Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. 01, 1995, pp. 17–26 Submitted: June 11, 1995. Published: September 5, 1995.

STOCHASTIC CALCULUS AND INITIAL VALUE

ANALYSIS ON THE ONE DIMENSIONAL DIFFUSIONS

Dražen Pantić

Abstract. Stochastic calculus and initial value analysis are used for complete description of distribution type of diffusion processes with Lipschitz coefficients. Sufficient conditions so that the solutions of stochastic differential equations posses absolutely continuous one dimensional distribution are given. For stochastic differential equations with uniformly elliptic coefficients probability density is investigated in detail. Also, the distribution of inverse process is given.

(1991) A.M.S. Subject Classification Codes. 60H07, 60H10.

Key Words and Phrases. Malliavin calculus, solutions of stochastic differential equations, point distributions.

INTRODUCTION

Let {X_{t,x}, t ≥ 0}, x ∈ ℝ, be solution of SDE

X_{t,x} − x = ∫_{0}^{t} a(X_{s,x})dB_{s} + ∫_{0}^{t} b(X_{s,x})ds  (1)

where coefficients a and b have bounded and continuous first order derivatives (a, b ∈ ℂ_{b}^{1}). Under those conditions for p > 1 there exists L_{p} derivative DX_{t,x} of X_{t,x} w.r.t. initial condition x, so that, as in [2, §8],

DX_{t,x} − 1 = ∫_{0}^{t} a′(X_{s,x})DX_{s,x}dB_{s} + ∫_{0}^{t} b′(X_{s,x})DX_{s,x} ds.  (2)

Using Ito formula, DX_{t,x} could be computed explicitly as

DX_{t,x} = exp\{∫_{0}^{t} a′(X_{s,x})dB_{s} + ∫_{0}^{t}(b′(X_{s,x}) − \frac{1}{2}a′(X_{s,x}))^{2}ds\}.  (3)

The boundeness of derivatives implies that, for p > 1, DX_{t,x} as well as its inverse (DX_{t,x})^{−1} are in L_{p}, [6], while (DX_{t,x})^{−1} satisfies following SDE, [10, Lemma (3.7)]:

(DX_{t,x})^{−1} − 1 = − ∫_{0}^{t} a′(X_{s,x})(DX_{s,x})^{−1}dB_{s} + ∫_{0}^{t}(a′(X_{s,x})^{2} − b′(X_{s,x}))(DX_{s,x})^{−1} ds.

Typeset by AMS-TEX

Department of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Yugoslavia

E-mail Address: epantidr@ubbg.etf.bg.ac.yu

\begin{matrix}c \\ ◯ \end{matrix}1995 Cameron University

Also, for a Skorohod integrable process u = {u_{s}, 0 ≤ s ≤ T } a directional derivative D_{u}X_{t,x} of functional X_{t,x} could be expressed in terms of derivatives {DX_{t,x}, t ≥ 0} in the following way, [12],

D_{u}X_{t,x} = DX_{t,x} ∫_{0}^{t}(DX_{s,x})^{−1}a(X_{s,x}) u_{s} ds.  (4)

First we shall treat the general case. Defining process \underline{u} = \{\underline{u}_{s}, 0 ≤ s ≤ t\} as

\underline{u}_{s} = a(X_{s,x})DX_{s,x}

we have

D\underline{u}X_{t,x} = DX_{t,x} ∫_{0}^{t} a(X_{s,x})^{2} ds.  (5)

Process \underline{u} is Ito integrable, so for a continuously differentiable function with compact support f it follows, by integration by parts formula, [5], and relation (5), that

𝔼\{f′(X_{t,x})DX_{t,x} ∫_{0}^{t} a(X_{s,x})^{2}ds\} = 𝔼\{D\underline{u}(f(X_{t,x}))\} = 𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})DX_{s,x}dB_{s}\}.  (6)

Knowing that DX_{t,x} is a.s. positive, we have to find conditions so that ∫_{0}^{t} a(X_{s,x})^{2}ds is a.s. positive in order to apply next Lemma.

