Let (2, F, P) be a probability space and let {W,:t>0} be a standard Wiener process....

 

Transcribed Image Text:

Let (2, F, P) be a probability space and let {W,:t>0} be a standard Wiener process. The following stochastic Ito integral with respect to a standard Wiener process can be defined as n-1 I₁ = - [ f(W., s)dW. = lim [ƒ(We, ts) (Wers. — W₁.) 11-700 i=0 where f is a simple process, and t; = it/n, 0 = to <t₁ <t₂ <..., <t₁_1 < t₁ = t₁ nEN. Using the properties of a standard Wiener process, show that Elf(W,, s)dw.) = 0 and deduce that if {M, :t> 0} is a martingale then El * 9(M₂, s)dM₂] = 0, where g is a simple process. (b) (10p) Suppose X, follows the generalized SDE dX = (X₁, t)dt + o(X₁, t)dWt where μ and o are functions of X, and t. Show that X, is a martingale if μ(X₁, t) = 0. Let (2, F, P) be a probability space and let {W,:t>0} be a standard Wiener process. The following stochastic Ito integral with respect to a standard Wiener process can be defined as n-1 I₁ = - [ f(W., s)dW. = lim [ƒ(We, ts) (Wers. — W₁.) 11-700 i=0 where f is a simple process, and t; = it/n, 0 = to <t₁ <t₂ <..., <t₁_1 < t₁ = t₁ nEN. Using the properties of a standard Wiener process, show that Elf(W,, s)dw.) = 0 and deduce that if {M, :t> 0} is a martingale then El * 9(M₂, s)dM₂] = 0, where g is a simple process. (b) (10p) Suppose X, follows the generalized SDE dX = (X₁, t)dt + o(X₁, t)dWt where μ and o are functions of X, and t. Show that X, is a martingale if μ(X₁, t) = 0.

Answer rating: 100% (QA)

To prove the given statement integral0s gWt dWt 0 We can start by expanding the stochastic ... View the full answer