By A. E. Bashirov (auth.), Uluğ Çapar, Ali Süleyman Üstünel (eds.)

Over the final years, stochastic research has had a huge development with the impetus originating from varied branches of arithmetic: PDE's and the Malliavin calculus, quantum physics, direction house research on curved manifolds through probabilistic tools, and more.

This quantity comprises chosen contributions which have been provided on the eighth Silivri Workshop on Stochastic research and similar issues, held in September 2000 in Gazimagusa, North Cyprus.

The subject matters comprise stochastic regulate thought, generalized capabilities in a nonlinear environment, tangent areas of manifold-valued paths with quasi-invariant measures, and purposes in video game concept, theoretical biology and theoretical physics.

Contributors: A.E. Bashirov, A. Bensoussan and J. Frehse, U. Capar and H. Aktuglul, A.B. Cruzeiro and Kai-Nan Xiang, E. Hausenblas, Y. Ishikawa, N. Mahmudov, P. Malliavin and U. Taneri, N. Privault, A.S. Üstünel

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Viot, Optimal control of stochastic linear distributed parameter systems, SIAM J. Control, 13 (1975), 904-926. [5] RF. Curtain and A. Ichikawa, The separation principle for stochastic evolution equations, SIAM J. Control and Optimization, 15 (1977), 367-383. E. Bashirov, Optimal control of partially observable systems with arbitrarily dependent noises, Stochastics, 17 (1986), 163-205. [7] E. Hille and RS. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. ColI. , Vol. , 1957. [8] RS.

9) then the system fl, A, P, Px,v, wx,v(t) forms a probability system in which wx,v(t) is an P standardized Wiener process. (t))dt + dWx,v(t), x(O) = x. 10) p. 11) and let Tx = inf{tlx(t) ~ O}. 12) We shall stop the process x(t) at the exit of the domain 0, and to save the notation, we shall still denote by x(t) the stopped process. Let also fv(x) be a scalar measurable bounded function. 14) where VV(t) represents all components different from VV. (t). )) rx 1 = J1 log Ex,v exp15 [10 Uv(x(t)) + "2lvv(tW + (}vv(t) .

Bensoussan and J. 3) Pv = LPw The first step is to consider, for a given p, a Nash point in v for the function Lv(v,p). L Pv + (N -1)(}) - 1- (). 7) (). 9) and also Pv = -(N - l)(}vv(p) + (-N() + 2(} - l)vv(p). 10) In particular, we can write 1 Lv(p) = -"2lvv(p)1 2 _ + Pv . vv(p). L 1-2() 2 2(}-1 " + 2(1- (})2 IPvl + (1- (})2(1 + (N _l)(})Pv. 2. More developments on Lagrangians We continue some useful developments on Lagrangians. 13) 2(1 - 0)2(1 + (N _ 1)0)2 ~ p" " 2 (3N - 4)0 - 2(N - 3)0 - 2 ' " +2 2(1 _ 0)2(1 + (N _ 1)0)2 ~ Pll-· p".