Is the following process Xt a BM?X0=0, Xt= Bt + Z, where Z is independent of Ft for any t>0, and Z is a standard normal.

no

It's stationary + independent increments. So, why not?

It is not a martingale.E(Xt) <> X0 if Bt<>0

It is a Brownian Motion. It's not a standard BM though.

What is your definition of a BM?

its (a.s.) not continuous at zero: it jumps by Z.the increments are not stationary: B(h)-B(0) and B(t+h)-B(t) do not have the same distribution, t>0 & h>0.but it is a MG.

What is a MG?In your formula is Z constant or dependant of t?

It is Brownian motion with a jump discontinuity at zero. We can get the same process by defining Yt as Brownian motion with Ya = 0, where a is the opposite of the square of the standard deviation of Z divided by volatility of B. Then Xt = Yt for t > 0 and Xt = Ya for t = 0.

Brownian motion has continuous paths so it can't be a Brownian motionalso var(B_t) = t which is not true either

helloIn foreign exchange market. I use Brownian motion to simulate the price in the future(for example 1 month later).dS =(r- rf)*S*dt +sigma*S*dzI put the spot price for EURUSD as S right now,and the rate difference between USD and EUR,what does the result mean???I s it the one month forward price for EURUSD??Or it is the spot price 1 month later??Could anybody help me??THANKS

QuoteOriginally posted by: mjBrownian motion has continuous paths so it can't be a Brownian motionalso var(B_t) = t which is not true eitherNo. Var(Bt)=t for a standard BM. As for a definition of BM, you can define it as a continuous path stochastic process with independent + stationary increments. then normality will follow from CLT. Then standard BM is the one with Var(Xt)=t.JungixIn my example Z is a standard normal, where Z is independent of a standard BM Bt.But it is true, it does have discontinuity at zero.

QuoteOriginally posted by: wayneleeIn foreign exchange market. I use Brownian motion to simulate the price in the future(for example 1 month later).dS =(r- rf)*S*dt +sigma*S*dzI put the spot price for EURUSD as S right now,and the rate difference between USD and EUR,what does the result mean? Is it the one month forward price for EURUSD? Or it is the spot price 1 month later?You are in a not uncommon predictament, you did something to get a number, now you wonder what number you have. It's often easier to work the other way around, figure out what you want to know, then compute the number.Clearly you don't have the forward price in general. For one thing, you have a lot of prices, there is only one forward price. If you set sigma to zero and get things measured and defined properly, you will have the forward exchange rate.You also don't have the spot price one month later, unless you're in the Twilight Zone. If your sigma is good, you have a simple estimate of the probability distribution of the spot price one month from now. However, there are more sophisticated interest rate models that will do a better job of this, whether you are concerned with pricing derivatives or making predictions.

By the way, I have the following as an exercise but not sure if I've got a correct answer. For the standard Brownian motion Bt, what is the average time it spends in the interval [0, delta], where delta > 0. ??Now, my calculations are direct and rather long. I am pretty sure there should be some smart way of doing this.