There are multiple examples of conditioning a univariate Lévy process in some limiting sense, which alternatively can be described by Doob h-transforms, see [4, 12, 11] and references therein. Most often the focus is on establishing properties directly related to these conditional processes. The case of conditioning to stay positive or negative is special in the sense that it is intimately related to the post- and pre-infimum processes [4], leading to various important applications. Further links to path decomposition results can be found in [13].

Local behavior of a univariate Lévy process at its extremal points is studied in [16], see also [4] for a self-similar case and [3] for a linear Brownian motion. It is shown that zooming in at the point of infimum results in a pair of processes obtained from the underlying self-similar Lévy process conditioned to stay positive and negative. Further applications of this theory in the setting of high-frequency statistics include estimation of threshold exceedance in [7] and optimal estimation of extremes in [17]. Bertoin’s pathwise construction of conditioned processes in [4] plays a fundamental role in these works. For yet another application see [2] where the discretization error in the two-sided Skorokhod reflection map is studied.

In this work we extend Bertoin’s construction to the multivariate setting to define a Lévy process conditioned to stay in a half-space specified by some normal vector $\eta \ne 0$, see Section 3. Importantly, the link to post- and pre-extremum processes is preserved, where extrema are understood with respect to the direction η. Furthermore, in Section 4 we establish an associated invariance principle which, in particular, yields a limit result when zooming in on a Lévy process at the point of directional extremum. This is achieved via a short and direct argument relying on the path-wise construction. Applications of this result to high frequency statistics and the study of discretization errors in problems related to directional extrema and exceedance are anticipated.

In the multivariate case we have a continuum of possible directions, and the effect of linear transformations is studied in Section 5. It is shown that conditioning with respect to any direction η can be reduced to, say, conditioning an appropriately rotated process so that its first component stays positive. Furthermore, we provide a simple expression for the conditioned correlated Brownian motion in terms of a certain linear transformation of independent standard Brownian motions and a Bessel-3 process. In Section 6 we present the semigroup of the conditioned process in the general case, which turns out to have an intuitive structure. In Section 7 we utilize the arguments and insights from [9] to establish some important properties of the conditioned process. This leads to a natural definition of the respective Feller process started from an arbitrary point in the closed half-space.

We have attempted to present the multivariate theory in a streamlined and concise form, while emphasizing the main novelties stemming from the multivariate setting. Finally, in Section 8 we state a conjecture related to the local behavior at the point furthest from the origin, which hints at even greater application potential of the multivariate theory.

Fix an integer $d\ge 1$ and let $\mathcal{D}$ denote the space of càdlàg functions $\omega :\mathbb{R}\to {\mathbb{R}^{d}}\cup \{\dagger \}$, where † is an isolated absorbing state. As usually we equip the path space $\mathcal{D}$ with the Skorokhod topology and let $\mathcal{F}$ denote the Borel σ-field. Furthermore we denote the coordinate process by $X=({X_{t}})$ and its natural completed filtration by $({\mathcal{F}_{t}})$. Unless stated otherwise we work with a subclass of processes satisfying ${X_{t}}=0$ for $t<0$, and let $\zeta :=\inf \{t\ge 0\hspace{0.2222em}|\hspace{0.2222em}{X_{t}}=\dagger \}\in [0,\infty ]$ be the lifetime.

We start with a fundamental representation of the law of the pair $(\underset{\gets }{X},\underrightarrow{X}):{\mathbb{P}^{T}}$, which extends a univariate construction by Bertoin [4] based, in turn, on an implicit identity for random walks appearing in [14, Lem. XII.8.3]. Our representation is in terms of time-changed stochastic integrals, since the construction in [4] in terms of the local time at 0 does not have a simple analogue in the multivariate setting.