Lemma 1. Let X be a random variable such that there exists integrable and almost sure positive random variable Y and a constant C > 0 so that for each f ∈ ℂ_{b}^{1} we have

| 𝔼{f′(X) · Y } |≤ C sup{| f(x) |: x ∈ ℝ}.

Then probability law generated by the random variable X is absolutely continuous with respect to Lebesgue measure.

Proof. Let us define positive bounded linear functional on the space of the continuously differentiable functions with compact support with

L(f) = 𝔼{f(X)Y } = 𝔼{f(X)𝔼{Y |X}} = ∫ f(x)𝔼{Y |X = x}G(dx)

where G is probability measure generated by X. Then, we have

| ∫ f′(x)𝔼{Y |X = x}G(dx) |≤ C sup{| f(x) |: x ∈ ℝ}

so using [4, §1] it follows that there exists density g so that

∫ f(x)𝔼{Y |X = x}dG(x) = ∫ f(x)g(x) dx.

Clearly, the previous relation could be extended to the space of bounded measurable functions, so for a Borel set B we have

𝔼{𝕀{X ∈ B}Y } = ∫ 𝕀{x ∈ B}g(x) dx.

If B is of Lebesgue measure zero then

𝔼{𝕀{X ∈ B}Y } = 0.

Knowing that Y > 0 almost sure one easily gets that ℙ{X ∈ B} = 0, i.e. probability law generated by the random variable X is absolutely continuous with respect to Lebesgue measure. □

GENERAL CASE

In the next theorem we shall completely describe the first moment τ from which ∫_{0}^{t} a(X_{s,x})^{2}ds is a.s. positive, as well as the behavior of diffusion process (1) up to that moment.

Theorem 1. Let {X_{t,x}, t ≥ 0}, x ∈ ℝ, be solution of SDE (1) with coefficients in ℂ_{1}^{b}, and {Y_{t,x}, t ≥ 0} deterministic process defined by

Y_{t,x} − x = ∫_{0}^{t} b(Y_{s,x}) ds.

Define

τ_{x} = inf{t ≥ 0 :| a(X_{t,x}) |> 0} and D_{x} = inf{t ≥ 0 :| a(Y_{t,x}) |> 0}.

Then, with probability one, τ_{x} = D_{x} and X_{t,x} = Y_{t,x} for t ≤ D_{x} .

Proof. Assume first that 0 < D_{x} < ∞, and let 0 < t < D_{x}. Then

X_{t,x} − Y_{t,x} = ∫_{0}^{t}(a(X_{s,x}) − a(Y_{s,x}))dB_{s} + ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds, and

𝔼{(X_{t,x} − Y_{t,x})^{2}}^{1/2} ≤
≤ 𝔼{(∫_{0}^{t}(a(X_{s,x}) − a(Y_{s,x}))dB_{s})^{2}}^{1/2} + 𝔼{( ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds)^{2}}^{1/2} ≤
≤ 𝔼{∫_{0}^{t}(a(X_{s,x}) − a(Y_{s,x}))^{2}ds}^{1/2} + 𝔼{t ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))^{2}ds}^{1/2} ≤
≤ K(∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2}}ds)^{1/2} + t^{1/2}K( ∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2}}ds)^{1/2} ≤
≤ K(1 + D_{x}^{1/2})( ∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2}}ds)^{1/2},

where K is upper bound for first derivatives of a and b. Then, using Gronwall inequality, we can conclude that with probability one X ≡ Y on [0, D_{x}], and hence D_{x} ≤ τ_{x} a.s.

