Let h(t, x) be defined as in the statement of the above proposition. Then h is a solution of the Kolmogoro ff backward equation

Characterization of Markov processes and reciprocal classes

Lemma 7.52. Let h(t, x) be defined as in the statement of the above proposition. Then h is a solution of the Kolmogoro ff backward equation

· · ·`(tm, x + q1+ · · · + qm−1, qm)e⁻

[tm−1,tm]`(s,x+q¯ 1+...qm−1)ds

e⁻

[tm,1]`(s,x+q¯ 1+···+qm)ds

1^{t1<···<tm}dt₁· · ·dt_m,

for any x ∈ R^d, {(t₁, q1), . . . , (tm, qm)} ∈∆I×Qand m ∈ N.

7.3.2. Comparison of the reciprocal classes of nice jump processes.

The next proposition generalizes the result of Lemma 7.16 to h-transforms of nice jump processes.

Proposition 7.50. Let P_`be the law of a nice jump process and h : R^d→ R+such that E_`(h(X₁))= 1. Then the h-transform hP_`is a Markov jump process on J(I, Q) with intensity

(7.51) k(t, Xt−, q)dtΛ(dq) = `(t, Xt−, q)h(t, Xt−+ q)

h(t, Xt−) dtΛ(dq), where h(t, Xt) := E_`(h(X₁)|Xt).

Proof. Clearly the h-transform hP` has the Markov property, see also Lemma 4.4. Define the space-time harmonic function h(t, y) = E` h(X₁)|X_t= y and put Gt := h(t, Xt). Then h(t, y) is differentiable in time and solution of a Kolmogoroff backward equation.

Lemma 7.52. Let h(t, x) be defined as in the statement of the above proposition. Then h is a solution of the Kolmogoroff backward equation

(7.53)

∂th(t, x) = −Z

h(t, x + q) − h(t, x) `(t, x, q)Λ(dq), holds P_`(X_t∈dx)-a.s. for any t ∈ I, in particular h(., x) ∈ C¹(I).

Proof. We omit the proof, since it works in the same way as the proof of Lemma 6.55 in the unit jump case. Just use Remarks 7.5 and 7.48 on the explicit distribution of P_`.

By regularity of h we can apply the It ˆo formula h(t, Xt)= h(s, Xs)+

[s,t]∂th(r, Xr−)dr+ Z

[s,t]×Qh(r, Xr−+ q) − h(r, Xr−) N^X(drdq).

But since the Kolmogoroff backward equation holds, we deduce that Gt= h(t, Xt) is solution of the exponential stochastic integral equation:

G_t= 1 +Z

[0,t]×QG_s− h(s, Xs−+ q) h(s, Xs−) − 1

(N^X(dsdq) −`(s, Xs−, q)dsΛ(dq)), which has the explicit solution

Gt= exp − Z

[0,t]×Q

h(s, Xs−+ q) h(s, Xs−) − 1

`(s, Xs−, q)dsΛ(dq)

!^η

(Q)

i=1

h(T_i, XTi−+ Vi) h(T_i, XTi−) . We just have to multiply the densities G^k₁P = G1G^`₁P, which gives

G₁G^`₁= exp − Z

I×Q

`(s, Xs−, q)h(s, Xs−+ q) h(s, Xs−) − 1

dsΛ(dq)

! η

i=1

`(Ti, XTi−, Vi)h(T_i, XT_i−+ Vi) h(T_i, XTi−) ,

and invoke the Girsanov theorem to identify

k(s, Xs−, q)dsΛ(dq) = `(s, Xs−, q)h(s, Xs−+ q)

h(s, Xs−) dsΛ(dq), P_`-a.s.

as intensity of hP_`.

The fact, that h(t, x) is a solution of the Kolmogoroff backward equation is now used to compute reciprocal invariants for the law P_`. Our main result in this paragraph is the fact, that these invariants are indeed characteristics of R(P_`).