Consider the nonkilled process X and let ${\tilde{X}_{t}}:={X_{(-t)-}}$ be its time-reversal, which is a process with stationary and independent increments for negative times. Define two $({\mathcal{F}_{t}})$-adapted càdlàg processes ${Y^{\pm }}$ by and ${Y_{\infty }^{\pm }}:=\dagger $. These stochastic integrals can be understood intuitively as ${\textstyle\int _{0}^{t}}{\text{1}_{\{\langle {X_{s}},\eta \rangle >0\}}}\hspace{0.1667em}\mathrm{d}{X_{s}}$ and ${\textstyle\int _{0}^{t}}{\text{1}_{\{\langle {X_{s}},\eta \rangle \le 0\}}}\hspace{0.1667em}\mathrm{d}{X_{s}}$, where the integrands are not predictable.

The cumulative times when X is and is not in S are denoted by ${A^{+}}$ and ${A^{-}}$ respectively. That is, Consider now the right-continuous inverses ${\alpha _{t}^{\pm }}:=\inf \{s\ge 0\hspace{0.2222em}|\hspace{0.2222em}{A_{s}^{\pm }}>t\}$ of ${A^{\pm }}$, and define The processes ${X^{\uparrow }}$ and ${X^{\downarrow }}$ under ${\mathbb{P}^{T}}$ are obtained by killing ${X^{\uparrow }}$ and ${X^{\downarrow }}$ at the times ${A_{T}^{+}}$ and ${A_{T}^{-}}$ under $\mathbb{P}$, respectively. The times ${A_{T}^{\pm }}$ are nondecreasing in T, which results in longer lifetimes ${\zeta ^{\uparrow }}$ and ${\zeta ^{\downarrow }}$ for larger time horizons T.

Importantly, the above construction of the pair $({X^{\downarrow }},{X^{\uparrow }}):{\mathbb{P}^{T}}$ depends on T via the killing times ${A_{T}^{\pm }}$ alone. In particular, for $0<{T_{1}}<{T_{2}}$ the paths of ${X^{\uparrow }}:{\mathbb{P}^{{T_{1}}}}$ and ${X^{\uparrow }}:{\mathbb{P}^{{T_{2}}}}$ coincide up to the time ${A_{{T_{1}}}^{+}}$ when the former is sent to †, whereas the latter is killed at ${A_{{T_{2}}}^{+}}\ge {A_{{T_{1}}}^{+}}$. It is convenient to think of paths growing as T increases. As $T\to \infty $ we obtain $({X^{\downarrow }},{X^{\uparrow }})$.

It is noted that the above weak convergence statement can be strengthened, see [4, Cor. 3.2], but we prefer using Theorem 2 directly when needed. The pair $({X^{\downarrow }},{X^{\uparrow }})$ is our main object of interest. According to Corollary 3, the process ${X^{\uparrow }}:\mathbb{P}$ can be called a limiting post-infimum process. In analogy to the univariate case we instead call it X conditioned to stay in the half-plane S, and provide a justification below.

Observe that $-{X_{t}^{\downarrow }},{X_{t}^{\uparrow }}\in S\cup \{\dagger \}$ for $t>0$ a.s., whereas the initial values are classified according to the trichotomy in Lemma 1. In particular, ${X_{0}^{\uparrow }}=0$ in cases $(\updownarrow ),(\uparrow )$, and ${X_{0}^{\downarrow }}=0$ in cases $(\updownarrow ),(\downarrow )$. Importantly, the projected conditioned processes $\langle {X^{\uparrow }},\eta \rangle $ and $\langle {X^{\downarrow }},\eta \rangle $ coincide with the univariate Lévy process Z conditioned to stay positive and negative, respectively. In particular, the lifetimes ${\zeta ^{\uparrow }}$ and ${\zeta ^{\downarrow }}$ can be studied using the univariate theory, and so with probability 1. Furthermore, ${\zeta ^{\uparrow }}>0$ unless Z is a nonincreasing process and then ${\zeta ^{\uparrow }}=0$ a.s. Yet another useful observation is given by the following result.

