An extension of P. Lévy’s distributional properties to the case of a Brownian motion with

drift.

S.E. Graversen A.N. Shiryaev MaPhySto^{∗}

1 Introduction

A classical result of Paul Lévy states that if B = (B_{t})_{0≤t≤1} is a standard Brownian motion (B_{0} = 0, EB_{t} = 0, EB_{t}^{2} = t) then

(M − B, M) =^{law} (|B|, L(B))  (1)

i.e. ((M_{t} − B_{t}, M_{t}); 0 ≤ t ≤ 1) =^{law} (|B_{t}|, L(B)_{t}; 0 ≤ t ≤ 1) where M = (M_{t})_{0≤t≤1},

M_{t} = max_{0≤s≤t} B_{s}, and L(B) = (L(B)_{t})_{0≤t≤1} is the local time of B at zero:

L(B)_{t} = lim_{ϵ↓0} \frac{1}{2ϵ} ∫_{0}^{t} 1_{(|B_{s}|≤ϵ)} ds.  (2)

( See, for example, [1;Ch.VI].)

The main aim of this note is to give an extension of the distributional property (1) to the case of a Brownian motion with drift B^{λ} where B^{λ} = (B_{t}^{λ})_{0≤t≤1}, B_{t}^{λ} = B_{t} + λt. Let’s denote M^{λ} = (M_{t}^{λ})_{0≤t≤1}, M_{t}^{λ} = max_{0≤s≤t} B_{s}^{λ}.

For our presentation the following process X^{λ} = (X_{t}^{λ})_{0≤t≤1} defined as the unique strong solution of the stochastic differential equation

dX_{t}^{λ} = −λ sgnX_{t}^{λ} dt + dB_{t}, X_{0}^{λ} = 0,  (3)

plays a key role. (Here sgn x is defined to be 1 on ℜ_{+}, −1 on ℜ_{−} and 0 at 0.) In particular we shall see that the process |X^{λ}| = (|X_{t}^{λ}|)_{0≤t≤1} realizes an explicit construction of the process RBM(−λ) i.e. a reflecting Brownian motion with drift (−λt).

∗MaPhySto -Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish National Research Foundation

2 Main result

Theorem 1 For any λ ∈ ℜ

(M^{λ} − B^{λ}, M^{λ}) =^{law} (|X^{λ}|, L(X^{λ}))  (4)

i.e. (M_{t}^{λ} − B_{t}^{λ}, M_{t}^{λ}); 0 ≤ t ≤ 1) =^{law} (|X_{t}^{λ}|, L(X^{λ})_{t}); 0 ≤ t ≤ 1) where

L(X^{λ})_{t} = lim_{ϵ↓0} \frac{1}{2ϵ} ∫_{0}^{t} 1_{(|X_{s}^{λ}|≤ϵ)} ds.

Proof. Denote by (Ω, F, (F_{t})_{0≤t≤1}, P) a filtered probability space and let B = (B_{t})_{0≤t≤1} be a standard Brownian motion on (Ω, F, (F_{t})_{0≤t≤1}, P). Define on (Ω, F) a new probability measure P^{λ}:

dP^{λ} = e^{−λB_{1}−λ^{2}/2} dP (= e^{−λB_{1}^{λ}+λ^{2}/2} dP).  (5)

By Girsanov’s theorem ([1],[3])

Law(B^{λ} | P^{λ}) = Law(B | P).  (6)

Denoting by C^{+}[0, 1] the space of non-negative continuous functions on [0, 1] we obtain, using (5), (6) and (1) that for any non-negative measurable functional G = G(x, y), (x, y) ∈ C^{+}[0, 1] × C^{+}[0, 1]:

E[ G(M^{λ} − B^{λ}, M^{λ})] = E^{λ}[ e^{λB_{1}^{λ}−λ^{2}/2}G(M^{λ} − B^{λ}, M^{λ})]
= E[ e^{λB_{1}−λ^{2}/2}G(M − B, M)] = E[ e^{λ(L(B)_{1}−|B_{1}|)−λ^{2}/2}G(|B|, L(B))].  (7)

¿From another side let us introduce a new measure P̃^{λ}:

dP̃^{λ} = e^{λ ∫_{0}^{1} sgnX_{s}^{λ} dB_{s}−λ^{2}/2} dP (= e^{λ ∫_{0}^{1} sgnX_{s}^{λ} dX_{s}^{λ}+λ^{2}/2} dP)  (8)

Again by Girsanov’s theorem

Law(X^{λ} | P̃^{λ}) = Law(B | P).  (9)

¿From (8) and (9) we find that (with Ẽ^{λ} denoting expectation w.r. to P̃^{λ})

E[ G(|X^{λ}|, L(X^{λ}))] = Ẽ^{λ}[e^{−λ ∫_{0}^{1} sgnX_{s}^{λ} dX_{s}^{λ}−λ^{2}/2} G(|X^{λ}|, L(X^{λ}))]
= E[ e^{−λ ∫_{0}^{1} sgnB_{s} dB_{s}−λ^{2}/2} G(|B|, L)].  (10)

Now we note that by Tanaka’s formula [1; Ch. VI]

|B_{t}| = ∫_{0}^{1} sgnB_{s} dB_{s} + L(B)_{t}.

