Stochastic differential - notation

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Assume we have processes $A,B,C$ s.t. $A=\int_0^tB_udC_u$ where $C$ is of bounded variation. Now, there is a notation to write this equation as $dA_t=B_tdC_t$. I saw somewhere that one can now go on to define B as $B_t=\frac{dA_t}{dB_t}$. How is this justified and what does $\frac{dA_t}{dC_t}$ mean?