Problem 3 in the introduction is a special case of the following general filtering problem: Suppose the state X
t
∈ R
n at time t of a system is given by a stochastic differential equation
EquationSource%MathType!MTEF!2!1!+-
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% aacaWG0baabeaaaaa!4B88!EquationSource$$\[\frac{{d{X_t}}}{{dt}} = b(t,{X_t}) + \sigma (t,{X_t}){W_t}\]$$
((6.1))
where b : R
n+1 → R
n,σ : R
n+1 → R
n×p satisfy conditions (5.14), (5.15) and W
t
is p-dimensional white noise. As discussed earlier the Ito interpretation of this equation is
EquationSource%MathType!MTEF!2!1!+-
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% amiDaaqabaaaaa!5449EquationSource$$\[(SYSTEM)\quad d{X_t} = b(t,{X_t})dt + \sigma (t,{X_t})d{U_t}\]$$
((6.2))
where U
t
is p-dimensional Brownian motion. We also assume that the distribution of X
0 is known and independent of U
t
. Similarly to the 1-dimensional situation (3.20) there is an explicit several-dimensional formula which expresses the Stratonovich interpretation of (6.1):
EquationSource% MathType!MTEF!2!1!+-
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% GaamiDaaqabaaaaa!4D86!EquationSource$$\[d{X_t} = b(t,{X_t})dt + \sigma (t,{X_t}) \circ d{B_t}\]$$
in terms of Ito integrals as follows:
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EquationSource$$\[\begin{array}{l} d{X_t} = \widetildeb(t,{X_t})dt + \sigma (t,{X_t})d{U_t},\;where \\ {\widetildeb_i}(t,x) = {b_i}(t,x) + \frac{1}{2}\sum\limits_{j = 1}^p {\sum\limits_{k = 1}^n {\frac{{\partial {\sigma _{ij}}}}{{\partial {x_k}}}} } {\sigma _{kj}};1 \le i \le n \\ \end{array}\]
((6.3))
(See Stratonovich (1966)). From now on we will use the Ito interpretation (6.2).
