Download Forward-Backward Stochastic Differential Equations and Their by Jin Ma PDF
By Jin Ma
This quantity is a survey/monograph at the lately constructed conception of forward-backward stochastic differential equations (FBSDEs). simple innovations resembling the strategy of optimum keep an eye on, the "Four Step Scheme", and the tactic of continuation are provided in complete. comparable issues similar to backward stochastic PDEs and lots of purposes of FBSDEs also are mentioned intimately. the quantity is appropriate for readers with uncomplicated wisdom of stochastic differential equations, and a few publicity to the stochastic keep an eye on thought and PDEs. it may be used for researchers and/or senior graduate scholars within the parts of chance, keep an eye on concept, mathematical finance, and different comparable fields.
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Extra info for Forward-Backward Stochastic Differential Equations and Their Applications
Sample text
2) Y(T) = g(X(T)). 1). 2). 52 Chapter 3. Method of Optimal Control The above observation can be viewed in a different way using the stochastic control theory. ) being the control process, and (x,y) 9 ~ n x ~ m being the initial state. 3) T = {(x,g(x)) I x 9 ]RU}. P r o b l e m (C). s. 3), at time t = T, almost surely, by choosing a suitable control Z(-) 9 Z[0, T]. In the previous chapter, we have presented some results related to this aspect for linear FBSDEs. We point out that the above controllability problem is very difficult for nonlinear case.
8). Proof. 12). We now prove the uniqueness. 12). 12), respectively. Denote Y = Y - Y and Z = Z - Z. 16) '1 + (PC - C)Z]dt + [PBIY - (I - PC1)Z]dW(t), ( Y ( T ) = O. 18) - (PC - O)(I - P C l ) - l Z}dt + ZdW(t), A Y(T) = O. N It is clear that such a BSDE admits a unique adapted solution (Y, Z) = 0 (see Chapter 1, w Consequently, Z = 0. 19). 14). Hence, we obtain the uniqueness. [] The following result tells us something more. 2. 11) holds for t E [To, T] (with some To >_ 0). 12) is uniquely solvable on [0, T].
12). 12). Let us now carry out a heuristic derivation. 12). s. where P : [0, T] --+ ]Rm• is a deterministic matrix-valued function and p : [0, T] • ~ --+ ]Rm is an {Svt)t>_0-adapted process. ) and p(-). 2) g = P ( T ) X ( T ) +p(T). 3) P(T) = O, p(T) = g. ) E L~(0, T; Rm) being undetermined. 7) (PAl + P B I P ) X + PC1Z + P B l p + q = Z. 8) Z= (I-PC1)-I{(PA1 +PB1P)X+PBlp+q}. Chapter 2. 9) + [PB - B + (PC - C)(I - P C , ) - I p B , ] p + (PC - C)(I - PC1)-lq + a. 10) BR +(PC-C)(I-PC1)-I(pAI+PB1P)=O, P(T) = O.