By Jiongmin Yong, Xun Yu Zhou

The utmost precept and dynamic programming are the 2 most typically used methods in fixing optimum keep an eye on difficulties. those ways were constructed independently. The topic of this e-book is to unify those ways, and to illustrate that the viscosity resolution conception presents the framework to unify them.

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**Extra resources for Stochastic Controls: Hamiltonian Systems and HJB Equations (Stochastic Modelling and Applied Probability 43) **

**Example text**

S. , and X(t A 7- ) is Tt -measurable. 7]. 12). Sometimes, we need to generalize the notion of martingale. 9. Tt } t >0-martingale for each j. We point out that every martingale is a local martingale, but it is not necessarily true the other way around (see Karatzas-Shreve [3, p. 168] for an example). We will mainly restrict ourselves to the finite interval [0, T] in this book. A local martingale X(t) restricted to [0, T] becomes a local martingale on [0, T]. We may also define a local martingale on [0, T] directly.

For any w E St, the map t i-- X (t, w) is called a sample path. In what follows, we let I = [0 , 7] with T > 0, or I = [0 , oo) . ), X(t), or even X to denote a stochastic process. 1) 1 Ft1 (x 1 )--1P{X(t 1 ) xi}, Ft1 ,t2 (xi,x2)a-P{X(ti) < xi, X(t2) < x21, ,x3 ) 12' P{X(ti) < ,X(t 3 ) Here, ti E I, r• Rm , and X(t) < x, stands for componentwise inequalities. 1) are called the finite - dimensional distributions of the process X(t). This family of functions satisfies the following conditions: (a) Symmetry: If {i1,...

The following result of Kolmogorov gives a positive answer to this question. Let F = {Ft1 ,. (xi, , x3 ), j > 1} be a family of functions satisfying the symmetry and compatibility conditions. Then there exists a probability space ( 1 2, F , P) and a stochastic process X(t) whose finite-dimensional distributions coincide with F. 2. Chapter 1. Basic Stochastic Calculus 16 For a proof, see Parthasarathy [1, pp. 143-144]. In what follows, any stochastic process will be called simply a process if no ambiguity should arise, and the probability space (0, F , P) and the time interval [0, T] will be fixed unless otherwise stated.