Assume now that 0 < τ_{x} < ∞, and let 0 < t < τ_{x}. As τ_{x} is optional time the set

A = {ω : t < τ_{x}(ω)}

is F_{t} measurable, where {F_{t}, t ≥ 0} is standard filtration generated by {B_{t}, t ≥ 0}. Then we have

𝔼{∫_{0}^{t} a(X_{s,x})^{2}ds 𝕀_{A}} = 0.

It is obvious that

(∫_{0}^{t} a(X_{s,x}) dB_{s})^{2} 𝕀{t < τ_{x}} ≤ (∫_{0}^{t∧τ_{x}} a(X_{s,x}) dB_{s})^{2}.

Having in mind that t ∧ τ_{x} is optional time we have

𝔼{(∫_{0}^{t} a(X_{s,x}) dB_{s})^{2} 𝕀{t < τ_{x}}} ≤ 𝔼{(∫_{0}^{t∧τ_{x}} a(X_{s,x}) dB_{s})^{2}} = 𝔼{∫_{0}^{t∧τ_{x}} a(X_{s,x})^{2} ds} = 0,

and consequently ∫_{0}^{t} a(X_{s,x})dB_{s} = 0 a.s. on set A. Hence for each t > 0 on the set A we have

X_{t,x} − Y_{t,x} = ∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds, and
𝔼{(X_{t,x} − Y_{t,x})^{2} 𝕀{t < τ_{x}}} =
= 𝔼{(∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))ds)^{2} 𝕀{t < τ_{x}}} ≤
≤ t 𝔼{∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))^{2}ds 𝕀{t < τ_{x}}} ≤
≤ t 𝔼{∫_{0}^{t}(b(X_{s,x}) − b(Y_{s,x}))^{2} 𝕀{s < τ_{x}} ds} ≤
≤ tK^{2} ∫_{0}^{t} 𝔼{(X_{s,x} − Y_{s,x})^{2} 𝕀{s < τ_{x}}} ds

Let us mark Δ_{t} = 𝔼{(X_{t,x} − Y_{t,x})^{2}𝕀{t < τ_{x}}}. Then previous relation could be written as

Δ_{t} ≤ tK^{2} ∫_{0}^{t} Δ_{s} ds,

from where it follows that Δ_{t} = 0 for each t > 0. Hence we can conclude that for each t > 0

ℙ{ω : t < τ_{x}(ω), X_{t,x} ≠ Y_{t,x}} = 0.

Using the continuity of paths we have that

ℙ{ω : (∃t > 0)(t < τ_{x}(ω), X_{t,x} ≠ Y_{t,x})} = 0,

so with probability one X ≡ Y on [0, τ_{x}], i.e. τ_{x} ≤ D_{x} a.s. and the proof is complete. □

The following theorem completely describes the type of one dimensional distributions of the solutions of SDE :

Theorem 2. Under conditions from Theorem 1. the distribution of random variable X_{t,x} is either degenerate or absolutely continuous, whether t ≤ D_{x} or t > D_{x}.

Proof. The first part when t ≤ D_{x} is described in Theorem 1 so let t > D_{x}. Then ∫_{0}^{t} a(X_{s,x})^{2}ds is positive a.s. and from relation (6) and Lemma 1 we can reach our conclusion. □

UNIFORMLY ELLIPTIC CASE

Now, we shall treat an important special case when that coefficient a is uniformly elliptic in sense that

inf{| a(x) |: x ∈ ℝ} > 0.  (EL)

Theorem 3. Let the diffusion process {X_{t,x}, t ≥ 0} defined by SDE (1) has continuously differentiable coefficients a and b with bounded derivatives and let

inf{|a(x)| : x ∈ ℝ} > 0.