So, from (10)

E[G(|X^{λ}|, L(X^{λ}))] = E[e^{λ(L(B)_{1}−|B_{1}|)−λ^{2}/2} G(|B|, L(B))].  (11)

Comparing (7) and (11) we get the statement (4).

3 Study of X^{λ}

In this section we consider some properties of the processes X^{λ} and |X^{λ}|. If λ = 0 then X^{0} = B, |X^{0}| = |B| and as it is well-known Law(|B|)=Law(RBM(0)), where RBM(0) is a Brownian motion reflecting in zero; [1;Ch.III], [2;Ch.IV]. In this sense the process |B| gives an explicit construction of the reflecting Brownian motion. We shall see below that for reflecting Brownian with drift the process |X^{λ}| plays the corresponding role.

Let’s describe first of all some properties of X^{λ} and |X^{λ}| from the point of view of the general theory of Markov processes.

On a filtered probability space (Ω, F, (F_{t})_{t≥0}, P) for given λ ∈ ℜ and every x ∈ ℜ we consider the stochastic process X^{x,λ} = (X_{t}^{x,λ})_{t≥0} which satisfies the stochastic differential equation

dX_{t}^{x,λ} = −λsgnX_{t}^{x,λ} dt + dB, X_{0}^{x,λ} = x.  (12)

This equation has a unique strong solution and as a corollary (see [1;Ch.IX, Th. 1.11]) we also have uniqueness in law. Denote the corresponding distribution of X^{x,λ} on the space (C, C) of continuous functions by P^{x,λ}:

Law(X^{x,λ}| P) = P^{x,λ}.  (13)

Denote also by (T_{t}^{λ}, t ≥ 0) the set of operators given by

T_{t}^{λ}f(x) = ∫ f(c_{t}) P^{x,λ}(dc),  (14)

where f ∈ B_{b}(ℜ) (the set of bounded Borel measurable real valued functions defined on ℜ) and c = (c_{t})_{t≥0} denotes the coordinate process, c ∈ C.

If τ is a finite (F_{t})_{t≥0}-stopping time and A ∈ F_{τ} then

E[ f(X_{τ+t}^{x,λ}) · 1_{A}] = E[ T_{t}f(X_{τ}^{x,λ}) · 1_{A}].  (15)

Indeed, from (12)

X_{τ+t}^{x,λ} = X_{τ}^{x,λ} − λ ∫_{0}^{t} sgn(X_{τ+u}^{x,λ}) du + (B_{τ+t} − B_{τ}).  (16)

But Law(B_{τ+t} − B_{τ}, t ≥ 0| P) =Law(B_{t}, t ≥ 0| P) and (B_{τ+t} − B_{τ})_{t≥0} is independent of F_{τ} and so by uniqueness in law of equation (12) we get (15).

Thus the process X^{x,λ} = (X_{t}^{x,λ})_{t≥0} is a time homogeneous Markov process with transition functions (T_{t}^{λ}, t ≥ 0). ¿From [4; Ch.6,6.5] it follows that transition densities p_{t}^{λ}(y|x) such that

T_{t}^{λ}(x, dy) = p_{t}^{λ}(y|x) dy

do exist and, for example, for x ≥ 0, λ ≥ 0 are given by the following formula:

p_{t}^{λ}(y|x) = \frac{1}{\sqrt{2πt}} (e^{− \frac{(x−y−λt)^{2}}{2t}} + λe^{−2λy} ∫_{x+y}^{∞} e^{− \frac{(v−λt)^{2}}{2t}} dv) , y ≥ 0,

p_{t}^{λ}(y|x) = \frac{1}{\sqrt{2πt}} (e^{−(2λx− \frac{(x−y+λt)^{2}}{2t})} + λe^{2λy} ∫_{x−y}^{∞} e^{− \frac{(v−λt)^{2}}{2t}} dv) , y < 0. (17)

This explicit form of the transition density can be used to show that X^{x,λ} is a Feller process (indeed that can also be deduced using Zvonkin’s method [1;Ch.IX, (2.11)].)

Now we show that |X^{x,λ}| is also a time homogeneous Markov process.

Indeed sgn x is an odd function and {t | X_{t}^{x,λ} = 0} is P -a.s. a Lebesgue null set (it is clearly true for λ = 0, that is for (x + B_{t})_{t≥0}, but the measures P^{x,0} and P^{x,λ} are locally equivalent so it holds, in fact, for any λ ∈ ℜ). Thus it follows that P -a.s.

