ ∆)[0,∞) be the space of pathsω : [0,∞)→ E ∪ ∆, for which ω is cadlag on [0, ζ(ω)), and forwhich ω(t) = ∆ for all t ≥ ζ(ω).For all t ≥ 0 define Xt(ω) = ω(t) and the sigma algebrasF0t = σ(Xs : s ∈ [0, t]) and Ft =

⋂s>t F0

s . Moreover, setF =

∨t≥0F0

t =∨

t≥0Ft .

τZn = inft ≥ 0 : Zt ≥ n ∧ n; τZ = limn↑∞

τZn .

ζ(ω) = inft ≥ 0 : ω(t) = ∆.

⋂s>t F0

∨t≥0F0

t =∨

t≥0Ft .

τZn .

ζ(ω) = inft ≥ 0 : ω(t) = ∆.

⋂s>t F0

∨t≥0F0

t =∨

t≥0Ft .

τZn .

ζ(ω) = inft ≥ 0 : ω(t) = ∆.

⋂s>t F0

∨t≥0F0

t =∨

t≥0Ft .

τZn .

ζ(ω) = inft ≥ 0 : ω(t) = ∆.

⋂s>t F0

∨t≥0F0

t =∨

t≥0Ft .

τZn .

Existence and uniqueness: c.a. case (theorem)Under Assumption (P), suppose that one of the followingconditions hold:

• Z is a P–local martingale;

• P satisfies EP [Zζ1ζ<∞] = 0.

Then there exist τ and Q such that (1) holds. If Z is a P–localmartingale, then we can use τ = τZ . Moreover:

(I) The following conditions are equivalent:(a) the set τ < ζ is Q|Fτ−–negligible;(b) there is a unique Follmer countably additive measure for (Z , τ).

(II) If τ is a stopping time such that the pair (Q, τ) alsosatisfies (1), then Q[τ = τ ] = 1.

(III) The following statement in (c) always implies the one in (d).The reverse implication holds if E is uncountable.(c) The P–supermartingale Z is a P–local martingale and the set

τZ < ζ is QZ |FτZ−

–negligible;

(d) there is a unique Follmer pair for Z .

–negligible;

Existence and uniqueness: c.a. case (proof)

Rough outline:

• Existence:• Motivation: Interpret Z as the reciprocal of a local martingale

that can jump to zero under Q. (e.g. Zt = e−t).• Use multiplicative decomposition: Z = MD.• Construct measure for local martingale M (use an extension

theorem).• Interpret D as survival function (proceed as when constructing

killed diffusions).

• Uniqueness:• Read lots of papers and don’t give up.

Rough outline:

killed diffusions).

Rough outline:

killed diffusions).

Rough outline:

killed diffusions).

Rough outline:

killed diffusions).

Rough outline:

killed diffusions).

Rough outline:

killed diffusions).

Existence and non-uniqueness: f.a. case

Assumption B: (Ω,F , (Ft)t≥0,P) supports a Brownian motionW = (Wt)t≥0.

Under Assumption (B), there exists a Follmer finitely additivemeasure for Z . The Follmer finitely additive measure is neverunique.

Existence and non-uniqueness: f.a. case

Assumption B: (Ω,F , (Ft)t≥0,P) supports a Brownian motionW = (Wt)t≥0.

Under Assumption (B), there exists a Follmer finitely additivemeasure for Z . The Follmer finitely additive measure is neverunique.

Existence and uniqueness: f.a. case (proof)

Rough outline:

• Existence:• Prove: There exists a family (L(i,j,k,m,n))i,j,k,m,n∈N of

u.i. nonnegative P–martingales with EP [L(i,j,k,m,n)0 ] = 1, s.t.

limi↑∞

limj↑∞

limk↑∞

limm↑∞

limn↑∞

L(i,j,k,m,n)ρ = Zρ

for all finite s.t. ρ. (Use suitable “suicide strategies” andlocalize.)

• Q is then a cluster point in the family of the (c.a.) probability

measures generated by L(i,j,k,m,n)∞ .

