Anticipative analogues of diffusion processes

Andrey Yu. Pilipenko Department of Mathematics and Mechanics, Kiev University, 252033 Kiev, Ukraine

Abstract. Linear stochastic differential equations with respect to the logarithmic process of smooth measure are studied. The theorem of existence and uniqueness is proved. Sufficient conditions for smoothness of solutions’s trajectories are given.

This paper studies the properties of anticipative analogues of diffusion processes in non-Gaussian case.

It is well known that under sufficiently general assumptions, the onedimensional diffusion process can be obtained as a solution of Ito stochastic differential equation

\{ \begin{matrix}dx(t) = a(x(t), t)dt + b(x(t), t)dw(t) \\ x(0) = x_{0} \end{matrix}  (1)

Developed theory of stochastic differential equations essentially used the Ito construction of stochastic integral. For this reason the initial condition x_{0} from (1) and Wiener process {w(t) : t ≥ 0} are independent. In attempt to refuse the requirement of independence we get the new class of problems. It connected with a fact that the solution x(t), now, can not be adapted to a current of σ-algebras, which are generated by w(t). In 70th this reason yielded [1,2] to the construction of extended stochastic integral w.r.t. Wiener process. The analogues of (1) with extended integral were considered by many authors [3 – 8]. Particularly it was observed that solutions of anticipative s.d.e. have some new properties not typical to the diffusion processes. For example, the solution of anticipative s.d.e. can possess discontinuous trajectories. Really, consider the following equation

\{ \begin{matrix}dx(t) = x(t)dw(t), t ∈ [0, 1] \\ x(0) = sign (w(1)). \end{matrix}  (2)

There the initial condition x_{0} = sign (w(1)) depends on {w(t), t ∈ [0, 1]} and stochastic integral in (2) is Skorohod extended integral.

Then a solution x(t) has a form:

x(t) = sign(w(1) − t)e^{w(t)−t/2}

and there is not any modification of x(t) with continuous trajectories.

The aim of this paper is to consider a situation when extended stochastic integral with respect to the Wiener process is replaced by extended stochastic integral with respect to the logarithmic process of smooth measure.

Let us describe the following construction.

Consider a Banach space X = C_{0}([0, 1]) of continuous functions starting at zero and probability measure μ on Borel σ-algebra B(X). Suppose μ is quasiinvariant and Fomin differentiable [9] with respect to the directions of Hilbert space H = L_{2}([0, 1]) which is continuously and densely embedded to X by the inclusion map

i : H ∋ f ↦ ∫_{0}^{.} f(s) ds ∈ X.

It will be supposed that the logarithmic derivative λ of measure μ [9] satisfies the following condition

∀p ≥ 1 ∀h ∈ H < λ, h >∈ L_{p}(X, μ).

Consider a space FC_{b}^{∞} of cylindrical infinitedifferentiable functions

FC_{b}^{∞} = {f = f(< e_{1}^{∗}, · >, …, < e_{n}^{∗}, · >)|
n ∈ N, f ∈ C_{b}^{∞}(R^{n}), e_{k}^{∗} ∈ X^{∗}},

and finitedimensional smooth processes

FC_{b}^{∞}(H) = {g = ∑_{k=1}^{n} g_{k}e_{k}|

n ∈ N, g_{k} ∈ FC_{b}^{∞}, e_{k} ∈ H}.

There we denote by < ·, · > the pairing between X^{∗} and X, or H and H. Definition 1. The stochastic derivative D,

D : W_{p}^{1} ⊂ L_{p}(X, μ) → L_{p}(X, μ, H)

is called the closure of a Frechet derivative ∇,

∇ : FC_{b}^{∞} ⊂ L_{p}(X, μ) → L_{p}(X, μ, H)

which is taken along the space H, i.e.

∇ : f(< e_{1}^{∗}, · >, …, < e_{n}^{∗}, · >) ↦ ∑_{k=1}^{n} \frac{∂f}{∂x_{k}} i^{∗}(e_{k}^{∗}).