−X_{t}^{x,λ} = −x − λ ∫_{0}^{t} sgn(−X_{s}^{x,λ}) ds − B_{t},  (18)

and by the uniqueness in law we then obtain

Law(−X^{x,λ} | P) = Law(X^{−x,λ} | P).  (19)

Using the Markov property of X^{x,λ}-processes this implies that for all s, t ≥ 0, x ∈ [0, ∞) and all bounded real valued Borel function f on [0, ∞) we have for any A^{x} ∈ σ(|X_{u}^{x,λ}| | u ≤ s)

E[ f(|X_{s+t}^{x,λ}|), A^{x}] = E[ f̃(X_{s+t}^{x,λ}), A^{x}] = E[ T_{t}f̃(X_{s}^{x,λ}), A^{x}]

and

E[ f(|X_{s+t}^{x,λ}|), A^{x}] = E[ f(| − X_{s+t}^{x,λ}|), A^{x}] = E[ f(|X_{s+t}^{−x,λ}|), A^{−x}]
= E[ f̃(X_{s+t}^{−x,λ}), A^{−x}] = E[ T_{t}f̃(X_{s}^{−x,λ}), A^{−x}] = E[ T_{t}f̃(−X_{s}^{x,λ}), A^{x}].

Here we have used the notation f̃(x) for f(|x|), x ∈ ℜ. In other words

E[ f(|X_{s+t}^{x,λ}|), A] = E[ T_{t}f̃(|X_{s}^{x,λ}|), A]  (20)

showing that |X^{x,λ}| is indeed a Feller Markov process.

Theorem 2 For each x ∈ ℜ_{+} and λ ∈ ℜ

Law(|X^{x,λ}|) = Law(RBM^{x}(−λ))  (21)

Proof. In Markov theory, as it is well-known (see, for example, [2;Ch.IV,§5]), the process RBM^{x}(−λ), called a Brownian motion with drift (−λt) started at x ≥ 0 and reflected at zero, is a diffusion Markov process with infinitesimal operator A^{λ} acting on functions

D(A^{λ}) = \{f ∈ C_{b}^{2} ([0, ∞)) , \frac{df}{dx}|_{x↓0} = 0\}

by the formula

A^{λ}f(x) = \frac{1}{2}f′′(x) − λf′(x).  (22)

(It is well-known that the operator A^{λ} generates a unique (diffusion) family of measures Q^{x,λ}, x ≥ 0 and the corresponding Markov process, is by definition the process RBM^{x}(−λ), [2].)

Now let’s consider our process X^{x,λ}. By the Itô-Tanaka formula, [1;Ch.VI]

d|X_{t}^{x,λ}| = sgnX_{t}^{x,λ} dX_{t}^{x,λ} + dL(X^{x,λ})_{t}
= −λ dt + sgnX_{t}^{x,λ} dB_{t} + dL(X^{x,λ})_{t}  (23)

where L(X^{x,λ})_{t} is a local time at zero on the time interval [0, t] for the process X^{x,λ}. Suppose that f ∈ C_{b}^{2}([0, ∞)) with f′(0+) = \frac{df}{dx}|_{x↓0} = 0. Then by Itô’s formula

f(|X_{t}^{x,λ}|) − f(|X_{0}^{x,λ}) = ∫_{0}^{t} f′(|X_{s}^{x,λ}|) d|X_{s}^{x,λ}| + 1/2 ∫_{0}^{t} f′′(|X_{s}^{x,λ}|) ds
= ∫_{0}^{t} f′(|X_{s}^{x,λ}|) (−λ ds + sgnX_{s}^{x,λ} dB_{s} + dL(X^{x,λ})_{s}) + 1/2 ∫_{0}^{t} f′′(|X_{s}^{x,λ}|) ds (24)

= ∫_{0}^{t}(−λf′(X_{s}^{x,λ}|)f′′(|X_{s}^{x,λ}|) ds + M_{t}

where M_{t} = ∫_{0}^{t} f′(|X_{s}^{x,λ}|)sgnX_{s}^{x,λ} dB_{s} is a local martingale. Note that

∫_{0}^{t} f′(|X_{s}^{x,λ}|) dL(X^{x,λ})_{s} = 0

because f′(0+) = 0 and L(X^{x,λ}) increases only on the time set {t | X_{t}^{x,λ} = 0}. From (24) we see that

f(|X_{t}^{x,λ}|) − f(|X_{0}^{x,λ}|) − ∫_{0}^{t} A^{λ}f(|X_{s}^{x,λ}|) ds  (25)

is a local martingale and thus the infinitesimal operators for the two processes |X^{x,λ}| and RBM^{x}(−λ) are the same (acting on D(A^{λ})). Therefore (21) is proved.

References

[1] Revuz D., Yor M. Continuous Martingales and Brownian Motion, Second ed.,1994. Springer-Verlag.

[2] Ikeda N., Watanabe Sh. Stochastic Differential Equations and Diffusion Processes, 1981. North Holland.

[3] Liptser R., Shiryaev A.N. Statistics of Random Processes, Vol. 1, 1977. Springer-Verlag.

[4] Karatzas I., Shreve S. Brownian Motion and Stochastic Calculus, 1988. Springer-Verlag.