THE PRESENT STUDY INCLUDES INTRODUCE ITO’S stochastic differential EQUATION AND APPLICATION OF stochastic differential EQUATION IN METROLOGY. ESTIMATION OF stochastic PROCESS HAS BEEN UNDERTAKEN IN THE PRESENT INVESTIGATION. THE APPROXIMATE SOLUTION OF ITO’S FORMULA IS CALCULATED AND THE GRAPH OF SOLUTIONS HAVE BEEN SHOWN IN RESULTS. THE MODEL HAS BEEN COMPARED WITH THE ACTUAL PHENOMENA IN NATURE AND FOUND THAT, THE PRESENT MODEL IS COMPATIBLE WITH THEM.

IN THIS PAPER, WE STUDYG-BACKWARD stochastic differential equations WITH RANDOM TERMINAL TIME. WE EXPLAIN HOW TO EXTEND THE RESULTS OF THE CASE OF FIXED TERMINAL TIME TO THE CASE OF A RANDOM TERMINAL TIME. WE PRESENT THE EXISTENCE AND UNIQUENESS OF A SOLUTIONS FORG-BACKWARD stochastic differential equations WITH A RANDOM TERMINAL TIME.

IN THIS PAPER, WE WILL PRESENT A NEW ALGORITHM FOR SOLVING ITO stochastic differential equations (SDES) IN CONTINUOUS PIECEWISE POLYNOMIAL SPACE. IN THIS APPROACH, WE WILL EMPLOY PIECEWISE COLLOCATION METHOD FOR DRIFT AND DIFFUSION TERMS OF THE GIVEN EQUATION. CONVERGENCE ORDER OF THE METHOD IS INVESTIGATED AND SOME NUMERICAL EXAMPLES ARE CONSIDERED TO DEMONSTRATE THE EFFICIENCY AND ROBUSTNESS OF THE METHOD.

IN THIS PAPER FOR THE NUMERICAL APPROXIMATION OF stochastic ADVECTION DIFFUSION equations OF IT O TYPE, AN IMPLICIT stochastic FINITE DIFFERENCE SCHEME IS CONSTRUCTED. IN CONTINUATION THE MAIN PROPERTIES OF stochastic DIFFERENCE METHODS, I.E. CONSISTENCY, STABILITY AND CONVEGENCY, ARE ESTABLISHED FOR PROPOSED stochastic DIFFERENCE SCHEME (SDS).

AN EFFICIENT NUMERICAL METHOD IS PROPOSED FOR SOLVING NONLINEAR stochastic differential equations, USING OPERATIONAL MATRIX OF BLOCK PULSE FUNCTIONS. BY USING THIS APPROACH, THE stochastic differential EQUATION REDUCES TO A NONLINEAR SYSTEM OF ALGEBRAIC equations WHICH CAN BE SOLVED BY NEWTON, S ITERATIVE METHOD. ACCURACY AND EFFICIENCY OF THE METHOD ARE SHOWN WITH AN EXAMPLE.

THE PRESENT ARTICLE FOCUSES ON THE USE OF A DIFFERENCE SCHEME IN ORDER TO APPROXIMATE THE SOLUTIONS OF stochastic PARTIAL differential equations (SPDES) OF ITO {TYPE. THE MAIN PROPERTIES OF DETERMINISTIC DIFFERENCE METHODS, I.E. CONSISTENCY AND STABILITY ARE INVESTIGATED FOR THE stochastic CASE. FROM stochastic VERSION OF LAX-RICHTMYER THEOREM THE CONVERGENCE OF THIS SCHEME IS ESTABLISHED.