**3. Fluctuation Theory**

In the following section, to fully appreciate the similarity (and eventual differences) with the spectrally negative case, the reader is invited to directly compare the exposition of this subsection with that of (Bertoin 1996, sct. VII.1) and (Kyprianou 2006, sct. 8.1).

### *3.1. Laplace Exponent, the Reflected Process, Local Times and Excursions from the Supremum, Supremum Process and Long-Term Behaviour, Exponential Change of Measure*

Since the Poisson process admits exponential moments of all orders, it follows that <sup>E</sup>[*eβXt* ] < ∞ and, in particular, <sup>E</sup>[*eβXt* ] < ∞ for all {*β*, *t*} ⊂ [0, <sup>∞</sup>). Indeed, it may be seen by a direct computation that for *β* ∈ C<sup>→</sup>, *t* ≥ 0, <sup>E</sup>[*eβXt* ] = exp{*tψ*(*β*)}, where *ψ*(*β*) := R(*eβ<sup>x</sup>* − 1)*λ*(*dx*) is the Laplace exponent of *X*. Moreover, *ψ* is continuous (by the DCT) on C→ and analytic in C→ (use the theorems of Cauchy (Rudin 1970, p. 206, 10.13 Cauchy's theorem for triangle), Morera (Rudin 1970, p. 209, 10.17 Morera's theorem) and Fubini).

Next, note that *ψ*(*β*) tends to +∞ as *β* → ∞ over the reals, due to the presence of the atom of *λ* at *h*. Upon restriction to [0, <sup>∞</sup>), *ψ* is strictly convex, as follows first on (0, ∞) by using differentiation under the integral sign and noting that the second derivative is strictly positive, and then extends to [0, ∞) by continuity.

Denote then by Φ(0) the largest root of *ψ*|[0,∞). Indeed, 0 is always a root, and due to strict convexity, if Φ(0) > 0, then 0 and Φ(0) are the only two roots. The two cases occur, according as to whether *ψ*(0+) ≥ 0 or *ψ*(0+) < 0, which is clear. It is less obvious, but nevertheless true, that this right derivative at 0 actually exists, indeed *ψ*(0+) = R *x<sup>λ</sup>*(*dx*) ∈ [−∞, <sup>∞</sup>). This follows from the fact that (*eβ<sup>x</sup>* − 1)/*β* is nonincreasing as *β* ↓ 0 for *x* ∈ R− and hence the monotone convergence applies. Continuing from this, and with a similar justification, one also gets the equality *ψ*(0+) = *x*2*λ*(*dx*) ∈ (0, <sup>+</sup>∞] (where we agree *ψ*(0+) = +∞ if *ψ*(0+) = −<sup>∞</sup>). In any case, *ψ* : [Φ(0), ∞) → [0, ∞) is

continuous and increasing, it is a bijection and we let Φ : [0, ∞) → [Φ(0), ∞) be the inverse bijection, so that *ψ* ◦ Φ = idR+ .

With these preliminaries having been established, our first theorem identifies characteristics of the reflected process, the local time of *X* at the maximum (for a definition of which see e.g., (Kyprianou 2006, p. 140, Definition 6.1)), its inverse, as well as the expected length of excursions and the probability of an infinite excursion therefrom (for definitions of these terms see e.g., (Kyprianou 2006, pp. 140–47); we agree that an excursion (from the maximum) starts immediately after *X* leaves its running maximum and ends immediately after it returns to it; by its length we mean the amount of time between these two time points).

**Theorem 1** (Reflected process; (inverse) local time; excursions)**.** *Let qn* := *λ*({−*nh*})/*λ*(R) *for n* ∈ N *and p* := *<sup>λ</sup>*({*h*})/*λ*(R)*.*


$$-\log \mathbb{E}\left[\exp(-\theta L\_1^{-1})\mathbb{I}\_{\{L\_1^{-1} < +\infty\}}\right] = \theta + \lambda((-\infty,0))\left(1 - \sum\_{k=1}^{\infty} \mathbb{P}(N=k)\left(\frac{\lambda(\mathbb{R})}{\lambda(\mathbb{R}) + \theta}\right)^k\right).$$

