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Exp_OU_Process

*Exp OU Process* is a Type that represents an exponential Ornstein Uhlenbeck stochastic process with jumps.

It is driven by 3 independent stochastic factors, *x, J* and *N*, representing respectively the continuous diffusion part, the random jump size and random timing of each jump.

Formally the diffusion equation of the stochastic process *s* is:

*s = exp(x + y)*

where *x* is an Ornstein Uhlenbeck process *dx = θ(μ-x)dt + σdw* as described in Extended OU Process and *y* is a jump diffusion following the equation:

*dy(t) = -βy(t-)dt + J(t)dN(t)*

where β is constant, y(t-) is the prior to jump value of y at time t, J(t) is an independent identically distributed (iid) process representing the jump size and N is a Poisson-process with intensity *λ*

More specifically, at each t, J(t) is exponentially distributed with rate η, with the probability density function *ηexp(-ηJ)*

Those *1/η* is interpreted as the mean jump size in the evolution of y.

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