Then we have :

1^{∘} for each t > 0 and x ∈ ℝ the probability law of random variable X_{t,x} is absolutely continuous w.r.t. Lebesgue measure with transition density p_{t}(x, ·); 2^{∘} for any bounded, measurable function f and t > 0 the mapping x → 𝔼{f(X_{t,x})} is continuously

differentiable and

\frac{∂}{∂x}𝔼\{f(X_{t,x})\} = t^{−1} 𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}\};  (7)

3^{∘} for each y ∈ ℝ and t > 0 transition density p_{t}(x, y) is differentiable w.r.t. initial condition x and

\frac{∂}{∂x}p_{t}(x, y) = t^{−1} 𝔼\{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x} = y\} p_{t}(x, y).  (8)

Proof. 1^{∘} Taking ũ_{s} = a(X_{s,x})^{−1}DX_{s,x}, one has, from (4)

DũX_{t,x} = DX_{t,x} t.  (9)

Process {ũ} is Ito integrable, so for a continuously differentiable function with compact support f it follows by integration by parts formula and relation (9), that

𝔼{f′(X_{t,x})DX_{t,x} t} = 𝔼{Dũ(f(X_{t,x}))} = 𝔼{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}}.  (10)

Now, DX_{t,x} is a.s. positive random variable and its inverse (DX_{t,x})^{−1} is in L_{p} for each p > 1. Also the assumptions we have made on coefficient a assure, via a Burgholder theorem, that ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s} is in L_{2}, hence by Lemma 1 assertion 1^{∘} holds. 2^{∘} First, take f ∈ ℂ_{b}^{1}. Then by [2] and relation (10) one has

\frac{∂}{∂x}𝔼\{f(X_{t,x})\} = 𝔼\{f′(X_{t,x})DX_{t,x})\} = t^{−1}𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}\}.

Now, for x, y ∈ ℝ we have

𝔼\{f(X_{t,y})\} − 𝔼\{f(X_{t,x})\} = ∫_{x}^{y} 𝔼\{f(X_{t,z}) ∫_{0}^{t} a(X_{s,z})^{−1}DX_{s,z} dB_{s}\} \frac{dz}{t} .  (11)

Both sides of relation (11) are continuous linear functionals on the set M_{b} of measurable and bounded functions, so we can conclude that (11) is valid for each f ∈ M_{b}. Also, for f ∈ M_{b} and t > 0 mapping x → 𝔼{f(X_{t,x})} is absolutely continuous.

If we define semigroup {P_{t}, t ≥ 0} of operators on the space M_{b} of bounded measurable functions as P_{t}f(x) = 𝔼{f(X_{t,x})}, then Markov property of diffusion {X_{t,x}} yields, for δ > 0

P_{t+δ}f(x) = 𝔼{P_{δ}f(X_{t,x})}.

Substituting t with t + δ and f with P_{δ}f, relation (11) reads as

P_{t+δ}f(y) − P_{t+δ}f(x) = ∫_{x}^{y} 𝔼\{P_{δ}f(X_{t,z}) ∫_{0}^{t} a(X_{s,z})^{−1}DX_{s,z} dB_{s}\} \frac{dz}{t} .

Then, from continuity of the function P_{δ}f(x) it follows that for each x ∈ ℝ there exists

lim_{y→x} \frac{P_{t+δ}f(y) − P_{t+δ}f(x)}{y − x}, and

\frac{∂}{∂x}P_{t+δ}f(x) = t^{−1} 𝔼\{P_{δ}(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}\}

hence \frac{∂}{∂x}P_{t+δ}f(x) is continuous function in x.

Now relation (11) yields that, for each x

\frac{∂}{∂x}P_{t}f(x) = t^{−1} 𝔼\{f(X_{t,x}) ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x} dB_{s}\}.

3^{∘} Let us mark

W_{t,x} = t^{−1} ∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}.  (12)

Then relation (10) reads as

𝔼{f′(X_{t,x})DX_{t,x}} = 𝔼{f(X_{t,x})W_{t,x}},

or, having in mind 1^{∘},

∫_{−∞}^{∞} f′(u) 𝔼{DX_{t,x}|X_{t,x} = u}p_{t}(x, u) du = ∫_{−∞}^{∞} f(u) 𝔼{W_{t,x}|X_{t,x} = u}p_{t}(x, u) du.