*In particular, L*−<sup>1</sup> *has a killing rate of <sup>λ</sup>*((−∞, <sup>0</sup>))*p*<sup>∗</sup>*, Lévy mass <sup>λ</sup>*((−∞, 0))(1 − *p*<sup>∗</sup>) *and its jumps have the probability law on* (0, <sup>+</sup>∞) *given by the length of a generic excursion from the supremum, conditional on it being finite, i.e., that of an independent N-fold sum of independent* Exp(*λ*(R))*-distributed random variables, conditional on N being finite. Moreover, one has, for k* ∈ N*,* P(*N* = *k*) = ∑*kl*=<sup>1</sup> *ql pl*,*k, where the coefficients* (*pl*,*<sup>k</sup>*)<sup>∞</sup>*l*,*k*=<sup>1</sup>*satisfy the initial conditions:*

$$p\_{l,1} = p\delta\_{l1\prime} \quad l \in \mathbb{N};$$

*the recursions:*

$$p\_{l,k+1} = \begin{cases} 0 & \text{if } l = k \text{ or } l > k+1\\ \sum\_{m=1}^{k-1} q\_m p\_{m+1,k} & \text{if } l = 1\\ p^{k+1} & \text{if } l = k+1\\ p p\_{l-1,k} + \sum\_{m=1}^{k-l} q\_m p\_{m+l,k} & \text{if } 1 < l < k \end{cases} \qquad \text{,} \qquad \{l,k\} \subset \mathbb{N};$$

*and pl*,*<sup>k</sup> may be interpreted as the probability of X reaching level* 0 *starting from level* −*lh for the first time on precisely the k-th jump (*{*l*, *k*} ⊂ N*).*

**Proof.** Theorem 1-1 is clear, since, e.g., *Y* transitions away from 0 at the rate at which *X* makes a negative jump; and from *s* ∈ Z+*h* \{0} to 0 at the rate at which *X* jumps up by *s* or more etc.

Theorem 1-2 is standard (Kyprianou 2006, p. 141, Example 6.3 & p. 149, Theorem 6.10).

We next establish Theorem 1-3. Denote, provisionally, by *β* the expected excursion length. Furthermore, let the discrete-time Markov chain *W* (on the state space N0) be endowed with the initial distribution *wj* := *qj* 1−*p* for *j* ∈ N, *w*0 := 0; and transition matrix *P*, given by *P*0*i* = *δ*0*i*, whereas for *i* ≥ 1: *Pij* = *p*, if *j* = *i* − 1; *Pij* = *qj*−*i*, if *j* > *i*; and *Pij* = 0 otherwise (*W* jumps down with probability *p*, up *i* steps with probability *qi*, *i* ≥ 1, until it reaches 0, where it gets stuck). Further let *N* be the first hitting time for *W* of {0}, so that a typical excursion length of *X* is equal in distribution to an independent sum of *N* (possibly infinite) Exp(*λ*(R))-random variables. It is Wald's identity that *β* = (1/*λ*(R))E[*N*]. Then (in the obvious notation, where ∞ indicates the sum is inclusive of ∞), by Fubini: E[*N*] = ∑<sup>∞</sup>*n*=<sup>1</sup> *n* ∑∞*<sup>l</sup>*=<sup>1</sup> *wl*P*l*(*<sup>N</sup>* = *n*) = ∑∞*<sup>l</sup>*=<sup>1</sup> *wlkl*, where *kl* is the mean hitting time of {0} for *W*, if it starts from *l* ∈ N0, as in (Norris 1997, p. 12). From the skip-free property of the chain *W* it is moreover transparent that *ki* = *<sup>α</sup>i*, *i* ∈ N0, for some 0 < *α* ≤ ∞ (with the usual convention 0 · ∞ = 0). Moreover we know (Norris 1997, p. 17, Theorem 1.3.5) that (*ki* : *i* ∈ N0) is the minimal solution to *k*0 = 0 and *ki* = 1 + ∑∞*j*=<sup>1</sup> *Pijkj* (*i* ∈ N). Plugging in *ki* = *<sup>α</sup>i*, the last system of linear equations is equivalent to (provided *α* < ∞) 0 = 1 − *pα* + *αζ*, where *ζ* := ∑∞*j*=<sup>1</sup> *jqj*. Thus, if *ζ* < *p*, the minimal solution to the system is *ki* = *<sup>i</sup>*/(*p* − *ζ*), *i* ∈ N0, from which *β* = *ζ*/(*λ*((−∞, <sup>0</sup>))(*p* − *ζ*)) follows at once. If *ζ* ≥ *p*, clearly we must have *α* = +<sup>∞</sup>, hence <sup>E</sup>[*N*]=+∞ and thus *β* = +<sup>∞</sup>.