Importantly, (under Assumption A) the process ${X^{\downarrow }}$ is a.s. the same if the nonstrict inequalities in its definition are replaced by strict inequalities, which follows from basic properties of Lévy processes. In particular, we find that ${X^{\downarrow }}=-{(-X)^{\uparrow }}$ a.s. The respective equality in distribution can also be seen using the representation in Theorem 2 and the standard time-reversal argument. Finally, observe a close link to the classical Sparre Andersen identity [5, Lem. VI.15]: ${A_{T}^{+}}$ has the same law as the time of the supremum of Z on $[0,T]$, which by time-reversal coincides with the law of the lifetime of the respective post-infimum process.

Bertoin’s representation and its above stated generalization are indispensable in the study of Lévy processes around their extremes. In the one-dimensional setting it has been fundamental for the results in [7, 17]. We further demonstrate its usefulness by establishing an invariance principle, see [10] and [16] for alternative approaches in the univariate case (the latter needs a better justification of convergence of Markov processes). The following short proof requires certain assumptions, and for simplicity we consider only the case of an oscillating ${Z_{t}}=\langle {X_{t}},\eta \rangle $: Recall that this assumption implies that both ${X^{\uparrow }}$ and ${X^{\downarrow }}$ have infinite lifetimes.

The above argument can be adapted to include the case where ${\lim \nolimits_{t\to \infty }}{Z_{t}}=\infty $ and ${\lim \nolimits_{t\to \infty }}{Z_{t}^{(n)}}=\infty $ for all large enough n, as well as the case with $-\infty $ limits. That is, the infinite-time behavior of Z and the approximating sequence ${Z^{(n)}}$ is the same. Otherwise, the proof becomes substantially more difficult and it is then required to work with a compactified space where † is a point at infinity.

Finally, we show that zooming in on X at the time-space location of the directional infimum results in the pair of conditioned processes corresponding to the underlying Brownian part. This limit law is studied in Proposition 9 below.

Linear transformations play an important role in the multivariate theory as demonstrated by the following result.

Consequently, it suffices to study conditioning for just one direction, say For any unit vector $\eta \in {\mathbb{R}^{d}}$ we may choose an orthogonal matrix R ($R{R^{\top }}=I$) such that $R\eta ={\eta _{1}}$. Then ${X^{\uparrow }}$ coincides with ${R^{\top }}{(RX)^{\uparrow }}$ where the latter is defined for the direction ${\eta _{1}}$. Our next result allows us to reduce certain multivariate cases to the univariate theory.

We are now ready to treat the basic example of a conditioned Brownian motion. In this regard note that a univariate standard Brownian ${B^{(1)}}$ conditioned to stay positive is a Bessel-3 process which we denote by ${{B^{(1)}}^{\uparrow }}$.

In particular, simulation of ${X^{\uparrow }}$ over a grid is a trivial task when X is a driftless Brownian motion. We depict two independent sample paths in Figure 2.

Further insight can be obtained from Figure 3 consisting of three plots, each containing 500 simulations of ${X_{1}^{\uparrow }}$ for different values of ρ.

We need some additional notation. Consider an extension of the probability space $(D,\mathcal{F},\mathbb{P})$ supporting a standard exponential random variable ${e_{1}}$ independent of everything else. Define ${e_{q}}={e_{1}}/q$, an exponential random variable of rate $q>0$, and let $X:{\mathbb{P}^{{e_{q}}}}$ be the process $X:\mathbb{P}$ killed at ${e_{q}}$. Finally, the process $X:{\mathbb{P}_{x}^{{e_{q}}}}$ corresponds to the shifted process ${X_{t}}+x{\text{1}_{\{t\ge 0\}}}$ killed at ${e_{q}}$, and in the case of no killing we write ${\mathbb{P}_{x}}$.

Theorem 12 characterizes the law of ${X^{\uparrow }}$ in case $(\downarrow )$, but otherwise it lacks convergence of the semigroup as $x\to 0$. In this section we address this issue and also state a number of further useful properties. The proofs follow closely the univariate analogues in [9] and thus we only state the main steps and observations.