Taking, for ϵ > 0 and y ∈ ℝ,

f_{ϵ}(x) = ∫_{−∞}^{x} e^{−(t−y)^{2}/2ϵ} \frac{dt}{\sqrt{2πϵ}},

we get, using partial integration

∫_{−∞}^{∞} e^{−(u−y)^{2}/2ϵ} 𝔼\{DX_{t,x}|X_{t,x} = u\}p_{t}(x, u) \frac{du}{\sqrt{2πϵ}} =

= ∫_{−∞}^{∞}(∫_{−∞}^{u} e^{−(t−y)^{2}/2ϵ} \frac{dt}{\sqrt{2πϵ}})𝔼\{W_{t,x}|X_{t,x} = u\}p_{t}(x, u) du =

= − ∫_{−∞}^{∞} e^{−(t−y)^{2}/2ϵ}(∫_{−∞}^{t} 𝔼\{W_{t,x}|X_{t,x} = u\}p_{t}(x, u) du)\frac{dt}{\sqrt{2πϵ}}. (13)

¿From relation (11) follows, via [3, Theorem 1.28.], that 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) is continuous and bounded function so letting ϵ ↓ 0 one can conclude that

𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) = − ∫_{−∞}^{y} 𝔼{W_{t,x}|X_{t,x} = u}p_{t}(x, u)du = −𝔼{W_{t,x}𝕀{X_{t,x} ≤ y}}.  (14)

It is obvious that there exist

lim_{y→±∞} 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y)

and as 𝔼{DX_{t,x}|X_{t,x} = ·}p_{t}(x, ·) is nonnegative, integrable function it follows that

lim_{y→±∞} 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) = 0.

Relation (7) implies that

𝔼\{W_{t,x}𝕀\{X_{t,x} ≤ y\}\} = \frac{∂}{∂x}𝔼\{𝕀\{X_{t,x} ≤ y\}\} = \frac{∂}{∂x}ℙ\{X_{t,x} ≤ y\},

so relation (14) read as

𝔼\{DX_{t,x}|X_{t,x} = y\}p_{t}(x, y) = −\frac{∂}{∂x}ℙ\{X_{t,x} ≤ y\}.  (15)

Now, as ℙ{X_{t,x} ≤ y} is continuously differentiable in x, we have for x_{1} < x_{2}

ℙ{X_{t,x_{2}} ≤ y} − ℙ{X_{t,x_{1}} ≤ y} = − ∫_{x_{1}}^{x_{2}} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz
= ∫_{x_{1}}^{x_{2}} ∫_{−∞}^{y} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dudz,

and by Fubini theorem for almost all u function 𝔼{W_{t,·}|X_{t,·} = u}p_{t}(·, u) is integrable and

∫_{x_{1}}^{x_{2}} ∫_{−∞}^{y} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dudz = ∫_{−∞}^{y} ∫_{x_{1}}^{x_{2}} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dzdu.

Finally, as ℙ{X_{t,x} ≤ y} = ∫_{−∞}^{y} p_{t}(x, u) du we have that, for each y ∈ ℝ,

∫_{−∞}^{y} p_{t}(x_{2}, u) du − ∫_{−∞}^{y} p_{t}(x_{1}, u) du = ∫_{−∞}^{y} ∫_{x_{1}}^{x_{2}} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u) dzdu

so for almost all y we have

p_{t}(x_{2}, y) − p_{t}(x_{1}, y) = ∫_{x_{1}}^{x_{2}} 𝔼{W_{t,z}|X_{t,z} = u}p_{t}(z, u)dz

hence p_{t}(x, y) is absolutely continuous function in x and

\frac{∂}{∂x}p_{t}(x, y) = 𝔼\{W_{t,x}|X_{t,x} = y\} p_{t}(x, y). □

As an appendix to the previous theorem we shall study in detail partial derivative \frac{∂}{∂x}p_{t}(x, y). Precisely, we shall show that the process t^{−1}𝔼{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x}} is a backwards martingale.

