• B Rajeev

      Articles written in Proceedings – Mathematical Sciences

    • From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus

      B Rajeev

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      In this article we give a new proof of Ito's formula inRn starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.

    • Weak convergence of the past and future of Brownian motion given the present

      K B Athreya B Rajeev

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      In this paper, we show that for $t > 0$, the joint distribution of the past $\{W_{t−s} : 0 \leq s \leq t\}$ and the future $\{W_{t+s} : s \geq 0\}$ of a $d$-dimensional standard Brownian motion $(W_s)$, conditioned on $\{W_t\in U\}$, where $U$ is a bounded open set in $\mathbb{R}^d$, converges weakly in $C[0,\infty)\times C[0, \infty)$ as $t\rightarrow\infty$. The limiting distribution is that of a pair of coupled processes $Y + B^1$, $Y + B^2$ where $Y$, $B^1$, $B^2$ are independent, $Y$ is uniformly distributed on $U$ and $B^1$, $B^2$ are standard $d$-dimensional Brownian motions. Let $\sigma_t$, $d_t$ be respectively, the last entrance time before time $t$ into the set $U$ and the first exit time after $t$ from $U$. When the boundary of $U$ is regular, we use the continuous mapping theorem to show that the limiting distribution as $t\rightarrow\infty$ of the four dimensional vector with components $(W_{\sigma_t}, t − \sigma_t, W_{d_t}, d_t − t)$, conditioned on $\{W_t\in U\}$, is the same as that of the four dimensional vector whose components are the place and time of first exit from $U$ of the processes $Y + B^1$ and $Y + B^2$ respectively.

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