Theorem 7.54. Let P_`and Pkbe the laws of two nice jump processes. Then R(Pk)= R(P_`) if and only if

(i) log k − log` is the “gradient of a potential”: For every x ∈ R^d there exists a function ψ : I × R^d → R such that

log k(t, y, q) − log `(t, y, q) = ψ(t, y + q) − ψ(t, y)

holds dt ⊗ P^x_`(X_t∈dy)-a.e. for all q ∈ Q and such that e^ψ(t,X^t⁾∈ L¹(P^x_`) for all t ∈ I.

(ii) the “harmonic” invariants coincide: ∀q ∈ Q,Ξ^q_`(t, y) = Ξ^q_k(t, y), with Ξ^q_`(t, y) = ∂tlog`(t, y, q) +

`(t, y + q, ¯q) − `(t, y, ¯q) Λ(d¯q).

Proof. Assume first, that R(Pk) = R(P_`). Fix x ∈ R^d. We prove (i) and (ii) under P^x_`, the general result for (ii) follows by mixing over the initial condition. There exists h : R^d→ R+

that is everywhere positive such that P^x_k = h(X1)P^x_` is an h-transform, see Remark 4.18. We know that relation (7.51) holds between the intensities k and` with h(t, y) = E^x_`(h(X1)|Xt= y), which by Lemma 7.52 is a solution of the Kolmogoroff backward equation. This implies condition (i) with potential log h(t, y). Moreover for q, ¯q ∈ Q this implies

∂th(t, y + q)

h(t, y + q) = −Z

h(t, y + q + ¯q) − h(t, y + q) h(t, y + q)

`(t, y + q, ¯q)Λ(d¯q) and

∂th(t, y)

h(t, y) = − Z

h(t, y + ¯q) − h(t, y) h(t, y)

`(t, y, ¯q)Λ(d¯q).

We take the difference of these two equations and insert h(t, y + ¯q)

h(t, y) = k(t, y, ¯q)

`(t, y, ¯q), andh(t, y + q + ¯q)

h(t, y + q) = k(t, y + q, ¯q)

`(t, y + q, ¯q) to get the “harmonic” invariant, condition (ii).

For the converse assume that (i) and (ii) hold. We will show that P^x_k is an h-transform of P^x_` for arbitrary x ∈ R^d. Define

h(t, y) := e^ψ(t,y)c(t) with c(t)= ce⁻^R^[0,t]×Q(k(s,x,q)−`(s,x,q))dsΛ(dq),

where ψ is the potential from condition (i) and c is a normalization constant such that E^x_`(h(1, X1))= 1. The normalization constant exists by the boundedness of ` and k and the integrability condition e^ψ(t,X^t⁾ ∈ L¹(P_`) for all t ∈ I. Clearly the relation (7.51) holds since log h is a potential for log k(t, y, q) − log `(t, y, q) too since c(t) does not depend on y. We

insert this into the equality of the harmonic invariant, condition (ii), which results in

∂th(t, y + q) h(t, y + q) +

h(t, y + q + ¯q) − h(t, y + q) h(t, y + q)

`(t, y + q, ¯q)Λ(d¯q)

= ∂th(t, y) h(t, y) +

h(t, y + ¯q) − h(t, y) h(t, y)

`(t, y, ¯q)Λ(d¯q) (7.55)

for all q ∈ Q and y ∈ Qx, the support ofδ^{x}∗P^∞

m=0(Λ(Q))⁻^mΛ^∗^m

. In particular 0 = ∂tlog h(t, x) +

h(t, x + ¯q) h(t, x) − 1

`(t, x, ¯q)Λ(d¯q)

= − Z

(k(s, x, ¯q) − `(s, x, ¯q))Λ(d¯q) + Z

(k(s, x, ¯q) − `(s, x, ¯q))Λ(d¯q).

Since for every y ∈ Q_x there exist q₁, . . . qm ∈ Q such that y= x + q1+ · · · + qm we invoke equation (7.55) and deduce that h(t, y) is a solution of the Kolmogoroff backward equation with respect to`. Therefore h(t, Xt−) is a martingale under P^x_` and we may apply the first

part of the proof to identify P^x_k = hP^x_`.

