By Dr. Peter A. Ruymgaart, Professor Tsu T. Soong (auth.)

The moment variation has no longer deviated considerably from the 1st. The printing of this variation, even if, has allowed us to make a few corrections which escaped our scrutiny on the time of the 1st printing, and to often enhance and tighten our presentation of the cloth. a lot of those alterations have been urged to us via colleagues and readers and their kindness in doing so is tremendously preferred. Delft, The Netherlands and P. A. Ruymgaart Buffalo, big apple, December, 1987 T. T. Soong Preface to the 1st version seeing that their creation within the mid Nineteen Fifties, the filtering options constructed via Kalman, and through Kalman and Bucy were widely recognized and typical in all parts of technologies. beginning with functions in aerospace engineering, their influence has been felt not just in all parts of engineering yet as all additionally within the social sciences, organic sciences, clinical sciences, besides different actual sciences. regardless of all of the strong that has pop out of this devel opment, despite the fact that, there were misuses as the concept has been used regularly as a device or a strategy by means of many utilized employees with out totally realizing its underlying mathematical workings. This ebook addresses a mathematical method of Kalman-Bucy filtering and is an outgrowth of lectures given at our associations due to the fact that 1971 in a series of classes dedicated to Kalman-Bucy filters.

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11. s. differentiable at tel, then EX : I ~ R is differentiable at t and ; EX(t) Proof. = EX' (t) . 9) Following (A) of Sect. , EX(t + h) - EX(t) ~ EX' (t) . 9). o Let both X and Y be mappings of I into L2 (Q) whose random variables are, respectively, X(t) and Y(t), tel. 12. s. s. differentiable with :t [aX(t) + bY(t)] = aX' (t) + bY' (t) . Proof. 3 by replacing X(s) by [X(s) - X(t)]/(s - t) and Y(s) by [Y(s) - Y(t)]/(s - t) . 0 38 2. 13. If X: I ~ L2 (Q), then Xo = X - EX is a centered stochastic process.

If a :s:; c :s:; b and if c b f f(t)dX(t), f f(t) dX (t) , a and a b f f(t)dX(t) c exist, then b c a a ff(t)dX(t) = ff(t)dX(t) b fX(t)df(t) a b + ff(t)dX(t) c b a c = fX(t)df(t) + JX(t)df(t). 27) 44 2. Calculus in Mean Square Proof. Let {Pn [a, {Pn [c, bDneN and CDneN be convergent sequences of partitions of [a, c] and [c, b], respectively. Then, is a convergent sequence of partitions of [a, b]. 26). 27) is proved analogously. D We shall omit the easy proof of the next theorem. 19. s. integrals given below exist and b f[pf(t) + qg(t)]dX(t) a b ff(t)d[pX(t) + qY(t)] a b b b b a a = P ff(t)dX(t) + q f g(t)dX(t) a a = P ff(t)dX(t) + q ff(t)dY(t) Similar equalities hold for integrals of the type b f X(t)df(t) .

S. s. 22. Now, to show equality of the integrals, we may use the same convergent sequence {Pn}neN of partitions of [to both integrals. 16). Then, SX,f(Pn) is the R-S sum corresponding to the first integral. )[[(ti ) i= I [(ti- I)] k = L X (ti')1' (ti')(ti i= I ti-I) . ' (Pn) corresponding to the second integral. 44) follows as n _ 00 . 24. s. continuous on I = [a, b]. s. s. continuous) on I and d t -d f X(s)ds t a = X(t), tel. 45) Proof. Suppose t and t + h, h > 0, are in I. We have /I! [tyX(S)ds -iX(S)ds]-X(t) " !