Particularly, if μ is a distribution of the process {ξ(t) : t ∈ [0, 1]} and e_{k}^{∗} = δ_{t_{k}} is a unit measure concentrated in a point t_{k} ∈ [0, 1], then

D_{t}f(ξ(t_{1}), …, ξ(t_{n})) = ∑_{k=1}^{n} \frac{∂f}{∂x_{k}} χ_{[0,t_{k}]}(t).

Definition 2. The operator I,

I : D_{q} ⊂ L_{q}(X, μ, H) ↦ L_{q}(X, μ)

which is adjoint to operator D,

D : W_{p}^{1} ⊂ L_{p}(X, μ) ↦ L_{p}(X, μ, H), \frac{1}{p} + \frac{1}{q} = 1

is called extended stochastic integral.

Remark 1. If μ is Wiener measure then I is Skorohod extended integral, which coincides with Ito integral on the space of non-anticipative processes.

Remark 2. The space FC_{b}^{∞}(H) belongs to D_{q}, q ≥ 1, besides

I (∑_{k=1}^{n} g_{k}e_{k}) =

= − ∑_{k=1}^{n}(< Dg_{k}, e_{k} > +g_{k} < λ, e_{k} >),  (3)

where g_{k} ∈ FC_{b}^{∞}, e_{k} ∈ H.

Remark 3. The random element φ belongs D_{q}, q ≥ 1 iff φ ∈ L_{q}(X, μ, H) and exists a random variable g ∈ L_{q}(X, μ) such that for all f ∈ FC_{b}^{∞}(H) the formulae integration by parts holds:

∫_{X} < Df, φ > dμ = ∫_{X} fg dμ  (4)

besides g = I(φ),

Definition 3. The process m(t) = I_{(χ_{[0,t]})} = − < λ, χ_{[0,t]} > is called the logarithmic process of measure μ.

The process x ∈ L_{1}(X × [0, 1], dμ × dt) is said to be integrable w.r.t. logarithmic process if exists a random variable g ∈ L_{1}(X, μ) such that

∀f ∈ FC_{b}^{∞} : ∫_{X} (∫_{0}^{1} x(t)D_{t}fdt) dμ = ∫_{X} gfdμ.

Let us denote g by ∫_{0}^{1} x(t)dm(t) and the space of integrable processes w.r.t

m(t) by D, the integral ∫_{0}^{t} x(s)dm(s) means ∫_{0}^{1} x(s)χ_{[0,t]}(s)dm(s).

Remark. If x ∈ D such that x ∈ L_{q}(X, μ, H) and ∫_{0}^{1} x(s)dm(s) ∈

L_{q}(X, μ) then x ∈ D_{q} and Ix = ∫_{0}^{1} x(s)dm(s).

If x ∈ L_{2}([0, 1]) is non-random function then ∫_{0}^{1} x(s)dm(s) can be obtained

analogously Ito construction.

If x(t) is adapted process to a flow {ξ(t) : t ∈ [0, 1]}, x(t) = ∑_{k=0}^{n−1} x_{k}χ_{[t_{k},t_{k+1}]},

where x_{k} ∈ L_{∞}(X, μ), 0 = t_{0} < t_{1} < ⋯ < t_{n} = 1, then

∫_{0}^{1} x(t)dm(t) = ∑_{k=0}^{n−1} x_{k}(m(t_{k+1}) − m(t_{k})).

Example 1. If μ is a distribution of Wiener process {w(t) : t ∈ [0, 1]}

then m(t) = w(t) and ∫_{0}^{1} x(s)dw(s) is Skorohod extended integral, which

coincides with Ito integral on the space of non-anticipative processes.