Example 7.56. Let Q = {1, 2} and P_` the law of a compound Poisson process with intensity `, where`(1), `(2) > 0. Then Ξ^q_`(t, y) = 0, and a nice jump process Pk is in the reciprocal class R(P`) if

∂tlog k(t, y, q) + X

¯q=1,2

k(t, y + q, ¯q) − k(t, y, ¯q) = 0, for q = 1, 2 and (t, y) ∈ I × R^d.

and there exists aψ : R^d → R such that

k(t, y, q) = `(q)eψ(t,y+q)−ψ(t,y), for all t ∈ I, q ∈ {1, 2}, y ∈ R^d. If we fit the potentialψ into the harmonic invariant, we get the condition

0= ∂t ψ(t, y + q) − ψ(t, y) + X

¯q=1,2

`(¯q)eψ(t,y+q+¯q)−ψ(t,y+q)−`(¯q)eψ(t,y+¯q)−ψ(t,y)

⇔ ∂tψ(t, y + q) + Z

eψ(t,y+q+¯q)−ψ(t,y+q)`(¯q)Λ(d¯q) = ∂tψ(t, y) + Z

eψ(t,y+¯q)−ψ(t,y)`(¯q)Λ(d¯q), which reduces to the statement, that h(t, y) = log ψ(t, y) is a space-time harmonic function.

A “rotational” invariant is hidden in the “gradient of a potential” condition (i) of the above theorem. It is easy to check that if (i) holds, then the “rotational invariants”

Ψ^q,¯q_` (t, Xt−)= Ψ^q,¯q_k (t, Xt−) coincide, where

(7.57) Ψ^q,¯q_` (t, Xt−) := log `(t, Xt−+ q, ¯q) − log `(t, Xt−, ¯q) − (log `(t, Xt−+ ¯q, q) − log `(t, Xt−, q)).

This invariant is called “rotational invariant”, since if `(y) and k(y) only depend on the position of the process, the condition reduces to

`(y + q) − `(y) − (`(y + ¯q) − `(y)) = k(y + q) − k(y) − (k(y + ¯q) − k(y)),

and thus resembles the rotational condition (ii) of the continuous case presented in Theorem 5.26: On both sides we have the difference of discrete derivative operators.

The equivalence of this identity of rotationals to the gradient of a potential condition (i) of the above Theorem 7.54 depends strongly on the algebraic properties of the set of jump-sizes Q. See e.g. Borwein, Lewis [BL92], who examine the gradient of a potential condition if (Qx, +) has a group structure. Both conditions are meaningless if Q only contains one

element, therefore only a “harmonic” invariant appeared in the characterization of the reciprocal class of nice unit jump processes in Theorem 6.58. Let us present a simple example that implies that the rotational invariant is weaker than the gradient of a potential condition.

Example 7.58. Let Q ⊂ R^d∗ be any finite set of jump-sizes and`(t, x, q)dtΛ(dq) the intensity of a nice jump process P`. Let k(t, x, q) := `(t, x, q)e^φ(q)−x·qfor some functionφ : Q → R. It is easy to that this result is in accordance with the statement of Example 7.18, that the reciprocal classes of the compound Poisson processes Pk and P`coincide if and only if` ≡ k.

In the next proposition we present a duality formula satisfied by any element of the reciprocal class R(P_`) of a nice jump process. Note that neither the “rotational” invariant nor the “gradient of potential” condition appear in the formula.

Proposition 7.59. Let P_`be the law of a nice jump process, Q be any element of the reciprocal class of P`such thatη ∈ L¹(Q). Then the duality formula

Proof. The proof is similar to the proof of Lemma 6.65 in the unit jump setting. Using the product rule satisfied by the derivative operator and the duality formula for a compound Poisson process with intensityΛ we derive

E_` F(X) if log G^`₁is differentiable in direction ¯u. By (7.47) we have to differentiate

log G^`₁ = number of jumps, thus log G^`₁is differentiable and the derivative is

D_u_¯log G^`₁ = −

Thus the duality formula (7.60) holds under P_`. We extend this to any Q in R(P_`) with η ∈ L¹(Q) as usually, see also the proof of Theorem 6.69. Take bounded functions φ, ψ : R^d → R, then Du¯φ(X0) = 0 and Du¯ψ(X1) = 0. The duality formula still holds for the bridges of P_`since

