By Makoto Maejima, Tokuzo Shiga

A range of Hiroshi Tanaka's amazing works on stochastic techniques and similar themes. For researchers and graduate scholars in chance concept, research and mathematical physics.

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**Extra resources for Stochastic processes: selected papers of Hiroshi Tanaka**

**Sample text**

4] H U N T , G. : Some theorems concerning Brownian motion. Trans. Amer. m a t h . Soc. 81, 294—319 (1956). [5] I T 6 , K . : On stochastic differential equations. Mem. Amer. math. Soc. No. 4. [6] — On a formula concerning stochastic differentials. Nagoya math. J . 3, 55—65 (1951). [7] —, and H. P . M C K E A N : Diffusion. : Diffusion process corresponding to \ 2 d2Jdxi' + 2 bi{x)djdxi. Ann. Inst. statist. Math. 12, 37—61 (1960). [9] VENTSEL, A. D . : Additive functionals of several dimensional Wiener process.

D is now identified as a diffusion; that it has Brownian hitting probabilities is clear, and to complete the discussion, it suffices to verify that it has e as its speed measure. 15 mrf = min (t: x~\f) 6 BD) = f(m8D), and f has e as its associated measure. 13. GENERATORS. [min\x(t)\^>n] = P. [min | #(£) | > « ] tends to 1 at oo, '2:° <2:° 1 so that G t f / tends to ar ^oo), and, if a 6 i ? 3 constant x (1 — /*) < constant x r 1 . 4 Ga-G&+(a-^)GaG^ =0 a,j8>0, it is evident that Ga maps our space of fine continuous functions onto some subspace D(&) independent of oc and that its null-space CS^CQ) is likewise independent of a.

133, 2 6 9 - 2 7 2 (1960). [3] — Additive functionals of a Wiener process determined b y stochastic integrals. Teor. Verojatn. Primen. 5, 441—452 (1960). [4] H U N T , G. : Some theorems concerning Brownian motion. Trans. Amer. m a t h . Soc. 81, 294—319 (1956). [5] I T 6 , K . : On stochastic differential equations. Mem. Amer. math. Soc. No. 4. [6] — On a formula concerning stochastic differentials. Nagoya math. J . 3, 55—65 (1951). [7] —, and H. P . M C K E A N : Diffusion. : Diffusion process corresponding to \ 2 d2Jdxi' + 2 bi{x)djdxi.