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Differential Equations and Their Applications Part 2

Tham khảo tài liệu 'differential equations and their applications part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 10 Chapter 1. Introduction We are now ready to give the formal description of an FBSDE. Let US consider an FBSDE in its most general form idX t b t X t Y t z t dt ơ t xụ Y t Z t dW t dY t h t X t Y t Z t dt a t X t Y t z t dw t X x Y T g X T . Here the initial value X of X - is in IR and b ơ h d and g are some suitable functions which satisfy the following Standing Assumptions- denoting M ET X Hl 1 X Hlz one has b G Ljr o T- W1 OO M- Rn ơ G L o T w1 00 Rnx 2.6 h G L 0 71 W1 oo M R n â G L 0 T W1 oo M IRmx i e Definition 2.1. A process X - y - Z -Ỵ G M 0 T is called an adapted solution of 2.5 if the following holds for any t G 0 7 almost surely 2.7 X t x b s X s Y s Z s ds Jo ơ s X s Y s Z s dW s Jo Y t g X T - h s X s Y s Z s ds - Ị S s X s Y s Z s dW s . Furthermore we say that FBSDE 2.5 is solvable if it has an adapted solution. An FBSDE is said to be nonsolvable if it is not solvable. In what follows we shall try to answer the the following natural question for given b ơ h ơ and g satisfying 2.6 and for given X G Rn is 2.5 always solvable In fact what makes this type of SDE interesting is that the answer to this question is not affirmative although the standing assumption 2.6 is already quite strong from the standard SDE point of view. 3. Some Nonsolvable FBSDEs In this section we shall first present some nonsolvability results and then give some necessary conditions for the solvability. It is well-known that two-point boundary value problems for ordinary differential equations do not necessarily admit solutions. On the other hand an FBSDE can be viewed as a two-point boundary value problem for stochastic differential equations with extra requirement that its solution is adapted solely to the forward filtration. Therefore we do not expect the general existence and uniqueness result even under the conditions that are 3. Some nonsolvable FBSDEs 11 usually considered strong in the SDE literature for instance the uniform Lipschitz conditions. The following result is