• Uniqueness: Consider two cases:

(A) P[Z∞ > 0] > 0 (destroy some mass of Qr at time ∞)(B) P[ρ <∞] = 1, where ρ = inft ≥ 0 : Zt ≤ 1/2. (play around

with Qs after ρ — “supported on the set where L(i,j,k,m,n)ρ is

large”)

Rough outline:

limi↑∞

limj↑∞

limk↑∞

limm↑∞

limn↑∞

large”)

Rough outline:

limi↑∞

limj↑∞

limk↑∞

limm↑∞

limn↑∞

large”)

Rough outline:

limi↑∞

limj↑∞

limk↑∞

limm↑∞

limn↑∞

large”)

Rough outline:

limi↑∞

limj↑∞

limk↑∞

limm↑∞

limn↑∞

large”)

Rough outline:

limi↑∞

limj↑∞

limk↑∞

limm↑∞

limn↑∞

large”)

Definition: local martingale

A stochastic process Z is a local martingale if there exists asequence of stopping times (τn) with limn↑∞ τn =∞ such thatZ τn := (Z (t ∧ τn)) is a martingale.

Example (Stochastic exponential)

For a continuous local martingale Y , the process

E(Y )(·) = exp

(Y (·)− 1

2[Y ,Y ](·)

)is a nonnegative local martingale.

Definition: local martingale

A stochastic process Z is a local martingale if there exists asequence of stopping times (τn) with limn↑∞ τn =∞ such thatZ τn := (Z (t ∧ τn)) is a martingale.

Example (Stochastic exponential)

For a continuous local martingale Y , the process

E(Y )(·) = exp

(Y (·)− 1

2[Y ,Y ](·)

)is a nonnegative local martingale.

Stochastic exponentials

• Consider an arbitrary local martingale M with jump measureµ (having compensator ν).

• Thus,

M = M0 + Mc + x ∗ (µ− ν).

• Assume that M0 = 1 and jumps of M are bounded frombelow by −1.

• Define the stochastic exponential of M by

E(M)t = exp

(Mt −

2[M,M]ct − (x − log(1 + x)) ∗ µt

• Then E(M) is nonnegative local martingale with E(M)0 = 1since it is the unique strong solution of the SDE

Z = 1 + Z− ·M.

• Thus,

M = M0 + Mc + x ∗ (µ− ν).

E(M)t = exp

(Mt −

2[M,M]ct − (x − log(1 + x)) ∗ µt

Z = 1 + Z− ·M.

• Thus,

M = M0 + Mc + x ∗ (µ− ν).

E(M)t = exp

(Mt −

2[M,M]ct − (x − log(1 + x)) ∗ µt

Z = 1 + Z− ·M.

• Thus,

M = M0 + Mc + x ∗ (µ− ν).

E(M)t = exp

(Mt −

2[M,M]ct − (x − log(1 + x)) ∗ µt

Z = 1 + Z− ·M.

• Thus,

M = M0 + Mc + x ∗ (µ− ν).

E(M)t = exp

(Mt −

2[M,M]ct − (x − log(1 + x)) ∗ µt

Z = 1 + Z− ·M.

Novikov-Kazamaki-type conditions

• Provide sufficient conditions that E(M) is a uniformlyintegrable martingale.

• For sake of simplicity, assume that jumps of M are strictlylarger than −1.

• Conditions were first provided by Lepingle and Memin.

• E.g.:supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

where, for some a ∈ R \ 0,

A = (1− a)M +

(a− 1

)[M,M]c +

(log(1 + x)− (1− a)x2 + x

)∗ µ.

• Their proof is “mysterieuse”.

• Here, canonical proof including some slight generalizations.

• E.g.:supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

A = (1− a)M +

(a− 1

)[M,M]c +

(log(1 + x)− (1− a)x2 + x

)∗ µ.

• E.g.:supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

A = (1− a)M +

(a− 1

)[M,M]c +

(log(1 + x)− (1− a)x2 + x

)∗ µ.

• E.g.:supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

A = (1− a)M +

(a− 1

)[M,M]c +

(log(1 + x)− (1− a)x2 + x

)∗ µ.

• E.g.:supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

A = (1− a)M +

(a− 1

)[M,M]c +

(log(1 + x)− (1− a)x2 + x

)∗ µ.

• E.g.:supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

A = (1− a)M +

(a− 1

)[M,M]c +

(log(1 + x)− (1− a)x2 + x

)∗ µ.