E_` φ(X0)ψ(X1)F(X) Z

I×Q

u(s, q)N¯ ^X(dsdq)

= E` (φ(X0)ψ(X1)D_u_¯F(X))

−E` φ(X0)ψ(X1)F(X) Z

I×Q

Ξ^q_`(s, Xs−)h ¯u(., q)isN^X(dsdq)

! ,

But Q can be decomposed into a mixture of the bridges of P_` by Definition 4.12. Thus the

duality formula is also satisfied by Q.

The duality formula (7.35) does not contain sufficient information to characterize the reciprocal class of P_`: We already remarked, that neither a gradient of a potential condition, nor a rotational invariant appear in the formula. An example to refute the idea of a characterization of the reciprocal class R(P_`) by the duality formula (7.60) was given in Example 7.39, where two compound Poisson processes with different reciprocal classes where solution of the same duality formula.

Conclusion

In this thesis we studied the characterization of classes of stochastic processes by du-ality formulae in various settings. The first part was dedicated to the characterization of processes with independent increments by a duality formula that is well-established in Malliavin’s calculus. The second part was devoted to the characterization of the reciprocal classes of various processes by duality formulae. In particular we were able to present new results concerning the reciprocal classes of pure jump processes. Let us now point to various open questions related to this work.

Our first result was a characterization of infinitely divisible random vectors by an inte-gration by parts formula. Following the ideas of Stein’s calculus, this characterization is a key to the study of approximations of the distribution of infinitely divisible random vectors.

The next step in the derivation of such approximation results would be the computation of a solution to Stein’s equation

g_f(z)(z − b) − A∇g_f(z) − Z

R^d^∗

(g_f(z+ q) − gf(z))qL(dq)= f (z) − E ( f (Z)).

The distance between the distribution of a random vector V to an infinitely divisible random vector Z is then computed using



E

g_f(V)(V − b) − A∇g_f(V) −R

R^d^∗(g_f(V+ q) − gf(V))qL(dq)





< ε

⇒ |E ( f (V)) − E ( f (Z))| < ε, see also Remark 1.4 and the comments after Theorem 3.1.

Next we have shown, that processes with independent increments are the unique càdlàg processes satisfying a specific duality formula. On the one hand this indicates, that ap-proaches similar to Stein’s calculus could be applied in the setting of càdlàg processes, see e.g. Chen, Xia [CX04] for results on Poisson process approximation. On the other hand our characterization result could be used to construct Gibbs-states on the configuration space D(I)^Z^d. See e.g. Dai Pra, Rœlly and Zessin [DPRZ02], who use a duality formula to show that the set of weak solutions of a class of infinite dimensional stochastic differential equations coincides with a set of space-time Gibbs fields for a certain potential.

At the end of the first part of this thesis, we give a new proof of a characterization of infinitely divisible random measures on Polish spaces. Our approach implies that similar characterizations should indeed hold for arbitrary random objects whose distribution is fixed by projections on random vectors that have infinitely divisible distributions, e.g.

infinitely divisible random fields in the sense of Lee [Lee67] and Maruyama [Mar70].

The second part of this thesis begins with a recapitulation of a result by Rœlly and Thieullen concerning the characterization of the reciprocal class of Brownian diffusions by a duality formula, see Theorem 5.34. Our characterization of Brownian diffusions in the class of semimartingales with integrable increments presented in Theorem 5.14 indicates that the hypotheses connected to Rœlly, Thieullens characterization result might be relaxed.

In his original article [Cla90] Clark derived reciprocal invariants for diffusions with non-unit diffusion coefficient, that is solutions of the SDE

dXt= b(t, Xt)dt+ σ(t, Xt)dWt,

for some b ∈ C^∞_b(I × R^d, R^d) andσ ∈ C^∞_b (I × R^d, R^d×d) such thatσ^tσ − εId is potive definite for someε > 0. His invariants permit to distinguish reciprocal classes: We presented his result restricted to Brownian diffusions in Theorem 5.26. A corresponding characterization

of the reciprocal class of diffusions with non-unit diffusion coefficient by a duality formula does not yet exist. It would be interesting to note, if such a duality formula still exhibits the similarities to the deterministic Newton equation presented in Remark 5.72.