As {X_{t,x}, t ≥ 0} is a Markov process, for a f ∈ M_{b} and s < t we have

𝔼_{x}{f(X_{t})} = 𝔼_{x}{𝔼{f(X_{t})|F_{s}}} = 𝔼_{x}{𝔼_{X_{s}}{f(X_{t−s})}},

where {F_{t}, t ≥ 0} is standard filtration generated by {B_{t}, t ≥ 0}, and 𝔼_{x} is expectation operator generated by initial value X_{0} = x. Now, 1^{∘} of Theorem 3 yields

\frac{∂}{∂x}𝔼_{x}\{f(X_{t})\} = 𝔼_{x}\{f(X_{t})W_{t}\} = 𝔼_{x}\{f(X_{t})𝔼\{W_{t}|X_{t}\} \} as well as
\frac{∂}{∂x}𝔼_{x}\{f(X_{t})\} = \frac{∂}{∂x}𝔼_{x}\{𝔼_{X_{s}}\{f(X_{t−s})\}\} = 𝔼_{x}\{𝔼_{X_{s}}\{f(X_{t−s})\} W_{s} \} =
= 𝔼_{x}{ 𝔼_{X_{s}}{f(X_{t−s})} 𝔼{W_{s}|X_{s}} } = 𝔼_{x}{ f(X_{t}) 𝔼{W_{s}|X_{s}} },

hence, for s < t

𝔼{W_{t,x}|X_{t,x}} = 𝔼{ 𝔼{W_{s,x}|X_{s,x}} |X_{t,x}}.

If we mark

W̃_{t,x} = 𝔼{W_{t,x}|X_{t,x}}

then previous relation could be rewritten as

W̃_{t,x} = 𝔼{W̃_{s,x}|X_{s,x}}, s < t.

Let us define backwards flow of σ-algebras as

F^{←}_{t,x} = σ{X_{u,x}, u ≥ t}, t > 0, x ∈ ℝ,

and let for ϕ(x_{1}, …, x_{n}) be n-dimensional bounded Borel function. For s < t ≤ t_{1} < ⋯ < t_{n} we have

𝔼_{x}{W̃_{s,x}ϕ(X_{t_{1},x}, …, X_{t_{n},x})} = 𝔼_{x}{ 𝔼{W̃_{s,x}ϕ(X_{t_{1},x}, …, X_{t_{n},x})|F_{t}} } =
= 𝔼_{x}{ W̃_{s,x} 𝔼{ϕ(X_{t_{1},x}, …, X_{t_{n},x})|F_{t}} } =
= 𝔼_{x}{ W̃_{s,x} 𝔼_{X_{t,x}}{ϕ(X_{t_{1}−t,x}, …, X_{t_{n}−t,x})} } =
= 𝔼_{x}{ 𝔼{W̃_{s,x}|X_{t,x}} 𝔼_{X_{t,x}}{ϕ(X_{t_{1}−t,x}, …, X_{t_{n}−t,x})} } =
= 𝔼_{x}{ 𝔼{W̃_{s,x}|X_{t,x}} ϕ(X_{t_{1},x}, …, X_{t_{n},x}) },

hence

𝔼{W̃_{s,x}|F^{←}_{t,x}} = 𝔼{W̃_{s,x}|X_{t,x}} = W̃_{t,x}, i.e.

process

W̃_{t,x} = t^{−1}𝔼{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x}}

is a backward martingale, relative to flow of σ-algebras

F^{←}_{t,x} = σ{X_{u,x}, u ≥ t}, t > 0, x ∈ ℝ.