To establish the probability of an excursion being infinite, i.e., ∑∞*<sup>i</sup>*=<sup>1</sup> *qi*(<sup>1</sup> − *<sup>α</sup>i*)/ ∑∞*<sup>i</sup>*=<sup>1</sup> *qi*, where *αi* := <sup>P</sup>*i*(*<sup>N</sup>* < ∞) > 0, we see that (by the skip-free property) *αi* = *αi*1, *i* ∈ N0, and by the strong Markov property, for *i* ∈ N, *αi* = *p<sup>α</sup>i*−<sup>1</sup> + ∑∞*j*=<sup>1</sup> *qj<sup>α</sup>i*+*j*. It follows that 1 = *pα*<sup>−</sup><sup>1</sup> 1 + ∑∞*j*=<sup>1</sup> *qjαj*1, i.e., 0 = *ψ*(log(*α*<sup>−</sup><sup>1</sup> 1 )/*h*). Hence, by Theorem 2-2, whose proof will be independent of this one, *α*1 = *e* −<sup>Φ</sup>(0)*h* (since *α*1 < 1, if and only if *X* drifts to −<sup>∞</sup>).

Finally, Theorem 1-4 is straightforward.

We turn our attention now to the supremum process *X*. First, using the lack of memory property of the exponential law and the skip-free nature of *X*, we deduce from the strong Markov property applied at the time *Ta*, that for every *a*, *b* ∈ Z+*h* , *p* > 0: <sup>P</sup>(*Ta*+*<sup>b</sup>* < *ep*) = P(*Ta* < *ep*)P(*Tb* < *ep*). In particular, for any *n* ∈ N0: P(*Tnh* < *ep*) = P(*Th* < *ep*)*<sup>n</sup>*. And since for *s* ∈ Z+*h* , {*Ts* < *ep*} = {*Xep* ≥ *s*} (P-a.s.) one has (for *n* ∈ N0): P(*Xep* ≥ *nh*) = P(*Xep* ≥ *h*)*<sup>n</sup>*. Therefore *Xep*/*h* ∼ geom(<sup>1</sup> − P(*Xep* ≥ *h*)).

Next, to identify P(*Xep* ≥ *h*), *p* > 0, observe that (for *β* ≥ 0, *t* ≥ 0): <sup>E</sup>[exp{Φ(*β*)*Xt*}] = *e<sup>t</sup>β* and hence (exp{Φ(*β*)*Xt* − *βt*})*t*≥0 is an (<sup>F</sup>, P)-martingale by stationary independent increments of *X*, for each *β* ≥ 0. Then apply the optional sampling theorem at the bounded stopping time *Tx* ∧ *t* (*t*, *x* ≥ 0) to get:

$$\mathbb{E}[\exp\{\Phi(\beta)X(T\_x \wedge t) - \beta(T\_x \wedge t))\}] = 1.$$

Please note that *X*(*Tx* ∧ *t*) ≤ *hx*/*h* and Φ(*β*)*X*(*Tx* ∧ *t*) − *β*(*Tx* ∧ *t*) converges to <sup>Φ</sup>(*β*)*hx*/*h* − *βTx* (P-a.s.) as *t* → ∞ on {*Tx* < <sup>∞</sup>}. It converges to −∞ on the complement of this event, P-a.s., provided *β* + Φ(*β*) > 0. Therefore we deduce by dominated convergence, first for *β* > 0 and then also for *β* = 0, by taking limits:

$$\mathbb{E}[\exp\{-\beta T\_{\mathbf{x}}\}\mathbb{I}\_{\{T\_{\mathbf{x}}<\infty\}}] = \exp\{-\Phi(\beta)h[\mathbf{x}/h]\}.\tag{1}$$

Before we formulate our next theorem, recall also that any non-zero Lévy process either drifts to +<sup>∞</sup>, oscillates or drifts to −∞ (Sato 1999, pp. 255–56, Proposition 37.10 and Definition 37.11).

**Theorem 2** (Supremum process and long-term behaviour)**.**


**Remark 2.** *Unlike in the spectrally negative case (Bertoin 1996, p. 189), the supremum process cannot be obtained from the reflected process, since the latter does not discern a point of increase in X when the latter is at its running maximum.*

**Proof.** We have for every *s* ∈ Z+*h*:

$$\mathbb{P}(\overline{X}\_{t\_p} \ge s) = \mathbb{P}(T\_s < c\_p) = \mathbb{E}[\exp\{-pT\_s\} \mathbb{I}\_{\{T\_s < \infty\}}] = \exp\{-\Phi(p)s\}.\tag{2}$$

Thus Theorem 2-1 obtains.