Idea of a novel proof

• Recall Follmer countably additive measure Q along withstopping time τ (which can be interpreted as explosion time):

Q[ρ < τ ] = EP [E(M)ρ1ρ<∞]

• Thus: E(M) is a martingale if and only if E(M) does notexplode under a new measure Q.

• The new measure Q corresponds to the “Radon-Nikodymderivative” E(M).

• Thus, it is sufficient to check that Novikov-Kazamakiconditions guarantee that the Q–local martingale 1/E(M)does not tend to zero under Q.

Q[ρ < τ ] = EP [E(M)ρ1ρ<∞]

Convergence of local martingale

Theorem (Limiting behaviour of a local martingale)

Let τ be a positive predictable time, and X a local martingale on[0, τ) with ∆X ≥ −1. Denote the jump measure of X by µ, and letν be its compensator. The following equalities hold almost surely:

limt↑τ

Xt exists in R

lim inft↑τ

(Xt −

2[X ,X ]ct − (x − log(1 + x))1x 6=−1 ∗ µt

)> −∞

= [X ,X ]τ− <∞

⋂lim sup

t↑τXt > −∞

[X ,X ]cτ− + (|x | ∧ x2) ∗ ντ− <∞

lim inft↑τ

Xt > −∞.

Convergence of local martingale

Theorem (Limiting behaviour of a local martingale)

Let τ be a positive predictable time, and X a local martingale on[0, τ) with ∆X ≥ −1. Denote the jump measure of X by µ, and letν be its compensator. The following equalities hold almost surely:

limt↑τ

Xt exists in R

lim inft↑τ

(Xt −

2[X ,X ]ct − (x − log(1 + x))1x 6=−1 ∗ µt

)> −∞

= [X ,X ]τ− <∞

⋂lim sup

t↑τXt > −∞

[X ,X ]cτ− + (|x | ∧ x2) ∗ ντ− <∞

lim inft↑τ

Xt > −∞.

Novikov-Kazamaki conditions: outline of the proof

• Starting from the Novikov-Kazamaki condition

supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

change the measure from P to Q via “Radon-Nikodymderivative” E(M).

• Observe that 1/E(M) is a Q–local martingale, in particular, astochastic exponential E(N) of some Q–local martingale N.

• Assume that E(N) tends to zero.

• By means of the last theorem, obtain a contradiction to theNovikov-Kazamaki condition.

• Conclude that E(M) is a uniformly integrable P–martingale.

supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

supσ∈T

[exp(Aσ)1E(M)σ>0

]<∞,

Distribution of explosion time: setup

• Interval I = (`, r) =⋃

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

dX (t) = s(X (t)) (dW (t) + b(X (t))dt) , X (0) = ξ ∈ I

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

∣∣∣∣)dy <∞, for every compact set K ⊂ I .

• Engelbert & Schmidt prove the existence of a weak solutionX , unique in the sense of the probability distribution, and

• defined up until the “explosion time”

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

• Due to uniqueness, X is Markovian.

• “Local mean/variance ratio” function f(·) := b(·)s(·) = b(·)s(·)

s2(·) .

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

s2(·) .

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

s2(·) .

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

s2(·) .

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

s2(·) .

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

s2(·) .

n∈N(`n, rn) with −∞ ≤ ` < r ≤ ∞.

• Consider

• Assume that∫K

s2(y)+

∣∣∣∣b(y)

S := inf t ≥ 0 : X (t) /∈ (`, r) = limn↑∞

inf t ≥ 0 : X (t) /∈ (`n, rn) .

s2(·) .

Feller’s test of explosions

• “Feller’s test” function:

v(x) :=

(∫ y

zf(u)du

s2(z)dz

• Feller’s test states that P[S =∞] = 1 holds if and only ifv(`+) = v(r−) =∞ holds.

• Alternative characterizations?

• Yields directly that X leaves (`n, rn) almost surely.

v(x) :=

(∫ y

zf(u)du

s2(z)dz

v(x) :=

(∫ y

zf(u)du

s2(z)dz

v(x) :=

(∫ y

zf(u)du

s2(z)dz

Feller’s test: special cases

v(x) :=

(∫ y

zf(u)du

s2(z)dz

By simplifying v(·) via Tonelli, criterion of explosions is sometimeseasier to check:• If s(·) is differentiable and b(·) = s′(·)/2:

v(x) =

s(z)dz

(∫ x

s(z)dz

• If I = (0,∞) and b(·) = 0:

v(x) =

x − z

s2(z)dz .