Having in mind the definition of W̃_{t,x} and 3^{∘} of Theorem 3, we deduce

∫_{−∞}^{∞}|\frac{∂}{∂x}p_{t}(x, y)|dy = 𝔼\{|W̃_{t,x}|\} < ∞.

Moreover, for s < t

\begin{aligned}∫_{−∞}^{∞}|\frac{∂}{∂x}p_{t}(x, y)|dy & = 𝔼\{|W̃_{t,x}|\} = 𝔼\{ |𝔼\{W̃_{s,x}|F^{←}_{t,x}\}| \} \\ & ≤ 𝔼\{ 𝔼\{|W̃_{s,x}||F^{←}_{t,x}\} \}= 𝔼\{ |W̃_{s,x}| \}= \end{aligned}

= ∫_{−∞}^{∞}|\frac{∂}{∂x}p_{s}(x, y)|dy

We have proved the following Corollary: Corollary 1. For each t > 0 and x ∈ ℝ process

W̃_{t,x} = t^{−1}𝔼{∫_{0}^{t} a(X_{s,x})^{−1}DX_{s,x}dB_{s}|X_{t,x}}

is a backward martingale, relative to flow of σ-algebras

F^{←}_{t,x} = σ{X_{u,x}, u ≥ t}, t > 0, x ∈ ℝ.

Consequently, function

∫_{−∞}^{∞}|\frac{∂}{∂x}p_{t}(x, y)|dy

is finite and decreasing with t. □

POINT CUT AS SPATIAL HOMEOMORPHISM

It is well known [10, Theorem 2.20] that, for each t > 0, the mapping x → X_{t,x} is a.s. homeomorphism of ℝ onto ℝ, hence we can define inverse random transformation

X_{t}^{−1}(·) : ℝ → ℝ

such that

X_{t}^{−1}(X_{t,x}) = x, x ∈ ℝ.

In the next theorem we shall investigate the distribution of random variable X_{t}^{−1}(y), i.e. the random function X_{t}^{−1}(·) applied to the point y.

Theorem 4. Let everything be as in Theorem 3. Then:

1^{∘} for each t > 0 and y ∈ ℝ function 𝔼{DX_{t,·}|X_{t,·} = y}p_{t}(·, y) is integrable and

∫_{−∞}^{+∞} 𝔼{DX_{t,x}|X_{t,x} = y}p_{t}(x, y) dx = 1;  (16)

2^{∘} for each t > 0 and x, y ∈ ℝ

ℙ{X_{t}^{−1}(y) ≤ x} = ∫_{−∞}^{x} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz;  (17)

3^{∘} probability density

ρ_{t}(y, x) = 𝔼{DX_{t,x}|X_{t,x} = y} p_{t}(x, y)  (18)

of the random variable X_{t}^{−1}(y) satisfy Chapman-Kolmogorov equations:

ρ_{t+s}(y, x) = ∫_{−∞}^{+∞} ρ_{s}(y, z) ρ_{t}(z, x) dz  (19)

for t, s > 0 and x, y ∈ ℝ.

Proof. 1^{∘} As for each t > 0, X_{t,·} is a.s. homeomorphism of ℝ onto ℝ and DX_{t,x}, partial derivative in L_{2}(Ω) sense of random variable X_{t,x} w.r.t. x, is a.s. positive, we have

lim_{x→±∞} X_{t,x} = ±∞,

and from relation (15) we have, for x_{1} < x_{2}

ℙ{X_{t,x_{2}} ≤ y} − ℙ{X_{t,x_{1}} ≤ y} = − ∫_{x_{1}}^{x_{2}} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz.  (20)

Now, letting x_{1} → −∞ and x_{2} → +∞ one can conclude that

∫_{−∞}^{+∞} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y) dz = 1,

what was to be proved in 1^{∘}. 2^{∘} ¿From the fact that a.s. X_{t,x} is homeomorphism and that X_{t}^{−1}(y) is its inverse we have the following equivalence

X_{t}^{−1}(y) ≤ x ⇔ X_{t,x} ≥ y,

so, using relation (20), with x_{1} = −∞ and x_{2} = x

ℙ{X_{t}^{−1}(y) ≤ x} = ℙ{X_{t,x} ≥ y} = 1 − ℙ{X_{t,x} < y} = ∫_{−∞}^{x} 𝔼{DX_{t,z}|X_{t,z} = y}p_{t}(z, y)dz.