Example 2. a Let μ_{2}(dx) = ρ(x)μ_{1}(dx). Let us compare integrals I_{μ_{1}} and I_{μ_{2}}. If ρ ∈ W_{p}^{1}(X, μ_{1}) and ρ > 0 μ_{1}a.a. then

I_{μ_{2}}(x) = I_{μ_{1}}(x) − ⟨\frac{Dρ}{ρ}, x⟩ ,  (5)

if all variables in (5) has a proper moments. b If μ is a distribution of Wiener process w(t),

μ_{1}(dx) = \sqrt{2}e^{− \frac{w(1)^{2}}{2}} μ(dx)

then m_{1}(t) = w(t) + w(1)t,

∫_{0}^{t} x(s)dm(s) = ∫_{0}^{t} x(s)dw(s) + w(1) ∫_{0}^{t} x(s)ds.

This example shows that a logarithmic process is not always a martingale. But in [10] is shown that m(t) satisfies the following condition

E(m(t) − m(s)/F_{s}) = 0,

where 0 ≤ s ≤ t, F_{s} = σ(ξ(τ)|τ ≤ s).

Consider the following linear s.d.e.

\{ \begin{matrix}dx(t) = a(t)x(t)dt + b(t)x(t)d m(t) \\ x(0) = η \end{matrix}  (6)

where η is random variable; a, b are random processes.

Definition 4. The random process x(t) ∈ L_{1}(X × [0, 1], dμ × dt) is a solution of (6) if

1) x(t) ∈ L_{1}(X, μ), t ∈ [0, 1],

2) a(t)x(t), b(t)x(t) ∈ L_{1}(X × [0, 1], dμ × dt)

3) for all t ∈ [0, 1] we have

x(t) = η + ∫_{0}^{t} a(s)x(s)ds + ∫_{0}^{t} b(s)x(s)dm(s).

The condition 3) is equivalent to

3′) ∀t ∈ [0, 1], f ∈ FC_{b}^{∞} : ∫_{X} (∫_{0}^{t} b(s)x(s)D_{s}fds) dμ =

= ∫_{X} (x(t) − η − ∫_{0}^{t} a(s)x(s)ds) fdμ.  (7)

Theorem 1 (existence and uniqueness). Let b ∈ C([0, 1]) is non-random function; a ∈ L_{∞}(X × [0, 1], dμ × dt), η ∈ L_{∞}(X, μ). Then equation (6) has a unique solution. This solution is of the form

x(t) = η(A_{t}) exp {∫_{0}^{t} a(s, T_{s}A_{t})ds} L_{t},  (8)

where

T_{t}ω = ω + ∫_{0}^{.} χ_{[0,t]}(z)b(z)dz, ω ∈ X,

A_{t}ω = T_{t}^{−1}ω = ω − ∫_{0}^{.} χ_{[0,t]}(z)b(z)dz,

L_{t} = \frac{d(μ ∘ T_{t}^{−1})}{dμ}

is a density of shifted measure.

Example 3. Let m(t) is a process from example 2b. Then the stochastic exponent

\{ \begin{matrix}dE(s, t) = E(s, t)dm(t), t ≥ s \\ E(s, s) = 1 \end{matrix}

has a form

E(s, t) = exp\{m(t) − m(s) − \frac{1}{2}(t − s) − \frac{1}{2}(t − s)^{2}\}.

It is interesting to observe that E(s, t) has not a multiplicative property, i.e. for s < τ < t the equality

E(s, t) = E(s, τ)E(τ, t)

does not hold

Example 4. Let {ξ_{n} : n ≥ 0} is a sequence of i.i.d random variable, Eξ_{n} = 0 and exists ε > 0 such that E|ξ_{n}|^{2+ε} < ∞. Suppose that distribution of ξ_{n} (in R^{1}) is differentiable, λ(x), x ∈ R is a logarithmic derivative and E(λ(ξ_{n}))^{2} < ∞.

Then random process ξ(t) = ξ_{0}t + ∑_{n=1}^{∞} ξ_{n} \frac{sin nπt}{n}, t ∈ [0, 1] has a continuous

modification. The distribution of ξ(t) is differentiable with respect to the directions of i(L_{2}([0, 1])), i(f) = ∫_{0}^{.} f(s)ds. The logarithmic process m(t) has

a form

m(t) = λ(ξ_{0})t + ∑_{n=1}^{∞} λ(ξ_{n})\frac{sin nπt}{n}, t ∈ [0, 1].

