The exponential function is available in bipolar technology using the exponential characteristic of the bipolar transistor.
By virtue of the exponential martingale inequality, for any positive constants T, δ, β, we have.
By the exponential martingale inequality [3 10], we can show that for every integer, (4.9).
Consider the exponential Martingale process SDE (3): begin{aligned} {left{ begin{array}{ll} hbox {d}Y=lambda (t)Yhbox {d}W_{t}, Y 0)=Y_0.end{array}right.
end{aligned}From this equality we could conclude that (Xmathcal {Z}_t^{lambda }), is the exponential Martingale (mathcal {Z}_t^{lambda +sigma }).
By virtue of the exponential martingale inequality, for any positive constants T, δ, β, we have P { sup 0 ≤ t ≤ T [ M ( t ) − δ 2 〈 M ( t ), M ( t ) 〉 ] > β } ≤ e − δ β.
The exponential Martingale process associated with (lambda (t)) is defined as follows: begin{aligned} mathcal {Z}_t^{lambda }=exp left( int _0^tlambda (s),hbox {d}W_s-frac{1}{2}int _0^tlambda ^2(s),hbox {d}sright).
By the exponential martingale inequality (see, e.g., [36]), for each (kgeq1), P biggl{ sup_{0leq tleq k} biggl[M_{i}(t)- frac{varepsilon}{2} bigllangle M_{i}(t),M_{i}(t bigrrangle biggr]>frac{2ln k}{varepsilon} biggr} leq k^{-2}.
Using the reverse martingale can be risky if you lose.
Using the property of martingale and Gronwally' inequality, we obtain some conditions to guarantee that the complex network can realize mean square synchronization and mean square exponential synchronization, respectively.