For Theorem 2-2 note that letting *p* ↓ 0 in (2), we obtain *X*∞ < ∞ (P-a.s.), if and only if Φ(0) > 0, which is equivalent to *ψ*(0+) < 0. If so, *X*∞/*h* is geometrically distributed with failure probability exp{−Φ(0)*h*} and then (and only then) does *X* drift to <sup>−</sup>∞.

It remains to consider the case of drifting to +∞ (the cases being mutually exclusive and exhaustive). Indeed, *X* drifts to +<sup>∞</sup>, if and only if E[*Ts*] is finite for each *s* ∈ Z+*h* (Bertoin 1996, p. 172, Proposition VI.17). Using again the nondecreasingness of (*e*<sup>−</sup>*βTs* − 1)/*β* in *β* ∈ [0, <sup>∞</sup>), we deduce from (1), by the monotone convergence, that one may differentiate under the integral sign, to ge<sup>t</sup> E[*Ts* {*Ts*<∞}]=(*β* → − exp{−Φ(*β*)*s*})(0+). So the E[*Ts*] are finite, if and only if Φ(0) = 0 (so that *Ts* < ∞ P-a.s.) and Φ(0+) < ∞. Since Φ is the inverse of *ψ*|[Φ(0),<sup>∞</sup>), this is equivalent to saying *ψ*(0+) > 0.

Finally, Theorem 2-3 is clear.

Table 1 briefly summarizes for the reader's convenience some of our main findings thus far.

**Table 1.** Connections between the quantities *ψ*(0+), <sup>Φ</sup>(0), <sup>Φ</sup>(0+), the behaviour of *X* at large times, and the behaviour of its excursions away from the running supremum (the latter when *<sup>λ</sup>*((−∞, 0)) > 0).


We conclude this section by offering a way to reduce the general case of an upwards skip-free Lévy chain to one which necessarily drifts to +<sup>∞</sup>. This will prove useful in the sequel. First, there is a pathwise approximation of an oscillating *X*, by (what is again) an upwards skip-free Lévy chain, but drifting to infinity.

**Remark 3.** *Suppose X oscillates. Let (possibly by enlarging the probability space to accommodate for it) N be an independent Poisson process with intensity* 1 *and <sup>N</sup>t* := *Nt (t* ≥ 0*) so that N is a Poisson process with intensity , independent of X. Define X* := *X* + *hN. Then, as* ↓ 0*, X converges to X, uniformly on bounded time sets, almost surely, and is clearly an upwards skip-free Lévy chain drifting to* +<sup>∞</sup>*.*

The reduction of the case when *X* drifts to −∞ is somewhat more involved and is done by a change of measure. For this purpose assume until the end of this subsection, that *X* is already the coordinate process on the canonical space Ω = D*h*, equipped with the *σ*-algebra F and filtration F of evaluation maps (so that P *coincides* with the law of *X* on D*h* and F = *<sup>σ</sup>*(pr*s* : *s* ∈ [0, <sup>+</sup>∞)), while for *t* ≥ 0, F*t* = *<sup>σ</sup>*(pr*s* : *s* ∈ [0, *<sup>t</sup>*]), where pr*s*(*ω*) = *<sup>ω</sup>*(*s*), for (*s*, *ω*) ∈ [0, <sup>+</sup>∞) × D*h*). We make this transition in order to be able to apply the Kolmogorov extension theorem in the proposition, which follows. Note, however, that we are no longer able to assume the standard conditions on (<sup>Ω</sup>, F, F, <sup>P</sup>). Notwithstanding this, (*Tx*)*x*∈<sup>R</sup> remain F-stopping times, since by the nature of the space D*h*, for *x* ∈ R, *t* ≥ 0, {*Tx* ≤ *t*} = {*Xt* ≥ *<sup>x</sup>*}∈F*<sup>t</sup>*.

