y*******g
发帖数: 968 | 1 Y_t=exp(int_0^t(T_s d W_s)-1/2 int_0^t (T_s)^2 ds) | c*******e
发帖数: 150 | 2 note that the process Y_t satisfies dY_t = T_t Y_t dW_t
it is called the stochastic exponential of (T_t)_{t\in[0,T]}, sometimes
denoted as \epsilon(T)_t
If (T_t)_{t\in[0,T]} satisfies the Novikov condition, i.e.
\mathbb{E}[exp(1/2 \int_0^T T_s^2 ds)] < +\infty
then Y_t is also called an exponential martingale, since it is a true
martingale on [0, T]
【在 y*******g 的大作中提到】
: Y_t=exp(int_0^t(T_s d W_s)-1/2 int_0^t (T_s)^2 ds)
| c*******e
发帖数: 150 | 3 in finance and trading, we always have an expiration date for our contract,
so I turned lazy and just listed the situation for a compact time horizon.
You can extend the notions to the case of an infinite time horizon as well,
though not particularly interesting in finance applications.
have
higher
【在 c*******e 的大作中提到】
: note that the process Y_t satisfies dY_t = T_t Y_t dW_t
: it is called the stochastic exponential of (T_t)_{t\in[0,T]}, sometimes
: denoted as \epsilon(T)_t
: If (T_t)_{t\in[0,T]} satisfies the Novikov condition, i.e.
: \mathbb{E}[exp(1/2 \int_0^T T_s^2 ds)] < +\infty
: then Y_t is also called an exponential martingale, since it is a true
: martingale on [0, T]
| y*******g
发帖数: 968 | 4 thank you very much
【在 c*******e 的大作中提到】
: note that the process Y_t satisfies dY_t = T_t Y_t dW_t
: it is called the stochastic exponential of (T_t)_{t\in[0,T]}, sometimes
: denoted as \epsilon(T)_t
: If (T_t)_{t\in[0,T]} satisfies the Novikov condition, i.e.
: \mathbb{E}[exp(1/2 \int_0^T T_s^2 ds)] < +\infty
: then Y_t is also called an exponential martingale, since it is a true
: martingale on [0, T]
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