Suppose also that the distribution of ξ_{n} has a positive continuous density. Then the equation

\{ \begin{matrix}dx(t) = x(t)dm(t) \\ x(0) = f(ξ(t_{1}), …, ξ(t_{n})) \end{matrix}

has a solution

x(t) = f (ξ(t_{1}) − ∫_{0}^{t∧t_{1}}ds, …, ξ(t_{n}) − ∫_{0}^{t∧t_{n}}ds) ·

· exp \{∫_{0}^{1} λ(ξ_{0} − st)ds · t + ∑_{n=1}^{∞} ∫_{0}^{1} λ (ξ_{n} − s\frac{sin nt}{n}) ds\frac{sin nt}{n}\} .

Formulae for density in terms of logarithmic derivative follows from [11].

Proof of theorem 1. Existence. Note that μ ∘ T_{t}^{−1} ≪ μ because μ is quasiinvariant with respect to directions of H. Therefore L_{t} is well defined. Observe also that x(t) ∈ L_{1}(X × [0, 1], dμ × dt).

For proof it suffices to verify formulae (7). Let f ∈ FC_{b}^{∞} then

E (∫_{0}^{t} b(s)x(s)D_{s}fds) =

= E (∫_{0}^{t} b(s)η(A_{s}) exp {∫_{0}^{s} a(z, T_{z}A_{s})dz} L_{s}D_{s}fds) =

= E (∫_{0}^{t} b(s)η exp {∫_{0}^{s} a(z, T_{z})dz} D_{s}f(T_{s})ds) .  (9)

Lemma 1 [4]. If f ∈ FC_{b}^{∞} then for all s ∈ [0, 1] :

\frac{d}{ds}f(T_{s}) = b(s) < Df, χ_{[0,s]} > (T_{s}) = b(s)[D_{s}f](T_{s})

\frac{d}{ds}f(A_{s}) = −b(s)[D_{s}f](A_{s})

Using lemma 1 and integration by parts we will obtain the proof of existence.

Uniqueness. Let x(t) is a solution of equation (6), f ∈ FC_{b}^{∞}, t ∈ [0, 1]. It follows from (7) that

Ex(t)f(A_{t}) = Eηf(A_{t}) + E ∫_{0}^{t} b(s)x(s)D_{s}(f(A_{t}))ds+

+E ∫_{0}^{t} a(s)x(s)D_{s}(f(A_{t}))ds.  (10)

Note that f(A_{t}) belongs to FC_{b}^{∞} (t is fixed) and (lemma 1)

f(A_{t}) = f(A_{s}) − ∫_{s}^{t} b(u)D_{u}(f(A_{u}))du.  (11)

Substituting (11)into (10), we get that

Ex(t)f(A_{t}) = E(ηf) + E ∫_{0}^{t} a(s)x(s)f(A_{s})ds,

i.e.

Ex(t, T_{t})f \frac{dμ ∘ A_{t}^{−1}}{dμ} =

= E(ηf) + E (f ∫_{0}^{t} a(s, T_{s})x(s, T_{s})\frac{dμ ∘ A_{t}^{−1}}{dμ} ds) .

Since f ∈ FC_{b}^{∞} is arbitrary, we can conclude that for every t ∈ [0, 1],

x(t, T_{t}) · L_{t}^{−1} = η + ∫_{0}^{t} a(s, T_{s})x(s, T_{s})L_{s}^{−1}ds (mod μ),

where L_{t} = \frac{dμ∘T_{t}^{−1}}{dμ}. Therefore x(t, T_{t})L_{t}^{−1} = η exp \{∫_{0}^{t} a(s, T_{s})ds\} and x(t)

has a form (8). Theorem 1 is proved.

Remark. This method of proof was used in [4] in solving linear anticipating stochastic equations with respect to the Wiener process.

