Noncausal Problems In Stochastic Calculus
Noncausal Problems in Stochastic Calculus
Shigeyoshi OGAWA
Department of Mathematics, Ritsumeikan University, Kusatsu-shi, Shiga
525-8577, Japan
[ogawa-s@se.ritsumei.ac.jp]
The stochastic calculus is a calculus with respect to an underlying basic stochastic
process, like the Brownian motion, say Zt t ∈ I. It concerns the diferentiation
and integration with respect to the Zt of such random functions that appear as
functionals of the {Zt, t ∈ I}. The stochastic calculus originated by K.Itˆ
o in
1942 is founded on the fundamental Hypothesis of Causality, saying that; every
random function f (t, ω) should be adapted to the increasing familly of σ-fields Ft
generated by the Zt. The hypothesis seems well fit to the principle of causality in
physical sciences, where the variable ”t” appears as time parameter. Moreover it
endowes the theory a remarkable situation of being in natural concordance with
the notion of martingale which plays indeed an essential role in Itˆ
o’s Calculus.
Nevertheless the hypothesis of Causality gives a disagreable shade on the ap-
plicability of the causal theory of stochastic calculus. This can be seen immediately,
for example when we think of the case that ”t” stands for the space parameter,
or in such case where the parameter ”t” is multi-dimensional (that is, ”a stochas-
tic calculus” for the random field, [2]). The notion of Causality looses its sound
meaning in such cases because of the lack of natural sense of time direction. Even
in the case of physical problems where ”t” appears as time parameter, we can find
various situations of noncausal nature, such as the Cauchy problem in the theory
of Brownian particle equations [3], noncausal version of the Black-Sholes model in
Mathematical Finance [4],the White noise analysis [1] etc. These were the moti-
vations for the author to introduce the noncausal theory of stochastic calculus in
1979, based on the noncausal stochastic integral which is often refered by author’s
name.
In this talk we will give a unified sketch of the noncausal theory of stochastic
calculus as well as of its recent development. We will also refer to some typical
applications of the theory to mathematical sciences.
1. Ogawa, S. (1979). Sur le produit direct du bruit blanc par lui- mˆeme, C.R.Acad.Sci.Paris,
S´
erie-A t.288, 359–362.
2. Ogawa, S. (1991). On a stochastic integral equation for the random fields, S´eminaire
de Proba. vol.xxv, Springer Verlag, Berlin, 324–339.
3. Ogawa, S. (2001). On the Brownian particle equations and the noncausal stochastic
calculus, Rendiconti Acad.Nazionale delle Scienze detta dei XL,119 vol.XXV, 159–174.
4. Ogawa, S. (2004). On noncausal Cauchy problem for the noncausal SDEs, In: S.Watanabe
et al. (Eds.): Stochastic Processes and Applications to Mathematical Finance. World
Scientific, Singapore, 289–304.
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