6 votes
Accepted

Is first order stochastic dominance conserved under change of measure?

Consider a coin independently tossed 10 times. Assume under the measure $P$, $Pr(H)$ > 0.5 but not equal to 1. Let a risk neutral person be iteratively given the gamble between getting atleast $n$ ...
  • 1,662
4 votes
Accepted

Change of measure for a stochastic process to be a martingale

Let $Y_t= e^{B_t}$ and $Z_t = B_{t}-t / 2$. Then, \begin{align*} dX_t &= Z_t dY_t + Y_t dZ_t + d\langle Y, Z\rangle_t\\ &=(B_{t}-t / 2)e^{B_t}\big( dB_t + 1/2\,dt \big) + e^{B_t}\big(dB_t -1/2\...
  • 20.5k
2 votes

Change of measure for a stochastic process to be a martingale

(Now that I saw Gordon's solution, I can finish my attempt; I had noticed $dB_t +1/2dt$ immediately from product rule for $V_t$, zero quadratic covariation between $t/2$ and $e^{B_t}$, but hours later ...
  • 5,028
1 vote

Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup

The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions. I'm transcribing the abstract below: We show a class of stochastic ...
  • 5,028
1 vote

Poisson process under equivalent martingale measure

$N_t$ process comes with its own Poisson law (probability measure) $P$ defined via intensity $\lambda$. Under it, $N_t-\lambda t$ is a martingale wrt ${\cal F}_t =\sigma(N_u | u\in [0,t])$ (as $E^P[...
  • 5,028
1 vote

Poisson process under equivalent martingale measure

Consider a radon nikodym derivative, the Random variable: $Z(w)= 1_{N(T,w)=1}+1-Pr(N(T)=1)$. It is admissible since it is always positive and has an expectation 1. This will lead us to the formation ...
  • 1,662
1 vote
Accepted

EMM for Bachelier model

It sounds to me like you understand everything apart from how Girsanov's theorem defines the EMM. Girsanov's theorem tells us that if $B_t$ is standard Brownian motion under $P$, then for any adapted ...
  • 436
1 vote
Accepted

How to prove that the following is still a Brownian motion

I try to clarify. First, let us be under the real world probability measure $\mathbb{P}$. Say that the process $Y_t$ is a $\mathbb{P}$ standard Brownian motion. Then the following is true by ...
1 vote

We have a two LIBOR contracts, how to compare their values by change of change of numeraire

Let $X_{T_1}$ be a random quantity known (fixed) at $T_1$ (measurable wrt $T_1$-information), $B$ be the standard bank account and $P$ standard zero-coupon bond price. From standard pricing, for the ...
  • 5,028
1 vote
Accepted

Market Price of Risk for Consumption Asset - Hull's Example 28.1

Note that just before Equation (28.9), Hull writes $-$ my emphasis: The market price of risk of [asset] $\theta$ measures the trade-offs between risk and return that are made for securities ...

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