QUESTION 2 (23 marks) Suppose X, satisfies the Stochastic differential equation: dxt = (3 + 2t...
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QUESTION 2 (23 marks) Suppose X, satisfies the Stochastic differential equation: dxt = (3 + 2t - 3 cos 3t) dt + XdWt. (a) Derive a stochastic differential equation for the process Y₁ = 2t³e-3X₁ Use appropriate substitutions, where necessary, to find the simpliest stochastic differential in terms of Y, only. (7 marks) (b) Derive a stochastic differential equation for the process Y₁ = (X₁ + 1) ¹. Use appropriate substitutions, where necessary, to find the simpliest stochastic differential in terms of Y, only. (7 marks) (7 marks) (c) Find a solution for X, where X, is 2 at t = 0. (d) What is the solution for Y, for the stochastic differential equation derived in (b) where Yo = 81. (2 marks) QUESTION 2 (23 marks) Suppose X, satisfies the Stochastic differential equation: dxt = (3 + 2t - 3 cos 3t) dt + XdWt. (a) Derive a stochastic differential equation for the process Y₁ = 2t³e-3X₁ Use appropriate substitutions, where necessary, to find the simpliest stochastic differential in terms of Y, only. (7 marks) (b) Derive a stochastic differential equation for the process Y₁ = (X₁ + 1) ¹. Use appropriate substitutions, where necessary, to find the simpliest stochastic differential in terms of Y, only. (7 marks) (7 marks) (c) Find a solution for X, where X, is 2 at t = 0. (d) What is the solution for Y, for the stochastic differential equation derived in (b) where Yo = 81. (2 marks)
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