Stochastic differential equation with quadratic drift and volatility
up vote
2
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I am looking for an exact (closed-form) solution to the SDE:
begin{equation}
dX_t = alpha X_t(A_0 - X_t) dt + beta X_t(A_0 + X_t) dW_t
end{equation}
for a Wiener process $dW_t$ with initial condition $X(0) = X_0$. I have tried different substitutions such as
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right]
end{equation}
but after applying the Ito formula I always end up with an extra $X_t dt$ term on the right hand side:
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right] = C_1 z + C_2 W_t + C_3 X_t dt
end{equation}
where $C_j$ are constants. If I then integrate both sides I end up with an integral equation that has no solution. Is there any other way to simplify the expression, at least to eliminate the need for an integral equation? Any approximations/asymptotic expansions would also be useful
stochastic-calculus
asked Nov 14 at 5:00
OscarNieves
213
add a comment |
up vote
2
down vote
favorite
I am looking for an exact (closed-form) solution to the SDE:
begin{equation}
dX_t = alpha X_t(A_0 - X_t) dt + beta X_t(A_0 + X_t) dW_t
end{equation}
for a Wiener process $dW_t$ with initial condition $X(0) = X_0$. I have tried different substitutions such as
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right]
end{equation}
but after applying the Ito formula I always end up with an extra $X_t dt$ term on the right hand side:
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right] = C_1 z + C_2 W_t + C_3 X_t dt
end{equation}
where $C_j$ are constants. If I then integrate both sides I end up with an integral equation that has no solution. Is there any other way to simplify the expression, at least to eliminate the need for an integral equation? Any approximations/asymptotic expansions would also be useful
stochastic-calculus
asked Nov 14 at 5:00
OscarNieves
213
What makes you believe that a closed form exists...?
– saz
Nov 14 at 6:46
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am looking for an exact (closed-form) solution to the SDE:
begin{equation}
dX_t = alpha X_t(A_0 - X_t) dt + beta X_t(A_0 + X_t) dW_t
end{equation}
for a Wiener process $dW_t$ with initial condition $X(0) = X_0$. I have tried different substitutions such as
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right]
end{equation}
but after applying the Ito formula I always end up with an extra $X_t dt$ term on the right hand side:
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right] = C_1 z + C_2 W_t + C_3 X_t dt
end{equation}
where $C_j$ are constants. If I then integrate both sides I end up with an integral equation that has no solution. Is there any other way to simplify the expression, at least to eliminate the need for an integral equation? Any approximations/asymptotic expansions would also be useful
stochastic-calculus
asked Nov 14 at 5:00
OscarNieves
213
I am looking for an exact (closed-form) solution to the SDE:
begin{equation}
dX_t = alpha X_t(A_0 - X_t) dt + beta X_t(A_0 + X_t) dW_t
end{equation}
for a Wiener process $dW_t$ with initial condition $X(0) = X_0$. I have tried different substitutions such as
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right]
end{equation}
but after applying the Ito formula I always end up with an extra $X_t dt$ term on the right hand side:
begin{equation}
dleft[frac{1}{A_0}lnleft(frac{x}{x-A_0}right)right] = C_1 z + C_2 W_t + C_3 X_t dt
end{equation}
where $C_j$ are constants. If I then integrate both sides I end up with an integral equation that has no solution. Is there any other way to simplify the expression, at least to eliminate the need for an integral equation? Any approximations/asymptotic expansions would also be useful
stochastic-calculus
stochastic-calculus
asked Nov 14 at 5:00
OscarNieves
213
asked Nov 14 at 5:00
OscarNieves
213
asked Nov 14 at 5:00
OscarNieves
213
asked Nov 14 at 5:00
OscarNieves
213
asked Nov 14 at 5:00
OscarNieves
213
213
What makes you believe that a closed form exists...?
– saz
Nov 14 at 6:46
add a comment |
What makes you believe that a closed form exists...?
– saz
Nov 14 at 6:46
What makes you believe that a closed form exists...?
– saz
Nov 14 at 6:46
What makes you believe that a closed form exists...?
– saz
Nov 14 at 6:46
add a comment |
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What makes you believe that a closed form exists...?
– saz
Nov 14 at 6:46