Therefore, P[S =∞] = 1 holds if and only if∫ 1

s2(z)dz =∞.

v(x) :=

(∫ y

zf(u)du

s2(z)dz

v(x) =

s(z)dz

(∫ x

s(z)dz

• If I = (0,∞) and b(·) = 0:

v(x) =

x − z

s2(z)dz .

s2(z)dz =∞.

v(x) :=

(∫ y

zf(u)du

s2(z)dz

v(x) =

s(z)dz

(∫ x

s(z)dz

• If I = (0,∞) and b(·) = 0:

v(x) =

x − z

s2(z)dz .

s2(z)dz =∞.

A diffusion in natural scale

X o(·) = ξ +

∫ ·0s(X o(t))dW o(t)

• Again, a weak solution, unique in the sense of the probabilitydistribution, exists up until an explosion time So .

• Po [So =∞] = 1 holds if ` = −∞ and r =∞.

Transformation of probability

Theorem (Generalized Girsanov theorem (a special case))

Assume that the mean/variance ratio f(·) is locallysquare-integrable on I . For any given T ∈ (0,∞) and any Borelset ∆ ∈ BT (C ([0,∞))), we have

P [X (·) ∈ ∆, S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1X o(·)∈∆,So>T

where E(·) denotes the stochastic exponential.

• Appears in McKean (1969) under stronger assumptions.• Local square-integrability of f(·) and occupation time formula

yield that ∫ T

0b2(X o(t))dt <∞

on S > T.• Recall that E(...) is a local martingale, here on [0, So).

P [X (·) ∈ ∆, S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1X o(·)∈∆,So>T

yield that ∫ T

0b2(X o(t))dt <∞

P [X (·) ∈ ∆, S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1X o(·)∈∆,So>T

yield that ∫ T

0b2(X o(t))dt <∞

P [X (·) ∈ ∆, S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1X o(·)∈∆,So>T

yield that ∫ T

0b2(X o(t))dt <∞

Feynman-Kac representation

• With ∆ = C ([0,T ]):

P[S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1So>T

• Assume for the moment that f(·) is continuously differentiable.

• Denote by F (·) =∫ ·c f(x)dx its antiderivative.

• Then P[S > T ] =

exp(−F (ξ)) · Eo

(F (X o(T ))−

0V (X o(t))dt

)· 1So>T

where V (x) := 12

(b2(x) + f′(x)s2(x)

• Thus, the distribution of the explosion time is determinedcompletely by the joint distributions of X o(T ) and∫ T

0 V (X o(t))dt on So > T, for all T ∈ (0,∞).

• With ∆ = C ([0,T ]):

P[S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1So>T

• Then P[S > T ] =

(F (X o(T ))−

0V (X o(t))dt

)· 1So>T

where V (x) := 12

(b2(x) + f′(x)s2(x)

• With ∆ = C ([0,T ]):

P[S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1So>T

• Then P[S > T ] =

(F (X o(T ))−

0V (X o(t))dt

)· 1So>T

where V (x) := 12

(b2(x) + f′(x)s2(x)

• With ∆ = C ([0,T ]):

P[S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1So>T

• Then P[S > T ] =

(F (X o(T ))−

0V (X o(t))dt

)· 1So>T

where V (x) := 12

(b2(x) + f′(x)s2(x)

• With ∆ = C ([0,T ]):

P[S > T ] = Eo

[E(∫ T

0b (X o(t))dW o(t)

)1So>T

• Then P[S > T ] =

(F (X o(T ))−

0V (X o(t))dt

)· 1So>T

where V (x) := 12

(b2(x) + f′(x)s2(x)

Transformation of space

• Based on ideas of Lamperti, Doss, Sussmann.

• Assume that s(·) is continuously differentiable and s(·) > 0.

• Consider the function

h(x) =

s(z)dz .

• Then X explodes from I = (`, r) exactly when Y = h(X )explodes from I = (h(`), h(r)).

• Thus, X and Y have the same explosion time S .

• If b(·) = ν + s′(·)/2, then Y takes the simple form

Y (t) = h(ξ) + νt + W (t).

• Distribution of S for BM with drift is well known.

h(x) =

s(z)dz .

