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Local Time at Zero of Brownian Motion

Hard · Probability & Statistics · Quant Trader interview question · brownian-motion, local-time, tanaka-formula, stochastic-calculus

Suppose $W_t$ is a standard Brownian motion. Using Tanaka's formula, express the local time $L_t$ of $W_t$ at zero in terms of $|W_t|$. Recall that Tanaka's Formula states: $f(W_t) = f(W_0) + \int_0^t f'(W_s) dW_s + \frac{1}{2} \int_{-\infty}^{\infty} f''(x) L_t^x dx$, where $L_t^x$ is the local time of $W_t$ at $x$. In our case, we consider the local time at zero only, which is $L_t = L_t^0$. You are given that $L_t = |W_t| - \int_0^t \text{sgn}(W_s) dW_s$. Determine the correct exp