ADOMIAN DECOMPOSITION METHOD APPLIED TO CONTINUOUS-TIME BILINEAR STOCHASTIC PROCESSES WITH TIME-VARYING COEFFICIENTS

. In the present paper, we apply the Adomian decomposition method for bilinear stochastic processes with time-varying coefﬁcients in both time and frequency domain. More precisely, we derived an analytical approximate solution and we prove its convergence to the exact solution in time domain, furthermore we give an analytical approximate solution in frequency domain, i.e, we derived analytical approximate transfer functions which converge to the exact transfer functions.

The general principle is based on the so-called 'reversion method' which consists of decomposing the solution of a nonlinear functional equation in a series of functions elegantly computed [12,15]. The method can be applied directly for different type of (stochastic) differential equations providing a solution with fast speed of convergence [7,9,10,14,16]. So, the aim of this Note is to apply the ADM (as an alternative of Itô solution) for solving a continuous-time bilinear (COBL) equation defined as a stochastic differential equations (SDE) with time varying coefficients in time and frequency domain.

DECOMPOSITION METHOD APPLIED TO COBL(1, 1) IN TIME-DOMAIN
Without loss of generality we assume that β (t) = 0, for all t, since this assumption can be fulfilled by the transformation with some additional assumptions on the differentiability of the functions γ (t) and β (t). The Adomian's method suggest that the solution process X(t) of SDE (1) is sought in the form and L is the operator defined by L −1 [.] = t 0 (.)ds. So we have by identification (5) X 0 (t) = η + t 0 µ(s)ds and ∀n ≥ 1, More precisely, we have  (α(s 1 ) + γ(s 1 )w (1) (s 1 ))...(α(s n ) + γ(s n )w (1) (s n ))µ(s n+1 )ds n+1 ds n ds n−1 ...ds 2 ds 1 Finally, we approximate the solution by the truncated series In order to show the convergence of the series (7) to the exact solution processes we introduce the concept of the so-called stochastic exponential associated with a centred Gaussian random variable Z and defined by E {Z} = exp Z − 1 2 E Z 2 . First, we have the following lemma Proof. By induction, we have for n = 2, Theorem 3.2 (Theorem 1, El-Kalla [13] ). If q(t) is integrable and r(t) is derivable functions, We are now in a position to prove the convergence of the approximate solution Φ N (t) to the exact solution process is the approximate solution (7) given by Adomian decomposition method of the COBL(1,1) driven by the equation (1), then almost surely, we have Proof. Applying Lemma 3.1 on the first term of X n (t), we obtain and Applying Theorem 3.2 on the infinite sum of second term of X n (t), we obtain Then we can obtain where E t 0 (α(s) + γ(s)w (1) (s))ds denotes the stochastic exponential of t 0 (α(s) + γ(s)w (1) (s))ds, so we get which it is the exact Itô solution process (11) of SDE (1) and the proof is complete.

DECOMPOSITION METHOD IN FREQUENCY DOMAIN
In frequency domain, the process (X(t)) t∈R admits the so-called Wiener-Itô orthogonal representation (10) λ i and the integrals are multiple Wiener-Itô stochastic integrals with respect , (see Major [19] and Dobrushin [11]). and f (t, λ (r) ) are referred as the r − th evolutionary transfer functions of (X(t)) t∈R , uniquely determined and fulfill the condition ∑ r≥0 1 r! R r f (t, λ (r) ) 2 dF(λ (r) ) < ∞. As a property of the representation (10) is that for any f (t, λ (n) ) and f (s, λ (m) ), we have where δ m n is the delta function and f t, λ (r) is the symmetrized version of f by the vector λ (r) which is the average of those values of f taken by all possible permutations of entries of λ (r) .
We need also the diagram formula given as follow Theorem 4.1. Assume that the process (X(t)) t≥0 generated by the SDE (1) has a regular second-order solution. Then the evolutionary symmetrized transfer functions f (t, λ (r) ), (t, r) ∈ R × N of this solution are given by, Proof. See Bibi and Merahi [8].
Remark 4.4. Theorem 4.3 means that, ∀n ≥ 0 there exists a process (χ n (t)) t∈R admits the spectral representation such that

CONCLUSION
In this paper we propose the ADM for solving bilinear SDE driven by Brownian motion wich behaves as a nonlinear stochastic differential equation. This method is applied in time and frequency domain where we have proved the convergence of the approximate processes to the exact solution.