Y (t) = h(ξ) + νt + W (t).

h(x) =

s(z)dz .

Y (t) = h(ξ) + νt + W (t).

h(x) =

s(z)dz .

Y (t) = h(ξ) + νt + W (t).

h(x) =

s(z)dz .

Y (t) = h(ξ) + νt + W (t).

h(x) =

s(z)dz .

Y (t) = h(ξ) + νt + W (t).

h(x) =

s(z)dz .

Y (t) = h(ξ) + νt + W (t).

Analytic properties

Define the function U : (0,∞)× I → [0, 1] via

U(T , ξ) := Pξ[S > T ].

TheoremThe function U(·, ·) is jointly continuous in [0,∞)× I .

Analytic properties

Define the function U : (0,∞)× I → [0, 1] via

U(T , ξ) := Pξ[S > T ].

TheoremThe function U(·, ·) is jointly continuous in [0,∞)× I .

Parabolic PDE

U(T , ξ) := Pξ[S > T ]

TheoremAssume that the functions s(·) and b(·) are locally uniformlyHolder-continuous on I . Then the function U(·, ·) is of classC([0,∞)× I ) ∩ C1,2((0,∞)× I ) and satisfies the Cauchy problem

∂U∂τ

(τ, x) =s2(x)

∂2U∂x2

(τ, x) + b(x)s(x)∂U∂x

(τ, x)

U(0, x) = 1.

Moreover, the function U(·, ·) is dominated by every nonnegativeclassical supersolution of that Cauchy problem.

Parabolic PDE

U(T , ξ) := Pξ[S > T ]

TheoremAssume that the functions s(·) and b(·) are locally uniformlyHolder-continuous on I . Then the function U(·, ·) is of classC([0,∞)× I ) ∩ C1,2((0,∞)× I ) and satisfies the Cauchy problem

∂U∂τ

(τ, x) =s2(x)

∂2U∂x2

(τ, x) + b(x)s(x)∂U∂x

(τ, x)

U(0, x) = 1.

Moreover, the function U(·, ·) is dominated by every nonnegativeclassical supersolution of that Cauchy problem.

Second-order ODE

Consider the Laplace transform or “resolvent” of the functionU(·, ξ):

Uλ(ξ) =

∫ ∞0

exp(−λT )U(T , ξ)dT =1

λ(1− Eξ [exp(−λS)]) .

TheoremIf the functions b(·) and s(·) are continuous on I , then the functionUλ(ξ) is of class C2(I ) and satisfies the second-order ordinarydifferential equation

2u′′(x) + b(x)s(x)u′(x)− λu(x) + 1 = 0, x ∈ I .

Moreover, the function Uλ(·) is dominated by every nonnegativeclassical supersolution of this second-order ODE.

Second-order ODE

Uλ(ξ) =

∫ ∞0

λ(1− Eξ [exp(−λS)]) .

2u′′(x) + b(x)s(x)u′(x)− λu(x) + 1 = 0, x ∈ I .

Second-order ODE

Uλ(ξ) =

∫ ∞0

λ(1− Eξ [exp(−λS)]) .

2u′′(x) + b(x)s(x)u′(x)− λu(x) + 1 = 0, x ∈ I .

Equivalent formulations for the Feller test

Theorem (Equivalence of the following conditions)

(i) P[S =∞] = 1.

(ii) v(`+) = v(r−) =∞ for the “Feller test” function v .

If f(·) is locally square-integrable, (i)-(ii) are equivalent to:

(iii) the truncated exponential Po-supermartingale

E(∫ T

0 b(X o(t))dW o(t))· 1So>T is a Po-martingale.

If s(·) and b(·) are continuous on I , (i)–(iii) are equivalent to:

(iv) the smallest nonnegative classical solution of the second-orderdifferential equation above is u(·) ≡ 1/λ.

If s(·) and b(·) are locally uniformly Holder-cont., (i)–(iv) a.e.to:

(v) the smallest nonnegative classical solution of the Cauchyproblem above is U(·, ·) ≡ 1.

Circular proof??

(i) P[S =∞] = 1.

E(∫ T

Circular proof??

(i) P[S =∞] = 1.

E(∫ T

Circular proof??

(i) P[S =∞] = 1.

E(∫ T

Circular proof??