We characterized the reciprocal class of nice unit jump processes in the sense of a com-parison of invariants in Theorem 6.58 and by a duality formula in Theorem 6.69. Both results might be extended to the reciprocal classes of Markov unit jump processes that are not nice in the sense that the reference intensity`(t, Xt−)dt is unbounded,` has values in the whole of [0, ∞). The dynamical scope of such processes is larger: A unit jump process must leave a certain state y ∈ R almost surely until a given time t ∈ I, if ε →R

[0,t−ε]`(r, y)dr explodes forε → 0. On the other hand, the unit jump process is forced stay in a certain state y ∈ R almost surely in a given time interval [s, t] ⊂ I if `(r, y) = 0, ∀r ∈ [s, t].

In the last section of this thesis, we characterized the reciprocal class of compound Pois-son processes by a duality formula in Theorem 7.36 under the assumption of incommen-surable jump-sizes. We found a reciprocal invariant for nice jump processes in Theorem 7.54 without using the assumption of incommensurability. The first result indicates, that it should be possible to characterize even nice jump processes with incommensurable jumps by the duality formula (7.60). It is not verisimilar to prove such results for processes with commensurable jumps. In Example 7.39 we have given evidence, that the duality formula (7.35) containing the stochastic derivative of Definition 7.30 is not suited to characterize the reciprocal classes of jump processes with commensurable jumps. Alternative definitions of stochastic derivatives might be a key to find a duality formula that characterizes the reciprocal class of compound Poisson process with commensurable jump-sizes.

Appendix A. Stochastic calculus of pure jump processes

In this section we present a few aspects of the stochastic calculus associated to pure jump semimartingales on the space of c`adl`ag paths. This calculus is the basis of many results in Sections 2, 6 and 7. In particular we state the Girsanov theorem for pure jump processes, which is a key to the proof of duality formulae. Our presentation is based on the introduction to semimartingales by Jacod and Shiryaev in [JS03], the proofs are omitted.

Let X be the canonical process on D(I, R^d). The random jump measure N^X:= X

s:∆Xs,0

δ_{(s,∆X_s)}

on I × R^d∗ is well defined.

Definition A.1. Let P be a probability on D(I, R^d). Then X is called a pure jump process under P if

N^X(I × R^d∗)< ∞ = 1 and P ∃t ∈ I : X_t−X₀, Z

[0,t]×R^d^∗ qN^X(dsdq)

= 0.

The last condition is an equality of processes two up to evanescence. If X is a pure jump process under P, it is also a semimartingale since it has finite variation. The space-time version of It ˆo’s formula has a convenient form.

Theorem A.2. Let X be a pure jump process under P and f : I×R^d → R such that f (., x) ∈ C¹(I) for any x ∈ R^d. Then the Itˆo-formula

f (t, Xt)= f (s, Xs)+ Z

(s,t]∂tf (r, Xr−)dr+ Z

[s,t]×R^d^∗( f (r, Xr−+ q) − f (r, Xr−))N^X(drdq) P-a.s.

holds for any s< t ∈ I.

If f does not depend on time, the It ˆo-formula reduces to the canonical sum f (Xt) − f (Xs)=

(s,t]×R^d^∗ f (Xr−+ q) − f (Xr−) N^X(drdq)= X

s≤r≤t

f (Xr) − f (Xr−). Following [JS03, Theorem II.1.8] there exists a predictable random measure ¯A on I × R^d∗

such that

E Z

I×R^d^∗

u(s, q)N¯ ^X(dsdq)

= E Z

I×R^d^∗

u(s, q) ¯¯ A(dsdq)

for any predictable ¯u : I × R^d^∗ × D(I, R^d) → R₊. This implies that for any predictable u ∈ L¯ ¹( ¯A ⊗ P) the process

(A.3) t 7→ ˜X_t^u^¯ :=Z

[0,t]×R^d^∗ u(s, q)(N¯ ^X(dsdq) − ¯A(dsdq))

is a martingale. It is easy to see, that the solution of the Dol´eans-Dade integral equation Y_t = 1 +Z

[0,t]Y_s−dX_s is given by t 7→ Y_t=Y

s≤t

(1+ ∆Xs).