**Proposition 1** (Exponential change of measure)**.** *Let c* ≥ 0*. Then, demanding:*

$$\mathbb{P}\_{\varepsilon}(\Lambda) = \mathbb{E}[\exp\{cX\_{l} - \psi(\varepsilon)t\} \mathbb{I}\_{\Lambda}] \quad (\Lambda \in \mathcal{F}\_{l}, t \ge 0) \tag{3}$$

*this introduces a unique measure* P*c on* F*. Under the new measure, X remains an upwards skip-free Lévy chain with Laplace exponent ψc* = *ψ*(· + *c*) − *ψ*(*c*)*, drifting to* +<sup>∞</sup>*, if c* ≥ <sup>Φ</sup>(0)*, unless c* = *ψ*(0+) = 0*. Moreover, if λc is the new Lévy measure of X under* P*c, then λc λ and dλc dλ* (*x*) = *ecx λ-a.e. in x* ∈ R*. Finally, for every* F*-stopping time T,* P*c* P *on restriction to* <sup>F</sup>*T*:= {*A* ∩ {*T* < ∞} : *A* ∈ F*T*}*, and:*

$$\frac{d\mathbb{P}\_{\mathfrak{c}}|\_{\mathcal{F}\_T'}}{d\mathbb{P}|\_{\mathcal{F}\_T'}} = \exp\{cX\_T - \psi(c)T\}.$$

**Proof.** That P*c* is introduced consistently as a probability measure on F follows from the Kolmogorov extension theorem (Parthasarathy 1967, p. 143, Theorem 4.2). Indeed, *M* := (exp{*cXt* − *ψ*(*c*)*t*})*t*≥0 is a nonnegative martingale (use independence and stationarity of increments of *X* and the definition of the Laplace exponent), equal identically to 1 at time 0.

Furthermore, for all *β* ∈ C<sup>→</sup>, {*<sup>t</sup>*,*<sup>s</sup>*} ⊂ R+ and Λ ∈ F*t*:

$$\begin{split} \mathbb{E}\_{\mathbf{c}}[\exp\{\beta(\mathbf{X}\_{t+s}-\mathbf{X}\_{t}))\mathbbm{1}\_{\Lambda}] &=& \mathbb{E}[\exp\{c\mathbf{X}\_{t+s}-\psi(\mathbf{c})(t+s)\}\exp\{\beta(\mathbf{X}\_{t+s}-\mathbf{X}\_{t})\}\mathbbm{1}\_{\Lambda}] \\ &=& \mathbb{E}[\exp\{(\mathbf{c}+\beta)(\mathbf{X}\_{t+s}-\mathbf{X}\_{t})-\psi(\mathbf{c})\mathbf{s}\}]\mathbb{E}[\exp\{c\mathbf{X}\_{t}-\psi(\mathbf{c})t\}\mathbbm{1}\_{\Lambda}] \\ &=& \exp\{s(\psi(\mathbf{c}+\beta)-\psi(\mathbf{c}))\}\mathbb{P}\_{\mathbf{c}}(\Lambda). \end{split}$$

An application of the Functional Monotone Class Theorem then shows that *X* is indeed a Lévy process on (<sup>Ω</sup>, F, F, <sup>P</sup>*c*) and its Laplace exponent under P*c* is as stipulated (that *X*0 = 0 P*c*-a.s. follows from the absolute continuity of P*c* with respect to P on restriction to F0).

Next, from the expression for *ψ<sup>c</sup>*, the claim regarding *λc* follows at once. Then clearly *X* remains an upwards skip-free Lévy chain under P*<sup>c</sup>*, drifting to +<sup>∞</sup>, if *ψ*(*c*+) > 0.

Finally, let *A* ∈ F*T* and *t* ≥ 0. Then *A* ∩ {*T* ≤ *<sup>t</sup>*}∈F*<sup>T</sup>*∧*t*, and by the Optional Sampling Theorem:

$$\mathbb{P}\_{\mathfrak{c}}(A \cap \{T \le t\}) = \mathbb{E}[M\_t \mathbb{1}\_{A \cap \{T \le t\}}] = \mathbb{E}[\mathbb{E}[M\_t \mathbb{1}\_{A \cap \{T \le t\}} | \mathcal{F}\_{T \wedge t}]] = \mathbb{E}[M\_{T \wedge t} \mathbb{1}\_{A \cap \{T \le t\}}] = \mathbb{E}[M\_T \mathbb{1}\_{A \cap \{T \le t\}}].$$

Using the MCT, letting *t* → <sup>∞</sup>, we obtain the equality <sup>P</sup>*c*(*<sup>A</sup>* ∩ {*T* < ∞}) = E[*MT <sup>A</sup>*∩{*<sup>T</sup>*<∞}].