(i) P[S =∞] = 1.

E(∫ T

Circular proof??

(i) P[S =∞] = 1.

E(∫ T

Circular proof??

(i) P[S =∞] = 1.

E(∫ T

Circular proof??

Example (slightly cheating)With I = (0,∞), κ ∈ [1/2,∞), consider

X (·) = ξ −∫ ·

0(X (t))2dW (t) + κ

∫ ·0

(X (t))3dt.

• Corresponds to s(x) = −x2 and b(x) = −κx .• Thus, by the Feynman-Kac representation,

P[S > T ] = Eo

[(X o(T )

)κexp

(−κ

2 − κ2

0(X o(t))2dt

Tξ−ν exp

2T ξ2

)∫ ∞0

x1−ν exp

(−x2

where ν := κ− 1/2.• This is the smallest nonnegative solution of

∂U∂τ

(τ, x) =x4

∂2U∂x2

(τ, x) + κx3∂U∂x

(τ, x)

subject to U(0+, x) ≡ 1.

X (·) = ξ −∫ ·

0(X (t))2dW (t) + κ

∫ ·0

(X (t))3dt.

P[S > T ] = Eo

[(X o(T )

)κexp

(−κ

2 − κ2

0(X o(t))2dt

Tξ−ν exp

2T ξ2

)∫ ∞0

x1−ν exp

(−x2

∂U∂τ

(τ, x) =x4

∂2U∂x2

(τ, x) + κx3∂U∂x

(τ, x)

X (·) = ξ −∫ ·

0(X (t))2dW (t) + κ

∫ ·0

(X (t))3dt.

P[S > T ] = Eo

[(X o(T )

)κexp

(−κ

2 − κ2

0(X o(t))2dt

Tξ−ν exp

2T ξ2

)∫ ∞0

x1−ν exp

(−x2

∂U∂τ

(τ, x) =x4

∂2U∂x2

(τ, x) + κx3∂U∂x

(τ, x)

X (·) = ξ −∫ ·

0(X (t))2dW (t) + κ

∫ ·0

(X (t))3dt.

P[S > T ] = Eo

[(X o(T )

)κexp

(−κ

2 − κ2

0(X o(t))2dt

Tξ−ν exp

2T ξ2

)∫ ∞0

x1−ν exp

(−x2

∂U∂τ

(τ, x) =x4

∂2U∂x2

(τ, x) + κx3∂U∂x

(τ, x)

Special case of exampleSet κ = 1:

X (·) = ξ −∫ ·

0(X (t))2dW (t) +

∫ ·0

(X (t))3dt.

• Here:

P[S > T ] = 2

∫ 1/(ξ√T )

1√2π

(− r2

• Observe that

dX (t) = s(X (t))

(dW (t) +

2s′(X (t))dt

), X (0) = ξ.

• Thus, we can express X pathwise as

X (t) =1

W (t) + (1/ξ), 0 ≤ t < S .

X (·) = ξ −∫ ·

0(X (t))2dW (t) +

∫ ·0

(X (t))3dt.

• Here:

P[S > T ] = 2

∫ 1/(ξ√T )

1√2π

(− r2

• Observe that

dX (t) = s(X (t))

(dW (t) +

2s′(X (t))dt

), X (0) = ξ.

X (t) =1

W (t) + (1/ξ), 0 ≤ t < S .

X (·) = ξ −∫ ·

0(X (t))2dW (t) +

∫ ·0

(X (t))3dt.

• Here:

P[S > T ] = 2

∫ 1/(ξ√T )

1√2π

(− r2

• Observe that

dX (t) = s(X (t))

(dW (t) +

2s′(X (t))dt

), X (0) = ξ.

X (t) =1

W (t) + (1/ξ), 0 ≤ t < S .

X (·) = ξ −∫ ·

0(X (t))2dW (t) +

∫ ·0

(X (t))3dt.

• Here:

P[S > T ] = 2

∫ 1/(ξ√T )

1√2π

(− r2

• Observe that

dX (t) = s(X (t))

(dW (t) +

2s′(X (t))dt

), X (0) = ξ.

X (t) =1

W (t) + (1/ξ), 0 ≤ t < S .

Thank you!

Supermartingales as Radon-Nikodym densities, Novikov's and ... · Overview Supermartingales as RN...

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