3^{∘} Denote with ℙ_{x}{·} the probability measure generated by the Markov process {X_{t,x}, t ≥ 0} starting from x. Then for t, s > 0 the Markov property yields

ℙ{X_{t+s,x} ≤ y} = ℙ_{x}{X_{t+s} ≤ y} = ∫_{−∞}^{+∞} ℙ_{z}{X_{s} ≤ y}p_{t}(x, z) dz = 𝔼{ℙ_{X_{t,x}}{X_{s,z} ≤ y}}.

Using relation (15) we have

ρ_{t+s}(y, x) = −\frac{∂}{∂x}ℙ\{X_{t+s,x} ≤ y\} = −\frac{∂}{∂x}𝔼\{ℙ_{X_{t,x}}\{X_{s,z} ≤ y\}\} =

= −𝔼\{\frac{∂}{∂x}ℙ_{X_{t,x}}\{X_{s,z} ≤ y\} DX_{t,x}\} =

= −𝔼{−𝔼_{X_{t,x}}{DX_{s}|X_{s} = y}p_{s}(X_{t,x}, y) 𝔼{D_{t,x}|X_{t,x}}} =

= ∫_{−∞}^{+∞} 𝔼_{z}{DX_{s}|X_{s} = y}p_{s}(z, y) 𝔼{D_{t,x}|X_{t,x} = z}p_{t}(x, z) dz =

= ∫_{−∞}^{+∞} 𝔼{DX_{s,z}|X_{s,z} = y}p_{s}(z, y) 𝔼{D_{t,x}|X_{t,x} = z}p_{t}(x, z) dz =

= ∫_{−∞}^{+∞} ρ_{s}(y, z) ρ_{t}(z, x) dz,

and the theorem is completely proved. □

References

1. N. Bouleau and F. Hirsch, Formes de Dirichlet generales et densite des variables aleatoires reelles sur l’espace de Wiener., J. Func. Anal. 69 (1986), 229–259.

2. I.I. Gihman and A.V. Skorohod, Stochastic differential equations, Naukova Dumka, Kiev, 1968. (Russian)

3. S. Kusuoka and D. Stroock, Applications of the Malliavin Calculus. I , Proc of the Taniguchi Inter. Symp. on Stoch. Analysis. Katata, North-Holland, 1982, pp. 271–308.

4. P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976, pp. 195–263.

5. D. Nualart, Noncausal stochastic integrals and calculus, Proc. of Workshop on Stoch. Anal. and Rel. Topics. Silivri 1986, 1988, pp. 80–129.

6. D. Nualart and M. Zakai, Generalized stochastic integrals and Malliavin Calculus, Probability Theory and Rel. Fields 73 (1986), 255–280.

7. D. Pantić, Stochastic Calculus on Distorted Brownian Motion, J.Math.Phys. 29 (1988), 207-210.

8. D. Pantić, Stochastic Calculus and Schroedinger Semigroups Admitting Ground State, J. Math. Phys. 31 (1990), 1221-1225.

9. D. Pantić, Toward the Existence of Density for the Distribution of a Brownian Functional, submitted for publication.

10. D. W. Stroock, Topics in stochastic differential equations, Tata Institute of Fundamental Research, 1982.

11. S. Watanabe, Lectures on stochastic differential equations and Malliavin Calculus, Tata Institute of Fundamental Research, 1984.

12. M. Zakai, The Malliavin Calculus, Acta Appl. Math 3 (1985), 175–207.