Write down a formula for (ar) where X-N(4.0) and, by differentiating. or otherwise, derive an expression...

   

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Write down a formula for (ar) where X-N(4.0²) and, by differentiating. or otherwise, derive an expression for E(Xeax). (ii) Show that: [2] X₁ =(B, -at)e-²¹, is a martingale, where B, is a standard Brownian motion, and a is an arbitrary constant. You may assume that E[X]< [5] [Total 7] Let B, (120) be a standard Brownian motion process starting with B=0. Show that, when s<1,E(B₂B)=s. Hence find a general formula for the correlation coefficient p(B.B₁₂). (i) [3] [2] [Total 5] A student has said: "If you want to find the variance of X = B(s) + B(1), where s<, for a standard Brownian motion process, you can use the fact that B(s) and B(1) are independent to get var(X)=x+1." (i) Explain why the student's argument is not correct, and find a correct expression for var(X). [4] (ii) Hence show that the general formula for var B(4) +B(₂2)] when 4-12> 0 can be expressed as 1₁ +1₂ +2min(4, 42). [2] [Total 6] (i) What is meant by saying that the process (Y) is a martingale with respect to another process {X₂}? [2] Let B, (120) be a standard Brownian motion. Show that B, and B+ kr are both martingales with respect to B,, for a suitably chosen value of the constant k, which you should specify. [6] Show that there is a value of the constant c, which you should specify, such that (a+bB, )² +ct is a martingale with respect to B,, where a and b are constants. [4] [Total 12] Write down a formula for (ar) where X-N(4.0²) and, by differentiating. or otherwise, derive an expression for E(Xeax). (ii) Show that: [2] X₁ =(B, -at)e-²¹, is a martingale, where B, is a standard Brownian motion, and a is an arbitrary constant. You may assume that E[X]< [5] [Total 7] Let B, (120) be a standard Brownian motion process starting with B=0. Show that, when s<1,E(B₂B)=s. Hence find a general formula for the correlation coefficient p(B.B₁₂). (i) [3] [2] [Total 5] A student has said: "If you want to find the variance of X = B(s) + B(1), where s<, for a standard Brownian motion process, you can use the fact that B(s) and B(1) are independent to get var(X)=x+1." (i) Explain why the student's argument is not correct, and find a correct expression for var(X). [4] (ii) Hence show that the general formula for var B(4) +B(₂2)] when 4-12> 0 can be expressed as 1₁ +1₂ +2min(4, 42). [2] [Total 6] (i) What is meant by saying that the process (Y) is a martingale with respect to another process {X₂}? [2] Let B, (120) be a standard Brownian motion. Show that B, and B+ kr are both martingales with respect to B,, for a suitably chosen value of the constant k, which you should specify. [6] Show that there is a value of the constant c, which you should specify, such that (a+bB, )² +ct is a martingale with respect to B,, where a and b are constants. [4] [Total 12]

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