Comparing Perfectly Correlated Displaced Lognormal Diffusions

Summary

The document asks when a weighted difference between two perfectly correlated displaced lognormal diffusion processes stays nonnegative over time. The processes share a Brownian driver, have deterministic time-varying volatility coefficients, and have distinct positive displacement constants. The answer applies Itô’s formula to the logarithm of each process after subtracting its displacement, yielding an explicit exponential representation for each process.

It substitutes those representations into the weighted difference, separating the constant displacement contribution from the two stochastic exponential terms. This provides a way to examine positivity using the initial values and the paths of the shared Brownian motion. However, the response does not derive sufficient conditions on the coefficients or displacements, and it ends by leaving that condition-finding step open. The excerpt therefore supplies a useful transformation of the problem, not a guarantee of positivity or a complete stochastic dominance result.

Key ideas

• Taking the logarithm of a displaced process reduces its diffusion to a tractable stochastic integral.
• The solution expresses each process as its displacement plus an exponential stochastic term.
• The weighted difference can be written as a constant part and two correlated exponential parts.
• The shared Brownian driver links the two stochastic terms but does not by itself establish pathwise ordering.
• The response does not provide conditions that guarantee the difference stays nonnegative.

Tags

# stochastic dominance displaced diffusions


# stochastic dominance displaced diffusions












Suppose I have two processes both satisfying a displace lognormal diffusion: $$ dX(t) = \alpha(t)[X(t) - a] dW(t) $$ $$ dY(t) = \beta(t)[Y(t) - b] dW(t) $$ Note that the processes are perfectly correlated where $W(t)$ is a standard Brownian motion, $\alpha, \beta$ are deterministic functions of time, and $a,b$ are constants greater than zero.

Under what conditions will their difference be greater than zero at all times?: $$ p X(t) - q Y(t) \geq 0 \quad p,q \in \mathcal{R}, p > q > 0 $$ Initially at $t=0$ I know that their difference as written above is greater than zero. Are there simple conditions on $\alpha,\beta,a,b$ that ensure the difference is always positive?

## Answer by NN2 (score 1)

https://quant.stackexchange.com/a/49449

You can have explicite solution of $X_t$ and $Y_t$. Put $V_t = \ln{(X_t-a)}$, we can easily find the equation of $V_t$: $$dV_t = -\frac{1}{2}\alpha_t^2 dt+\alpha_tdW_t$$ So, $$V_t = V_0 -\frac{1}{2}\int_0^t{\alpha_s^2 ds}+\int_0^t{\alpha_sdW_s}$$ Hence $$X_t = a + (x_0-a)\exp{(-\frac{1}{2}\int_0^t{\alpha_s^2 ds}+\int_0^t{\alpha_sdW_s})}$$

The next step is to find the condition that $$Z_t = pX_t -qY_t=(pa-qb) +p(x_0-a)\exp{(-\frac{1}{2}\int_0^t{\alpha_s^2 ds}+\int_0^t{\alpha_sdW_s})} -q(y_0-b)\exp{(-\frac{1}{2}\int_0^t{\beta_s^2 ds}+\int_0^t{\beta_sdW_s})} \geq 0$$

I think that you can solve it easily.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.