As follows from (8), for research of smoothness for solution’s trajectories of s.d.e. (6) we need to study the trajectories of η(A_{t}), t ∈ [0, 1]. The next theorem gives us sufficient conditions for continuity of η(A_{t}) in terms of stochastic differentiability.

Theorem 2. Let there exists a modification of density ρ(x, t) = \frac{dμ∘T_{t}^{−1}}{dμ} (x), t ∈ [0, 1] such that ρ(x, t) > 0 and continuous in t for all x ∈ X, then for each η ∈ W_{p}^{1} there exists a process η̃(x, t) = η(A_{t}(x)), t ∈ [0, 1] (mod μ) with absolutely continuous trajectories.

Proof. Consider a space X×[0, 1] with measure P (dx, dt) = ρ(x, t)dμ(x)dt. Define a derivative with respect to t,

\frac{∂}{∂t} : D (\frac{∂}{∂t}) ⊂ L_{1}(X × [0, 1], P) → L_{1}(X × [0, 1], P),

where D (\frac{∂}{∂t}) consists of functions f which are continuously differentiable with respect to t for all fixed x ∈ X, besides f is bounded with \frac{∂f}{∂t}.

Lemma 2. The operator \frac{∂}{∂t} is closable.

Proof follows from the continuity ρ(x, t) in t when x is fixed.

Lemma 3. The mapping j which transfer the random variable η ∈ L_{1}(X, μ) to the random process η(A_{t}) ∈ L_{1}(X × [0, 1], P) is an isometry.

Lemma 4. The mapping j : W_{1}^{1} → D (\frac{∂^{c}}{∂t} ) is continuous. There W_{1}^{1} and

D (\frac{∂^{c}}{∂t} ) are provided with a norm

∥f∥_{W_{1}^{1}} = ∥f∥_{L_{1}(X,μ)} + ∥Df∥_{L_{1}(X,μ,H)},

∥g∥_{D(\frac{∂^{c}}{∂t})} = ∥g∥_{L_{1}(X×[0,1],P)} + ∥\frac{∂^{c}g}{∂t} ∥_{L_{1}(X×[0,1],P)} .

Proof follows from lemmas 1 and 3.

Lemma 5. If ρ satisfies the conditions of the theorem 2, f ∈ D (\frac{∂^{c}}{∂t} ) then exists f̃ = f (mod P) such that f̃(x, t) is absolutely continuous in t for all x ∈ X.

To prove this, observe that for μ-a.a. x ∈ X a function f(x, ·) belongs to W_{1}^{1}([0, 1], dt), consequently exists such a modification f̃ = f (mod P) that is absolutely continuous in t for all x ∈ X.

The proof of the theorem 2 follows from lemma 2-5 .

As a corollary of the theorem 2 we can deduce the following theorem. Theorem 3 (continuity of solution’s trajectories). Let conditions of the

theorems 1 and 2 be satisfied, x(t) is a solution of (6) and a_{t} ∈ W_{p}^{1} for all t ∈ [0, 1].

Then there exists a process x̃(t) = x(t) (mod μ), t ∈ [0, 1] which has continuous trajectories.

The next result shows that if stochastic derivative of random variable is essentially bounded then this random variable is “Lipschitz with respect to directions of H”

Definition 5. [12] A function F : X → R is called H-Lipschitz with a constant c > 0 if for all x ∈ X, h ∈ H:

|F (x + i(h)) − F (x)| ≤ c∥h∥_{H}.

A function F : X → R is called H-Lipschitz μ-a.a. with a constant c ≥ 0 if exists F̃ = F (mod μ) such that F̃ is H-Lipschitz with a constant c.

Theorem 4. Let a measure μ satisfies the following condition.

There exists an orthonormal basis {i^{∗}(e_{n}^{∗}) : n ≥ 1}, e_{n}^{∗} ∈ X^{∗} of the space H such that

1) (E^{σ_{n}}P_{n}λ − P_{n}λ) ∈ L_{∞}(X, μ, H),

∥E^{σ_{n}}P_{n}λ − P_{n}λ∥_{L_{∞}(X,μ,H)} → 0, n → ∞.