It is easy to see that the semigroup ${p_{t}^{\uparrow }}(x,\mathrm{d}y)$ of ${X^{\uparrow }}$, see Theorem 12, is conservative and satisfies the Feller properties on $\hat{S}:=S\cup \{\dagger \}$. Note that the hyperplane defining this half-space has been excluded. We write $X:{P_{x}^{\uparrow }}$ for the respective Feller process indexed by $[0,\infty )$ and started at $x\in \hat{S}$, and note that it satisfies the strong Markov property [18, Thm. 19.17].

Observe that the law of the Markov process with the semigroup in Proposition 11 when started in $x\in \hat{S}$ can be conveniently written as Furthermore, [9, Prop. 1] readily generalizes to which explains the name ‘conditioned to stay in a half-space’. Note that $\langle X,\eta \rangle :{P_{x}^{\uparrow }}$ is the univariate process $\langle X,\eta \rangle $ conditioned to stay positive and started from $\langle x,\eta \rangle $.

The above proof, in fact, shows that In cases $(\updownarrow ),(\uparrow )$ the process ${x_{0}}+{X^{\uparrow }}$ starts at ${x_{0}}$ and according to Lemma 4 it does not jump at fixed times. Hence in these cases we may extend our Feller process $X:{P_{x}^{\uparrow }}$ to the state space $\overline{S}\cup \{\dagger \}$, the closed half-space with an absorbing state, by setting Note that this definition coincides with the result of the construction presented in Section 3 if we take X started at ${x_{0}}$ and let ${Y_{t}^{+}}=-{\textstyle\int _{[-t,0]}}{\text{1}_{\{\langle {\tilde{X}_{s-}},\eta \rangle \ge 0\}}}\hspace{0.1667em}\mathrm{d}{\tilde{X}_{s}}$ which yields an a.s. identical process in the original case.

For a possible further application we turn our attention to the local behavior of a Lévy process at the time when it reaches the maximal distance from the origin. Assuming finite life time, $\zeta \in (0,\infty )$, we let be the (last) time when the Euclidean norm is maximal. Consider the respective position $M:={X_{\tau }}$ if $\| {X_{\tau }}\| \ge \| {X_{\tau -}}\| $ and $M:={X_{\tau -}}$ otherwise, and define the processes Observe that the pair $(\stackrel{\gets }{X},\overrightarrow{X})$ coincides with $(\underset{\gets }{X},\underrightarrow{X})$ studied above for the (path-dependent) direction $\eta =-M$, see also Figure 4 for a schematic illustration.

Inspired by Corollary 6 and using the intuition from the one-dimensional stable convergence in [16], we conjecture the following result; proving it seems to be exceedingly challenging at the moment. We anticipate that the convergence is again stable [1] but avoid complicating the statement.

We illustrate this conjecture by a simulation study where $X=B$ is a 2-dimensional Brownian motion with correlation $\rho =-0.8$ as in Section 5. We simulate K (approximate) copies of the random vector $\sqrt{n}{\overrightarrow{X}_{1/n}}$ under ${\mathbb{P}^{1}}$ for $n=1000$ using discretization with step size ${10^{-5}}$. The K samples of the limit quantity ${B_{1}^{\Uparrow }}$ are constructed by reusing the directions $\eta =-M:{\mathbb{P}^{1}}$ and then independently sampling ${B_{1}^{\uparrow }}$ according to Example 10.

Note that $\overrightarrow{X}$ may have a lifetime strictly smaller than $1/n$, making $\sqrt{n}{\overrightarrow{X}_{1/n}}$ undefined. In our simulation we exclude these cases, effectively conditioning $\overrightarrow{X}$ to have a lifetime larger than $1/n$. We simulated 5000 times, resulting in $K=4833$ (conditional) samples of $\sqrt{n}{\overrightarrow{X}_{1/n}}$. The respective bivariate densities are presented in Figure 5, and we observe that they are indeed rather close.

It is noted that the Brownian motion X with correlation ρ tends to achieve its maximal distance from the origin in the NW or SE direction, which leads to the two clusters of points in Figure 5.