Introduction to Infinite Dimensional Stochastic Analysis

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In addition, this type theorems are basis of applied mathematics such as a vector analysis and a foundation of modern geometry. This is often compared to scan a house only by walking around it after putting a suitable function. However, such situation completely differs in infinite dimensional spaces.

For example, we have no natural "volume" to define the integral.

Stochastic Optimal Control in Infinite Dimension

Also we cannot consider natural "shift" group action which fits to the volume-like concept. But they finally met together in the studies of probability theory in the late of 's, and we got a satisfactory differentialintegral calculus in infinite dimensional spaces based on measure theory. This theory is often named after a mathematician who made a breakthrough in this field, the Malliavin calculus. This theory states that it holds a formula of partial integration which is a general form of the elementary theorem of calculus but had difficulty for bounded domains to say something like divergence theorems.

Stochastic Optimal Control in Infinite Dimension |

It is quite contrastive that we first study for bounded sets and then had difficulty to extend to infinite region in finite dimensional cases. This comes from the fact that, in natural spaces say, we may consider the "size" of vectors , bounded closed sets are compact which means they behave nicely if and only if the space is finite dimensional, that makes completely difficult to formulate divergence theorems. Although we have had such a theorem for a smooth region like a ball, we want to formulate in a non-smooth region like a rectangle. When Einstein studied Brownian motion theoretically in the beginning of 20th century, they started to formulate the motion in a mathematical way.

Once we succeeded in constructing such a fundamental object like a Brownian motion, it is natural to ask for other diffusion processes to construct in a mathematically rigorous way.

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But it had been a hard task, and Kolmogorov tried to formulate it as a problem of certain partial differential equations but unfortunately unsatisfied. But in this moment they required some smoothness for coefficients.

Later, the conditions to have solutions are weakened by, among others, Maruyama and Skorokhod in late of 's. Statistical Analysis of Financial Data in R.

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nisgioukandeo.gastic analysis: Martingale processes (sub-supermartingale)