**Proposition 2** (Conditioning to drift to +∞)**.** *Assume* Φ(0) > 0 *and denote* P- := <sup>P</sup>Φ(0) *(see* (3)*). We then have as follows.*


**Proof.** With regard to Proposition 2-1, we have as follows. Let *t* ≥ 0. By the Markov property of *X* at time *t*, the process *X* := (*Xt*<sup>+</sup>*s* − *Xt*)*s*≥<sup>0</sup> is identical in law with *X* on D*h* and independent of F*t* under P. Thus, letting *Ty* := inf{*t* ≥ 0 : *Xt* ≥ *y*} (*y* ∈ R), one has for Λ ∈ F*t* and *n* ∈ N0, by conditioning:

$$\mathbb{P}(\Lambda \cap \{t < T\_{nh} < \infty\}) = \mathbb{E}[\mathbb{E}[\mathbb{I}\_{\Lambda}\mathbb{1}\_{\{t < T\_{nh}\}}\mathbb{1}\_{\{\overset{\wedge}{\bar{T}\_{nh-X\_{t}} < \infty}\}}|\mathcal{F}\_{t}]] = \mathbb{E}[e^{\Phi(0)(X\_{t} - nh)}\mathbb{1}\_{\Lambda \cap \{t < T\_{nh}\}}],$$

since {<sup>Λ</sup>, {*t* < *Tnh*}} ∪ *σ*(*Xt*) ⊂ F*<sup>t</sup>*. Next, noting that {*<sup>X</sup>*∞ ≥ *nh*} = {*Tnh* < ∞}:

$$\begin{split} \mathsf{P}(\Lambda|\overline{\mathcal{X}}\_{\infty} > nh) &= \ \mathsf{e}^{\Phi(0)n\mathbb{I}} \left( \mathsf{P}(\Lambda \cap \{T\_{n\mathbb{I}} \le t\}) + \mathsf{P}(\Lambda \cap \{t < T\_{n\mathbb{I}} < \infty\}) \right) \\ &= \ \mathsf{e}^{\Phi(0)n\mathbb{I}} \left( \mathsf{P}(\Lambda \cap \{T\_{n\mathbb{I}} \le t\}) + \mathbb{E}[\mathsf{e}^{\Phi(0)(X\_{\mathbb{I}} - nh)}\mathbbm{1}\_{\Lambda \cap \{t < T\_{n\mathbb{I}}\}}] \right) \\ &= \ \mathsf{e}^{\Phi(0)n\mathbb{I}} \mathsf{P}(\Lambda \cap \{T\_{n\mathbb{I}} \le t\}) + \mathsf{P}^{\dagger}(\Lambda \cap \{t < T\_{n\mathbb{I}}\}) . \end{split}$$

The second term clearly converges to P-(Λ) as *n* → ∞. The first converges to 0, because by (2) <sup>P</sup>(*Xe*1 ≥ *nh*) = *e*<sup>−</sup>*nh*Φ(1) = *<sup>o</sup>*(*e*<sup>−</sup>*nh*Φ(0)), as *n* → <sup>∞</sup>, and we have the estimate P(*Tnh* ≤ *t*) = P(*Xt* ≥ *nh*) = P(*Xt* ≥ *nh*|*<sup>e</sup>*1 ≥ *t*) ≤ <sup>P</sup>(*Xe*1 ≥ *nh*|*<sup>e</sup>*1 ≥ *t*) ≤ *<sup>e</sup>t*P(*Xe*1 ≥ *nh*).

We next show Proposition 2-2. Note first that *X* is F-progressively measurable (in particular, measurable), hence the stopped process *XTx* is measurable as a mapping into D*h* (Karatzas and Shreve 1988, p. 5, Problem 1.16).

Furthermore, by the strong Markov property, conditionally on {*Tx* < <sup>∞</sup>}, F*Tx* is independent of the future increments of *X* after *Tx*, hence also of {*Tx* < ∞} for any *x* > *x*. We deduce that the law of *XTx* is the same under P(·|*Tx* < ∞) as it is under <sup>P</sup>(·|*Tx* < ∞) for any *x* > *x*. Proposition 2-2 then follows from Proposition 2-1 by letting *x* tend to +<sup>∞</sup>, the algebra A being sufficient to determine equality in law by a *π*/*λ*-argument.