This is just an application of the It ô-formula with respect to the pure jump process Y. The Doléans-Dade exponential of the compensated process ˜Xû^¯ has the following convenient form

Theorem A.4. Let X be a pure jump process under P. If ˜Xû^¯ is the compensated process defined in (A.3) for someu ∈ L¯ ¹( ¯A ⊗ P), then the Doléans Dade expontial of ˜Xû^¯is given by

Y^u_t^¯ = 1 +R

[0,t]Y^u_s−^¯ d ˜X^u_s^¯, ∀t ∈ I

⇔ Y^u_t^¯ = exp

−R

[0,t]×R^d^∗u(s¯ , q) ¯A(dsdq)

s≤t(1+ ¯u(s, ∆Xs)), ∀t ∈ I.

This follows again with It ˆo’s formula, see [JS03, Theorem I.4.61]. The Dol´eans-Dade exponential is an important tool in the formulation of the Girsanov theorem for pure jump processes.

Theorem A.5. Let X be a pure jump process under P with intensity d ¯A. If Q P with density G and density process Gt = E (G|F_[0,t]), then X is a pure jump process under Q with intensity `d ¯A, where` : I × R^d∗× D(I, R^d) → R₊is predictable and satisfies

E Z

I×R^d^∗

u(s, q)`(s, q)Gs−N^X(dsdq)

= E Z

I×R^d^∗

u(s, q)GsN^X(dsdq)

for any predictableu : I × R¯ ^d∗× D(I, R^d) → R+.

We say that X is a pure jump process without fixed jumps under P if the intensity ¯A is of the form

A(dsdq)¯ = dsAs(dq).

In this case the density process G_tof the preceeding theorem has the following convenient representation.

Theorem A.6. Let X be a pure jump process without fixed jumps under P and with intensity dsA_s(dq). Let Q P with density G and density process Gt = E (G|F_[0,t]) such that X has intensity`(s, q)dsAs(dq) for a predictable function` ∈ L¹( ¯A ⊗ P) under Q. Then G is the Dol´eans-Dade exponential of t 7→R

[0,t](`(s, q) − 1)(N^X(dsdq) − dsA_s(dq)), that is G_t= 1^{Gt>0}exp −

[0,t]×R^d^∗(`(s, q) − 1)dsAs(dq)

s≤t

`(s, ∆Xs).

If X has fixed jumps, the form of G is somewhat more involved, although G is still a Dol´eans-Dade exponential. The following partial converse is one of the main tools in the derivation of duality formulae in Sections 2, 6 and 7.

Theorem A.7. Assume that the predictable function` : I × R^d∗× D(I, R^d) → R+is such that the Dol´eans-Dade exponential

G_t= 1 + Z

[0,t]×R^d^∗(`(s, q) − 1)Gs−(N^X(dsdq) − dsA_s(dq))

is well defined and a martingale. Then X is a pure jump process under the probability G₁P with intensity`(s, q)dsAs(dq).

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o(ε) - small Landau notation, xxii χ - cutoff function, 8

Canonical spaces

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J(I, Q) - pure jump paths, 116 Canonical setup

X - canonical process, xxi

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c(R^d, R^m) - compact support, xxii S_.- smooth and bounded functionals

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P0- initial distribution, 34 P1- final distribution, 34 P01- endpoint distribution, 34 P^x- pinned initial condition, 34 P^x,y- pinned endpoint conditions, 34 hP - h-transform, 36

R(P) - reciprocal class, 34 θ^ε_., π^ε_. - perturbation of...

θ^ε_u- paths, deterministic, 15 θ¯^ε_u_¯- paths, random, 17 π^ε_u- jump-times, 76

π^ε_u_¯- jump-times and -sizes, 128

135

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