Then it follows from Theorem 3.3 that the GRE (2.6) admits a regular solution (P. To determine u ∗, let (X ∗,Y ∗,Z ∗ be the adapted solution of (3.1).
In previous work, Ma et al. (1994) studied the solvability of the adapted solution to the FBSDEs, in particular, they designed a direct scheme, called the four step scheme to solve the FBSDEs explicitly.
Different from classical WM method, the weighting factor of retrieved case for each solution element adaptation is calculated by multiplying similarity matrix (SM) and relational matrix (RM), and the adapted solution values of new mechanical product are subsequently obtained by calculating the weighted average of solution values of K similar cases.
end{array} end{aligned} Since the adapted solution of (4.1) satisfies begin{aligned} left{!! begin{array}{ll} d eta,-,mathbb{E}[eta])!=-!left{(A,+,BTheta)^{top}(eta,-,mathbb{E}[eta]),+,(C,+,d eta,-,mathbb{E}[eta]athbb{E}[zeta]),+,(C,+,DTheta)^{top} P(sigma,-,mathbb{E}[sigma])right.
(ii) The adapted solution to the BSDE left{ begin{array}{ll} deta(s)=-left[(A+BTheta)^{top}eta+(C+DTheta)^{top}zeta+(C+DTheta)^{top} Psigmaright. left.qquadqquad~~~+Theta^{top}rho+Pb+qright]ds+zeta dW s),qquad sin[t,T], eta(T =g, end{array}right.
To transfer the developed technologies, the focus was set on the exemplary implementation of the adapted solutions.
The time-step is controlled by the interpolation error and preserves the accuracy of the mesh adapted solution.
Therefore, for every given (u(cdot)inmathcal{U}_{mathrm{ad}}), under assumptions (H3.1) and (H3.2), Eq. (3.1) admits the unique adapted solution (X cdot)in L_{mathbb{F}}^{2} -delta,T;mathbb{R} ^{2} -delta,
Then, if δ is sufficiently small, the anticipated SDE textstylebegin{cases} dX t) = b t,X t),X(t+delta)),dt + sigma t,X t),X(t+delta)),dW t), & 0leq tleq T; X 0) = x_{0},qquad X t) = lambda(t), & T< tleq T + delta, end{cases} (2.3) has <span class="lh">the unique adapted solution (X cdot)in L_{mathbb{F}}^{2}(0,T+ delta mathbb{R}^{n})).
Underlying this is the idea that the evolutionary forces of mutation, recombination, reproduction and selection can drive the population towards better adapted solution over time, effectively solving an optimization problem by navigating the fitness landscape they inhabit.
Given fixed ((t,x in[0, T]timesmathbb{R}), we note that (Phi (t,x =Y^{t,x}_{t}) is an adapted solution of the anticipated BSDE (14).