There E^{σ_{n}} is a mathematical expectation with respect to a σ-algebra, σ_{n} = σ(< e_{k}^{∗}, · > |1 ≤ k ≤ n), P_{n} is ortoprojection on the

L(i^{∗}(e_{k}^{∗})|1 ≤ k ≤ n),

P_{n}λ = ∑_{k=1}^{n} < λ, i^{∗}(e_{k}^{∗}) > i^{∗}(e_{k}^{∗}).

2) A measure μ_{n}, the joint distribution of < i^{∗}(e_{k}^{∗}), · >, 1 ≤ k ≤ n, has a continuous and positive density in R^{n}.

Let F ∈ W_{p}^{1}, p > 1 and DF ∈ L_{∞}(X, μ, H). Then F is H-Lipschitz μ-a.a. with a constant ∥DF ∥_{L_{∞}(X,μ,H)}.

For proof we need the following auxiliary lemma.

Lemma 6. [13] Let measure μ satisfies the condition 2) of the theorem 4, F ∈ W_{p}^{1}, p > 1.

Then E^{σ_{n}}F ∈ W_{p−}^{1} = ⋂_{α<p} W_{α}^{1} and the following formula holds

D(E^{σ_{n}}F) = E^{σ_{n}}(P_{n}DF)+
+ ⟨P_{n}λ − E^{σ_{n}}P_{n}λ, E^{σ_{n}}F ⟩ .

Proof of the theorem 4. Let F ∈ L_{∞}(X, μ) . There exists a Borel function φ_{n} : R^{n} → R such that

F_{n} = φ_{n}(< e_{1}^{∗}, · >, …, < e_{n}^{∗}, · >) (mod μ).

It follows from lemma 6 that φ_{n} ∈ W_{p}^{1}(R^{n}, μ_{n}), consequently exists φ̃_{n} = φ_{n} (mod μ_{n}) which is Lipschitz with a constant

∥DF ∥_{L_{∞}(X,μ,H)} + ∥F ∥_{L_{∞}(X,μ)} · ∥E^{σ_{n}}P_{n}λ − P_{n}λ∥_{L_{∞}(X,μ,H)}.

Therefore F_{n} is H-Lipschitz μ-a.a. with the same constant. An approximation of function F by E^{σ_{n}}F completes a proof in a case when F ∈ L_{∞}(X, μ).

The general case reduces to F ∈ L_{∞}(X, μ) by the approximation F = lim_{n→∞} ψ_{n}(F), where

ψ_{n} = \{ \begin{matrix}t, |t| ≤ n \\ n · sign t, |t| > n. \end{matrix}

Remark 1. This idea of the proof was realized in Wiener case in [12]. Remark 2. Wiener measure and the measure of example 4 are satisfied

the conditions of theorem 4. If we take in this case i^{∗}(e_{n}^{∗}) = \sqrt{2} cos nπt then E^{σ_{n}}P_{n}λ − P_{n}λ = 0.

Theorem 5. (Holder condition for solutions’s trajectories). Let conditions of theorem 1 and 4 are fulfilled and η ∈ W_{p}^{1}, a(t) ∈ W_{p}^{1}, t ∈ [0, 1],

Dη ∈ L_{∞}(X, μ, L_{∞}[0, 1]),

∫_{0}^{1} ∥Da(t)∥_{L_{∞}(X,μ,L_{∞}[0,1])}dt < ∞.

Suppose there exists α > 0 such that for all c > 0

sup_{∥h∥_{H}≤c} E |\frac{dμ(· − i(h))}{dμ}|^{1+α} < ∞.

Then there exists a modification of the solution (8) of the equation (6) with trajectories satisfying Holder condition with an index ρ, for every ρ < \frac{1}{2}.

Holder condition for L_{t} follows from Kolmogorov’s theorem, the formulae for the density of shifted measure in terms of logarithmic derivative[11] and assumption, that < λ, h >∈ L_{p}(X, μ) for all h ∈ H, p > 1. Application of the theorem 4 completes the proof.

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