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1) Ornstein–Uhlenbeck process

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(x,t) := x e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

The short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$d(X_te^{\theta t})=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen students mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

2) Geometric Brownian Motion

Even the very well-known SDE for the Geometric Brownian Motion written in short-hand notation leads to confusion:

$$dS_t=\mu S_tdt+\sigma S_tdW_t$$

Again, the terms on the RHS would normally be written as $S_h$ inside an integral, which makes it obvious that the terms cannot be "taken out" of the integral until they are integrated.

In the short-hand notation, I have seen far too often attempts to divide through by $S_t$ and write:

$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$$

The "next step" would then be to "integrate both side" (with respect to what variable?) and write:

$$\ln(S_t)-\ln(S_0) = \mu t+\sigma W_t$$

Which is obviously wrong (the solution is actually $\ln(S_t)-\ln(S_0) = \mu t + 0.5 \sigma^2 t + \sigma W_t$, again using Ito's lemma).

In conclusion, I would argue that the short-hand notation for SDEs is rather unfortunate, particularly for new students to the field. I would encourage anyone new to stochastic calculus to use the long-hand notation until they become very comfortable with the subject.

1) Ornstein–Uhlenbeck process

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(X_t,t) := X_t e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

The short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$d(X_te^{\theta t})=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen students mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

2) Geometric Brownian Motion

Even the very well-known SDE for the Geometric Brownian Motion written in short-hand notation leads to confusion:

$$dS_t=\mu S_tdt+\sigma S_tdW_t$$

Again, the terms on the RHS would normally be written as $S_h$ inside an integral, which makes it obvious that the terms cannot be "taken out" of the integral until they are integrated.

In the short-hand notation, I have seen far too often attempts to divide through by $S_t$ and write:

$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$$

The "next step" would then be to "integrate both side" (with respect to what variable?) and write:

$$\ln(S_t)-\ln(S_0) = \mu t+\sigma W_t$$

Which is obviously wrong (the solution is actually $\ln(S_t)-\ln(S_0) = \mu t - 0.5 \sigma^2 t + \sigma W_t$, again using Ito's lemma).

In conclusion, I would argue that the short-hand notation for SDEs is rather unfortunate, particularly for new students to the field. I would encourage anyone new to stochastic calculus to use the long-hand notation until they become very comfortable with the subject.

1) Ornstein–Uhlenbeck process

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(X_t,t):=X_t e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

The short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$dX_te^{\theta t}=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen students mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

2) Geometric Brownian Motion

Even the very well-known SDE for the Geometric Brownian Motion written in short-hand notation leads to confusion:

$$dS_t=\mu S_tdt+\sigma S_tdW_t$$

Again, the terms on the RHS would normally be written as $S_h$ inside an integral, which makes it obvious that the terms cannot be "taken out" of the integral until they are integrated.

In the short-hand notation, I have seen far too often attempts to divide through by $S_t$ and write:

$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$$

The "next step" would then be to "integrate both side" (with respect to what variable?) and write:

$$ln(S_t)=\mu t+\sigma W_t$$

Which is obviously wrong (the solution is actually $\frac{ln(S_t)}{ln(S_0)}=\mu t + 0.5 \sigma^2 t + \sigma W_t$, again using Ito's lemma).

In conclusion, I would argue that the short-hand notation for SDEs is rather unfortunate, particularly for new students to the field. I would encourage anyone new to stochastic calculus to use the long-hand notation until they become very comfortable with the subject.

1) Ornstein–Uhlenbeck process

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(x,t) := x e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

The short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$d(X_te^{\theta t})=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen students mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

2) Geometric Brownian Motion

Even the very well-known SDE for the Geometric Brownian Motion written in short-hand notation leads to confusion:

$$dS_t=\mu S_tdt+\sigma S_tdW_t$$

Again, the terms on the RHS would normally be written as $S_h$ inside an integral, which makes it obvious that the terms cannot be "taken out" of the integral until they are integrated.

In the short-hand notation, I have seen far too often attempts to divide through by $S_t$ and write:

$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$$

The "next step" would then be to "integrate both side" (with respect to what variable?) and write:

$$\ln(S_t)-\ln(S_0) = \mu t+\sigma W_t$$

Which is obviously wrong (the solution is actually $\ln(S_t)-\ln(S_0) = \mu t + 0.5 \sigma^2 t + \sigma W_t$, again using Ito's lemma).

In conclusion, I would argue that the short-hand notation for SDEs is rather unfortunate, particularly for new students to the field. I would encourage anyone new to stochastic calculus to use the long-hand notation until they become very comfortable with the subject.

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(X_t,t):=X_t e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

I would argue that in this specific case, the short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$dX_te^{\theta t}=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen people mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

1) Ornstein–Uhlenbeck process

Consider the SDE for the Ornstein–Uhlenbeck process:

$$X_t:=X_0+\int_{h=0}^{h=t}\theta(\mu- X_h)dh + \int_{h=0}^{h=t}\sigma dW_h.$$

Solution can be found by applying Itô's lemma to the function $F(X_t,t):=X_t e^{\theta t}$, which leads to:

$$X_te^{\theta t}=X_0+\int_{h=0}^{h=t}\left(e^{\theta h}\theta\mu\right)dh+\int_{h=0}^{h=t}\left(e^{\theta h} \sigma\right)dW_h.$$

The final step is then to isolate $X_t$ on the LHS by dividing through by $e^{\theta t}$:

$$X_t=X_0e^{-\theta t}+\int_{h=0}^{h=t}\left(e^{\theta(h-t)}\theta\mu\right)dh+\int_{h=0}^{h=t}\sigma e^{\theta(h-t)} dW_h.$$

The short-hand notation for the Orsntein–Uhlenbeck process leads to confusion:

$$dX_t:=\theta(\mu- X_t)dt + \sigma dW_t.$$

Applying Itô's lemma and continuing in the short-hand notation leads to:

$$dX_te^{\theta t}=\left(e^{\theta t}\theta\mu\right)dt+\left(e^{\theta t} \sigma\right)dW_t.$$

Using the short-hand notation, I have seen students mistakenly cancel out the terms $e^{\theta t}$ that would normally be written as $e^{\theta h}$ in the long-hand notation inside the integrals.

2) Geometric Brownian Motion

Even the very well-known SDE for the Geometric Brownian Motion written in short-hand notation leads to confusion:

$$dS_t=\mu S_tdt+\sigma S_tdW_t$$

Again, the terms on the RHS would normally be written as $S_h$ inside an integral, which makes it obvious that the terms cannot be "taken out" of the integral until they are integrated.

In the short-hand notation, I have seen far too often attempts to divide through by $S_t$ and write:

$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$$

The "next step" would then be to "integrate both side" (with respect to what variable?) and write:

$$ln(S_t)=\mu t+\sigma W_t$$

Which is obviously wrong (the solution is actually $\frac{ln(S_t)}{ln(S_0)}=\mu t + 0.5 \sigma^2 t + \sigma W_t$, again using Ito's lemma).

In conclusion, I would argue that the short-hand notation for SDEs is rather unfortunate, particularly for new students to the field. I would encourage anyone new to stochastic calculus to use the long-hand notation until they become very comfortable with the subject.