What is transition probability density function

What is transition probability density function? Explain the term with forward and Backward Equations.

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The transition probability density function p(y, t; y’, t’) is the function of four variables denoted by
Prob(a < y < b at time t’y at time t) This simply implies the probability as the random variable y lies among a and b at time t’ within the future, specified that this started out with value y at time t. You can think of y and t as being current or starting values with y’ and t’ being future values.

The transition probability density function is p(y, t; y’, t’) suits two equations, one involving derivatives regarding the future state and time (y’ and t’ ) and termed as the forward equation, and the other involving derivatives respecting the current state and time (y and t) and termed as the backward equation. Those two equations are parabolic partial differential equations different to the Black–Scholes equation.

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