Expected Value of a time varying diffusion
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I need to compute the following expected Value
$$V(w)=mathbb{E}left[left.int_{0}^{infty}e^{-rs}W_{t+s}dsright|W_{t}=wright]$$
where $W_{t}$ is a GBM whose diffusion is given by
$$dW_{t}=mu W_{t}dt+sigma W_{t}dZ_{t}$$
and $Z_{t}$ a Standard BM. In this exercise there is a Poisson Process with arrival rate $lambda$ such that when it jumps for the first time the diffusion changes to
$$dW_{t}=mu' W_{t}dt+sigma W_{t}dZ_{t}$$
I'm not sure how to compute the expected value in this case. I tried to use the Law of Iterated Expectations, compute the expected value given a fixed $tau$ in which the process jumps and then taking expectation of whatever I got. Is this ok?
Thanks in advance
stochastic-calculus stochastic-integrals expected-value
edited Nov 16 at 2:11
gt6989b
32.1k22351
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
add a comment |
up vote
0
down vote
favorite
I need to compute the following expected Value
$$V(w)=mathbb{E}left[left.int_{0}^{infty}e^{-rs}W_{t+s}dsright|W_{t}=wright]$$
where $W_{t}$ is a GBM whose diffusion is given by
$$dW_{t}=mu W_{t}dt+sigma W_{t}dZ_{t}$$
and $Z_{t}$ a Standard BM. In this exercise there is a Poisson Process with arrival rate $lambda$ such that when it jumps for the first time the diffusion changes to
$$dW_{t}=mu' W_{t}dt+sigma W_{t}dZ_{t}$$
I'm not sure how to compute the expected value in this case. I tried to use the Law of Iterated Expectations, compute the expected value given a fixed $tau$ in which the process jumps and then taking expectation of whatever I got. Is this ok?
Thanks in advance
stochastic-calculus stochastic-integrals expected-value
edited Nov 16 at 2:11
gt6989b
32.1k22351
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need to compute the following expected Value
$$V(w)=mathbb{E}left[left.int_{0}^{infty}e^{-rs}W_{t+s}dsright|W_{t}=wright]$$
where $W_{t}$ is a GBM whose diffusion is given by
$$dW_{t}=mu W_{t}dt+sigma W_{t}dZ_{t}$$
and $Z_{t}$ a Standard BM. In this exercise there is a Poisson Process with arrival rate $lambda$ such that when it jumps for the first time the diffusion changes to
$$dW_{t}=mu' W_{t}dt+sigma W_{t}dZ_{t}$$
I'm not sure how to compute the expected value in this case. I tried to use the Law of Iterated Expectations, compute the expected value given a fixed $tau$ in which the process jumps and then taking expectation of whatever I got. Is this ok?
Thanks in advance
stochastic-calculus stochastic-integrals expected-value
edited Nov 16 at 2:11
gt6989b
32.1k22351
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
I need to compute the following expected Value
$$V(w)=mathbb{E}left[left.int_{0}^{infty}e^{-rs}W_{t+s}dsright|W_{t}=wright]$$
where $W_{t}$ is a GBM whose diffusion is given by
$$dW_{t}=mu W_{t}dt+sigma W_{t}dZ_{t}$$
and $Z_{t}$ a Standard BM. In this exercise there is a Poisson Process with arrival rate $lambda$ such that when it jumps for the first time the diffusion changes to
$$dW_{t}=mu' W_{t}dt+sigma W_{t}dZ_{t}$$
I'm not sure how to compute the expected value in this case. I tried to use the Law of Iterated Expectations, compute the expected value given a fixed $tau$ in which the process jumps and then taking expectation of whatever I got. Is this ok?
Thanks in advance
stochastic-calculus stochastic-integrals expected-value
stochastic-calculus stochastic-integrals expected-value
edited Nov 16 at 2:11
gt6989b
32.1k22351
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
edited Nov 16 at 2:11
gt6989b
32.1k22351
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
edited Nov 16 at 2:11
gt6989b
32.1k22351
edited Nov 16 at 2:11
gt6989b
32.1k22351
edited Nov 16 at 2:11
gt6989b
32.1k22351
32.1k22351
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
asked Nov 16 at 1:43
Pedro Ignacio Martinez Bruera
11
11
add a comment |
add a comment |
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